Nuclear Safeguards Analysis - American Chemical Society

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Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on February 27, 2018 | https://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0079.ch004

D e c i s i o n A n a l y s i s f o r N u c l e a r Safeguards

JAMES P. SHIPLEY Los Alamos Scientific Laboratory of the University of California, P.O. Box 1663, Los Alamos, NM 87545

Materials accounting for safeguarding special nuclear mater i a l s (SNM) usually brings to mind instrumentation and measurement techniques for obtaining information on SNM locations and amounts. The emphasis frequently is on data collection, a broad, highly developed f i e l d that includes instrument design and the specification and operation of complete measurement systems (see Refs. 1,2,3,4,5 and the references therein). Just as important is the analysis of materials accounting data to detect diversion of SNM or process upset conditions. This paper is most concerned with examining some efficient methods for analyzing and interpreting safeguards data. Materials accounting for SNM currently relies heavily on material-balance accounting following perodic shutdown, cleanout, and physical inventory. The classical material balance associated with this system is drawn around the entire f a c i l i t y or a major portion of the process, and is formed by adding a l l measured receipts to the i n i t i a l measured inventory and subtracting a l l measured removals and the final measured inventory. During periods of routine production, control of materials is vested largely in administrative and process controls, augmented by secure storage for discrete items. Although conventional material-balance accounting is essent i a l to safeguards control of nuclear material, i t has inherent limitations in sensitivity and timeliness. The f i r s t l i m i t a tion results from measurement uncertainties that desensitize the system to losses of trigger quantities of SNM for large-throughput plants. The timeliness of traditional materials This chapter not subject to U.S. copyright. Published 1978 American Chemical Society Hakkila; Nuclear Safeguards Analysis ACS Symposium Series; American Chemical Society: Washington, DC, 1978.


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accounting i s l i m i t e d by the frequency at which the physical inventory i s taken. There are p r a c t i c a l l i m i t s on how often a f a c i l i t y can shut down i t s process and s t i l l be p r o d u c t i v e . These conventional methods can be augmented by u n i t process accounting i n which the f a c i l i t y is partitioned into discrete accounting envelopes c a l l e d u n i t process accounting areas. A u n i t process can be one or more chemical or p h y s i c a l processes, and i s chosen on the b a s i s of process l o g i c and whether a mater i a l balance can be drawn around i t . By dividing a facility into unit processes and measuring a l l significant material t r a n s f e r s , q u a n t i t i e s of m a t e r i a l much smaller than the total plant inventory can be c o n t r o l l e d on a timely b a s i s . Also, any d i s c r e p a n c i e s are l o c a l i z e d to that p o r t i o n of the process contained i n the u n i t process accounting area. M a t e r i a l balances drawn around such u n i t processes during the course of plant o p e r a t i o n are c a l l e d dynamic m a t e r i a l b a l ances to d i s t i n g u i s h them from balances drawn a f t e r a cleanout and p h y s i c a l inventory. I d e a l l y , the dynamic m a t e r i a l balances would a l l be zero unless nuclear m a t e r i a l had been stolen (diverted). In p r a c t i c e they never are, f o r two reasons. F i r s t , the measuring instruments always introduce e r r o r s , f o r example, random f l u c t u a t i o n s from e l e c t r o n i c n o i s e , or instrument m i s c a l i b r a t i o n s . Second, c o n s t r a i n t s on cost or impact on m a t e r i a l s processing operations may d i c t a t e that not a l l components of a m a t e r i a l balance be measured e q u a l l y o f t e n ; t h e r e f o r e , even i f the measurements were exact, the m a t e r i a l - b a l a n c e values would not be zero u n t i l closed by a d d i t i o n a l measurements . Use of dynamic m a t e r i a l s accounting implies that the operator of the safeguards system may be inundated with m a t e r i a l s accounting data. Furthermore, although these data c o n t a i n much p o t e n t i a l l y u s e f u l i n f o r m a t i o n concerning both safeguards and process c o n t r o l , the s i g n i f i c a n c e of any isolated ( s e t of) measurements i s seldom r e a d i l y apparent and may change from day to day depending on p l a n t operating c o n d i t i o n s . Thus, the safeguards system operator i s presented with an overwhelmingly complex body of information from which he must repeatedly d e t e r mine the safeguards status of the p l a n t . C l e a r l y , i t i s imperat i v e that he be a s s i s t e d by a coherent, logical framework of tools that address these problems. Decision analysis >§.»2.) i s such a framework, and i s w e l l suited for s t a t i s t i c a l treatment of the imperfect dynamic material-balance data that become a v a i l a b l e sequentially in time. I t s primary goals are (1) d e t e c t i o n of the event(s) that SNM has been d i v e r t e d , (2) e s t i m a t i o n of the amount(s) d i v e r t e d , and (3) determination of the s i g n i f i c a n c e of the estimates.

Hakkila; Nuclear Safeguards Analysis ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

36

NUCLEAR

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ANALYSIS


DECISION ANALYSIS D e c i s i o n a n a l y s i s combines techniques from estimation theory and hypothesis t e s t i n g , or d e c i s i o n theory, with systems analys i s t o o l s f o r t r e a t i n g complex, dynamic problems. The d e c i s i o n a n a l y s i s framework i s general enough to allow a wide range i n the level of s o p h i s t i c a t i o n i n examining nuclear m a t e r i a l s accounting data, while p r o v i d i n g g u i d e l i n e s f o r the development and a p p l i c a t i o n of a v a r i e t y of powerful methods. The d e c i s i o n - a n a l y s i s process i s i l l u s t r a t e d i n F i g . 1. For nuclear m a t e r i a l s accounting, the observed source generates true ( e r r o r - f r e e ) data according to the switch p o s i t i o n , which i s determined by some unknown f a c t o r . The observed source i s the nuclear m a t e r i a l s processing l i n e , and the unknown f a c t o r could be a d i v e r t o r , f o r example. I f the d i v e r t o r i s not s t e a l i n g nuclear m a t e r i a l , the switch i s i n the upper p o s i t i o n ; i f d i v e r s i o n i s o c c u r r i n g , the switch i s i n the lower p o s i t i o n . A major part of the nuclear m a t e r i a l s accounting problem i s to choose between the two s i t u a t i o n s ; the two choices are r e f e r r e d to as HQ, the n u l l hypothesis under which no d i v e r s i o n has occurred, and H]^, the a l t e r n a t i v e hypothesis that d i v e r s i o n has occurred. I t should be noted that other f a c t o r s besides d i v e r s i o n may cause SNM to be missing, which would appear as d i v e r s i o n . Part of the d e c i s i o n process c o n s i s t s of f u r t h e r i n v e s t i g a t i o n s to d i s c r i m i n a t e among p o s s i b l e causes. For the purposes of t h i s paper, no d i s t i n c t i o n i s made between d i v e r s i o n and (apparently) missing m a t e r i a l . The true m a t e r i a l s accounting data from the observed source, or e q u i v a l e n t l y the hypotheses HQ and H]_, are not observed d i r e c t l y ; otherwise, the d e c i s i o n problem would be trivial. Imperfect measurement devices ( p a r t of the d a t a - c o l l e c t i o n funct i o n ) provide corrupted data f o r the d e c i s i o n process. I f measurement e r r o r s can be t r e a t e d p r o b a b i l i s t i c a l l y , the r e s u l t i n g e r r o r s t a t i s t i c s can a i d subsequent a n a l y s i s . The e s t i m a t i o n p a r t o f the a n a l y s i s f u n c t i o n i s designed to take advantage of information i n a d d i t i o n to that a v a i l a b l e i n the measured data, with the o b j e c t i v e s of obtaining more accurate and p r e c i s e estimates of d i v e r t e d m a t e r i a l . I f no other information on the observed source i s available, the e s t i m a t i o n a l g o r i t h m simply passes the measured data and e r r o r statistics on to the d e c i s i o n f u n c t i o n , along with the i m p l i c i t assumption that when HQ i s true the m a t e r i a l - b a l a n c e values are a l l zero. Otherwise, e s t i m a t i o n i s based on more complicated models of source behavior, and estimate calculations assuming each hypothesis true are performed s e p a r a t e l y . The source models f o r HQ and true i n F i g . 1 represent the t r a n s l a t i o n i n t o mathematical terms of whatever additional information e x i s t s concerning the source. Accurate and p r e c i s e model c o n s t r u c t i o n i s extremely important; inaccurate models cause i n c o r r e c t , or biased, d e c i s i o n s , whereas imprecise models



SOURCE

1

I

h

OBSERVED I

r

H

0

R

U

E

1 TRUE

H

T

FACTOR

UNKNOWN

DESCRIPTION TRUE

FOR H

MODEL TRUE

FOR H

n

TRUE

ι

—

TRUE

ESTIMATION

H

ο Ν

I

s

D Ε C I

REQUIREMENTS

EXTERNAL

PROBABILITIES

FALSE-ALARM AND DETECTION

ESTIMATE..

[STATISTICS *

ESTIMATE

Figure 1. Structure of the decision analysis process

1

J STATISTICS

SOURCE

ERRORSμ

MEASUREMENT

MODEL

SOURCE n


H

1

TRUE

-SIGNIFICANCE

ESTIMATE A N D * UNCERTAINTY

" O R N O DECISION


38

NUCLEAR

SAFEGUARDS

ANALYSIS

make i t d i f f i c u l t to reach a d e c i s i o n having reasonable significance . The d e c i s i o n f u n c t i o n may be as informal as a perusal of the estimation r e s u l t s , or as s t r u c t u r e d as a s t a t i s t i c a l t e s t with parameters f i x e d by administrative fiat. For the p r a c t i c a l problems of nuclear materials accounting, a middle-ground approach i s appropriate. A battery of statistical tests will f a c i l i t a t e q u a n t i f i e d d e c i s i o n making, help eliminate personal b i a s e s , and form the b a s i s f o r e f f e c t i v e r e g u l a t i o n . However, a p p l i c a t i o n of the t e s t s and choice of t e s t parameters should not be r i g i d or a r b i t r a r y ; unforeseen circumstances and the poss i b i l i t y of hidden e r r o r s require f l e x i b i l i t y and subjective guidance i n the d e c i s i o n process. Although many d i f f e r e n t s t a t i s t i c a l t e s t s are suitable for use i n the d e c i s i o n process, they a l l have s e v e r a l characterist i c s i n common. Each operates on the estimation results to decide whether HQ or H^ i s true, and each r e q u i r e s some i n d i c a t i o n of d e s i r a b l e f a l s e - a l a r m and d e t e c t i o n p r o b a b i l i t i e s . One u s e f u l k i n d of t e s t compares a l i k e l i h o o d r a t i o to a thresho l d , the l i k e l i h o o d r a t i o being defined roughly as the ratio of the p r o b a b i l i t y that H^ i s true to the p r o b a b i l i t y that HQ i s true, with the threshold determined by the d e s i r e d f a l s e - a l a r m and d e t e c t i o n p r o b a b i l i t i e s . The v a r i e t y of t e s t s a v a i l a b l e to the d e c i s i o n process allows a wide range of t r a d e o f f s among comp l e x i t y , e f f e c t i v e n e s s , and a p p l i c a b i l i t y to s p e c i a l s i t u a t i o n s . D e c i s i o n a n a l y s i s based on mathematically derived decision functions i s appealing because i t can q u a n t i f y i n t u i t i v e f e e l i n g s and condense large c o l l e c t i o n s of data to a smaller set of more e a s i l y understood d e s c r i p t o r s , or s t a t i s t i c s . I t can a l s o e l i m i nate personal biases and other e r r o r s caused by s u b j e c t i v e e v a l u a t i o n of data, while p r o v i d i n g a degree of consistency for the d e c i s i o n process. However, d e c i s i o n a n a l y s i s should be considered as a management t o o l , not a management s u b s t i t u t e . Unreasoning faith in test r e s u l t s i s shortsighted for several reasons, the primary one being the inherent inadequacies of any t r a c t a b l e , mathematic a l formulation of the hypotheses ( i . e . , the s t a t i s t i c a l models). In other words, s t a t i s t i c a l treatments are always based on simp l i f i e d models derived from sometimes hidden assumptions that may not be v a l i d f o r a p a r t i c u l a r s i t u a t i o n , and p o s s i b l e e f f e c t s on the d e c i s i o n process must be c o n t i n u a l l y assessed. A r e l a t e d problem i s that a p a r t i c u l a r t e s t can be defeated by choosing a d i v e r s i o n scheme that does not match the statist i c a l model. A b a t t e r y of t e s t s and v a r i a b l e t e s t i n g procedures reduce the p r o b a b i l i t y of success of such schemes, e s p e c i a l l y i f the t e s t s and procedures are unknown to the divertor. This approach a l s o tends to suppress the "beat-the-system" a t t i t u d e e x h i b i t e d sometimes, which i s f o s t e r e d by r i g i d application and interpretation of statistical tests; it also provides w e l l - c h a r a c t e r i z e d information on which to base d e c i s i o n s .



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Analysis

As with a l l s t a t i s t i c a l procedures, a degree o f reasonableness must be e x e r c i s e d ; some r e s u l t s have s t a t i s t i c a l significance but no p r a c t i c a l s i g n i f i c a n c e , and v i c e v e r s a . Materials accounting data should c e r t a i n l y be examined c a r e f u l l y using s t a t i s t i c a l techniques, but the conclusions should be tempered by p r a c t i c a l experience and personal judgment. Therefore, decis i o n a n a l y s i s need not be regarded as l e a d i n g to an i r r e v e r s i b l e d e c i s i o n , but r a t h e r as an information-gathering procedure aimed at modifying a t t i t u d e s towards the hypotheses on the b a s i s o f experimental evidence.

PROBLEM STATEMENT M a t e r i a l s accounting data g e n e r a l l y c o n s i s t o f a set o f i n process inventory measurements at d i s c r e t e times, each denoted by l(k), and a set o f net m a t e r i a l t r a n s f e r measurements between those times, each represented by T(k) f o r those transfers o c c u r r i n g between times k and k+1. These measurements would s a t i s f y the c o n t i n u i t y equation f o r conservation o f mass i f the measurements were exact and a l l i n v e n t o r i e s and t r a n s f e r s were measured. However, SNM q u a n t i t i e s can never be measured e x a c t l y , and d i v e r s i o n o f SNM may have occurred or there may be other unobserved sidestreams, preventing measurement o f a l l SNM. Therefore, the measurements s a t i s f y a modified continuity equation: Kk+l) = I(k) + T(k) - M(k+1),

k = 0,1,2,...,

(1)

where M(k+1) i s the m a t e r i a l imbalance a t time k+1 caused by measurement e r r o r s , unmeasured SNM, and d i v e r s i o n . The q u a n t i t y M(k+1), c a l l e d the k+lst m a t e r i a l balance, can be determined from d i r e c t l y measurable q u a n t i t i e s by i n v e r t i n g Eq. 1: M(k+1) = - I(k+1) + I(k) + T(k)

.

(2)

C l e a r l y , M(k+1) i s a random v a r i a b l e , and the sequence {M(i), i = 1,2,...} i s a s t o c h a s t i c process having p r o b a b i l i s t i c propert i e s dependent on the inventory and t r a n s f e r measurements. For example, i f the measurement e r r o r s are unbiased, then each M ( i ) has a mean value equal to the amount o f missing (or extra) mater i a l at each time i . F u r t h e r , i f the measurement e r r o r s are Gaussian, then each M(i) a l s o has a Gaussian distribution with variance equal to the sum of the variances o f the measurement errors. Note that Eq. 2 shows that consecutive m a t e r i a l balances are c o r r e l a t e d , even i f i n d i v i d u a l measurements are not. The ending inventory measurement f o r one m a t e r i a l balance i s the beginning inventory measurement f o r the next, r e s u l t i n g i n a negative component of c o r r e l a t i o n between balances. Other correlations


NUCLEAR


40

SAFEGUARDS

ANALYSIS

between i n d i v i d u a l measurements (e.g., measurement b i a s e s caused by instrument c a l i b r a t i o n e r r o r ) y i e l d additional correlation components, which are u s u a l l y p o s i t i v e , between m a t e r i a l balances. A method f o r t r e a t i n g c o r r e l a t i o n s w i l l be discussed below. In an a c t u a l s i t u a t i o n , we collect the set of inventory measurements { l ( k ) , k = 0,1,...,N} f o r some time p e r i o d during which N(>0) m a t e r i a l balances have been drawn, the corresponding set of t r a n s f e r -^asurements {T(k), k = 0,1,...,N-1}, and some s t a t i s t i c a l information on the measurement e r r o r s . Denote the aggregation of these data by Z(N). The d e c i s i o n problem i s to determine by analyzing Z(N) whether d i v e r s i o n has occurred during the time i n t e r v a l , to estimate the amount of d i v e r s i o n , and to draw some conclusions about the s i g n i f i c a n c e of the estimate.

THE

LIKELIHOOD RATIO

Hypothesis t e s t i n g (10,11,12) provides a l o g i c a l method f o r analyzing Z(N) f o r p o s s i b l e d i v e r s i o n . To proceed i n a general way, we form the two mutually e x c l u s i v e , exhaustive hypotheses HQ:

d i v e r s i o n has

H]_ :

d i v e r s i o n has

not

occurred,

occurred.

In developing s p e c i f i c d e c i s i o n algorithms, more mathematically q u a n t i f i e d statements about the hypotheses w i l l be necessary, and the p a r t i c u l a r form of each t e s t w i l l be strongly dependent on the corresponding hypothesis statements. However, these vague statements are s u f f i c i e n t f o r the general development. For any p a r t i c u l a r Z(N) that i s observed, d i v e r s i o n may or may not have occurred, so that i f HQ i s true, Z(N) has the p r o b a b i l i t y density f u n c t i o n P[Z(N)|H ] 0

and

i f H^

,

i s true, Z(N)

p[Z(N)|H ] 1

has

the p r o b a b i l i t y density

function

.

These two c o n d i t i o n a l d e n s i t y functions are called the likelihood functions f o r the hypotheses HQ and H]^, r e s p e c t i v e l y . The values of the l i k e l i h o o d functions f o r a p a r t i c u l a r Z(N) are r e l a t i v e measures of the l i k e l i h o o d that Z(N) i s governed by one or the other d e n s i t y f u n c t i o n , or i n other words, that HQ or Hi i s true. In making the d e c i s i o n whether HQ or H^ i s true, we may commit e i t h e r of two e r r o r s : we may decide that HQ is true when i t i s not (a miss), or we may decide H^ i s true when i t


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Analysis

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i s not (a f a l s e alarm). Let the p r o b a b i l i t y of a miss be Ρ^> and the p r o b a b i l i t y of a f a l s e alarm be Pp. Decision algor ithms may be derived from s e v e r a l d i f f e r e n t c r i t e r i a concerning the s e l e c t i o n of Pp and P . One of the most common c r i t e r i a i s to f i x Pp and minimize Pjj, which i s known as the NeymanPearson c r i t e r i o n . Another method i s to assign costs to i n c o r r e c t d e c i s i o n s and minimize the expected value of the t o t a l cost of a d e c i s i o n . This criterion i s known as the Bayes risk c r i t e r i o n , and i t r e q u i r e s estimates of the p r i o r p r o b a b i l i t i e s that HQ and are t r u e . Whichever c r i t e r i o n i s chosen, the decision t e s t reduces to comparing the likelihood ratio, L [ Z ( N ) ] , to a t h r e s h o l d ; i . e . ,


M

p[Z(N)|H ] I f

L

Î

Z ( N )

where Τ is Roughly, Eq. occurred as then decide true.

^p[Z(N)lH ] 0

< Τ, accept HQ ·> T, accept

(3)

a threshold dependent on the c r i t e r i o n chosen. 3 says that i f Z(N) i s "enough" more l i k e l y to have a r e s u l t of HQ being true than of U\ being true, that HQ i s t r u e ; otherwise, decide that is

SEQUENTIAL DECISIONS So f a r , the l i k e l i h o o d r a t i o t e s t , Eq. 3, has been formu l a t e d as a f i x e d - l e n g t h t e s t ; that i s , a l l the data Z(N) i s col l e c t e d before the t e s t i s performed. In a c t u a l p r a c t i c e , how ever, the optimum length and the proper s t a r t i n g point for the t e s t w i l l be unknown beforehand because the pattern of d i v e r s i o n , which i s a l s o unknown, i s a determining f a c t o r i n t e s t charac t e r i s t i c s . Furthermore, the m a t e r i a l s accounting data n a t u r a l l y appear s e q u e n t i a l l y i n time so that a sequential t e s t procedure that s e l e c t s i t s own l e n g t h and s t a r t s from a l l p o s s i b l e initial points i s appropriate. Such t e s t s can be shown to require fewer data points on the average than f i x e d - l e n g t h t e s t s having the same c h a r a c t e r i s t i c s ( 1 0 , 1 3 ) . Because the s e q u e n t i a l l i k e l i h o o d r a t i o t e s t , or sequential p r o b a b i l i t y r a t i o t e s t (SPRT), determines i t s own length, there are three p o s s i b l e r e s u l t s at each time, rather than two as in Eq. 3. The form of the SPRT i s j < T , accept HQ , > T.,, accept fi^ , 0

I f L[Z(N)]

otherwise, take another

(4) observation,


NUCLEAR

42

SAFEGUARDS

ANALYSIS

and the t e s t i s repeated f o r a l l p o s s i b l e starting points. As already d i s c u s s e d , the thresholds TQ and can be found from a number of c r i t e r i a , but some necessary i n f o r m a t i o n may be u n a v a i l a b l e , making t h i s approach l e s s e f f e c t i v e . Given the f a l s e - a l a r m and miss p r o b a b i l i t i e s , Pp and P, r e s p e c t i v e l y , l e t the thresholds be defined by


M

(5)

These are approximations, f i r s t devised by Wald (13), that can be shown to be c o n s e r v a t i v e i n the sense that use of TQ and T^ i n a t e s t w i l l r e s u l t i n a c t u a l f a l s e - a l a r m and miss proba b i l i t i e s that are no l a r g e r than those o r i g i n a l l y given.

COMPOSITE HYPOTHESES In many problems, one or both hypotheses may r e s u l t i n likel i h o o d functions that c o n t a i n an unknown parameter y; such a hypothesis i s c a l l e d composite. For example, y might be the (unknown) mean value of the o b s e r v a t i o n s . Without a value f o r y, the l i k e l i h o o d ratio cannot be calculated. One possible approach i s to use estimates of y, under the corresponding hypotheses, f o r the a u a l y and proceed with the t e s t . The most common estimate i s the maximum l i k e l i h o o d estimate found by maximizing the a p p r o p r i a t e l i k e l i h o o d f u n c t i o n with respect to the unknown parameter. The resulting generalized likelihood ratio is max L[Z(N)]

=

p[Z(N)|H.,y]

— i 1 > max p[Z(N)|H ,y] 0

( 6 )

0

Y

where YQ and Y^ are the spaces of allowable values under the hypotheses HQ and H]_, r e s p e c t i v e l y (10,11,12).

for

y

SOME DECISION TESTS As i n d i c a t e d above, the formulation of s p e c i f i c decision t e s t s depends on more mathematically p r e c i s e statements of the hypotheses. In p a r t i c u l a r , we seek statements of hypotheses


4.

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43

Analysis

that allow us to condense the q u a n t i t y Z(N) to one number S(N) without l o s s of information The number S(N) i s c a l l e d a suffi c i e n t s t a t i s t i c (10) and i s equivalent to knowledge of Z(N). If such a S(N) e x i s t s (which i s u s u a l l y true f o r SNM accounting) and i f i t s form and d e n s i t y f u n c t i o n are known, then the SPRT, Eq. 4, can be replaced by


I f

L

t

S

(

N

)

]

=

< TQ, accept H

p[S(N)|H ]

Δ

Q

>_ T^, accept H^

p[S(N)|H ]

, ,

Q

otherwise, tion.

take another observa

This approach i s appealing because, now, the density functions are u n i v a r i a t e and much more t r a c t a b l e mathematically. However, the form of the s u f f i c i e n t s t a t i s t i c may not be r e a d i l y apparent without a l g e b r a i c r e d u c t i o n of the original likelihood ratio. Further, a guess about the form of S(N) may lead to a t e s t hav ing l e s s d e s i r a b l e p r o p e r t i e s . The technique of reducing the o r i g i n a l l i k e l i h o o d r a t i o i s more general and always y i e l d s an appropriate s u f f i c i e n t s t a t i s t i c whenever one e x i s t s . For any d e c i s i o n problem, a l a r g e number of d i f f e r e n t tests may be found, depending on the hypothesis statements. Following are some that have proven u s e f u l f o r SNM accounting.

The

and

One-State Kalman F i l t e r

Statistic

Assume that a l l measurement e r r o r s are Gaussian l e t the hypotheses be represented by HQ :

M(k)

= M

Q

+ v (k) M

,

M

Q

nd a d d i t i v e ,

< 0 k = 1,2,...,

Η : χ

M(k)

= Μ

χ

+ v (k) M

,

Μ

χ

(7)

> 0

where v^(k) i s the measurement e r r o r f o r the kth m a t e r i a l b a l ance. Then, the likelihood functions at any time k become (10,13)


NUCLEAR

44

SAFEGUARDS

ANALYSIS

p[Z(k)|Ho]= p[M(l),M(2),. . .,M(k)|HQ]

k -1/2 Π [2π V ( i ) ] exp 1—1 M

[M(i) - M Q F j ™ M

,


(8)

p[Z(k)|H ]= 1

k Π i=l

-1/2 [2π V ( i ) ] M

exp

[M(i) - M y 2

V

Y

( i

where V ^ ( i ) i s the e r r o r v a r i a n c e f o r the i t h m a t e r i a l balance, M(i). Note that the two l i k e l i h o o d functions have unknown parameters MQ and Mj_. From the h y p o t h e s i s statements, we must have MQ _< 0 and M^ 0. Thus, the maximum likelihood estimates f o r MQ and M^ (from the s e c t i o n on composite hypotheses) are, r e s p e c t i v e l y , Mn(k) = min {0, M(k) }

,

%(k)

,

(9) = max {0, M(k) }

where M(k) i s the one-state Kalman f i l t e r estimate (14,15) f o r the m a t e r i a l balance at time k. The estimate can be c a l c u l a t e d r e c u r s i v e l y from the equations M(k) = M(k-l) + B(k)[M(k) - M ( k - l ) ] , (10) «/.χ

=

V(k-l) V(k-l) + v (k) M

'

where B(k) i s c a l l e d the f i l t e r gain, and V ( k - l ) i s the v a r i a n c e of the e r r o r i n the estimate M ( k - l ) . V(k) i s a l s o given recur s i v e l y by V(k)

= [l-B(k)]V(k-D

.

(11)

I n i t i a l c o n d i t i o n s f o r the equations are M(0) = 0, V(0) = 0) or e x t r a ( i f M(k) < 0) m a t e r i a l per m a t e r i a l balance. However, t h i s does not mean that, f o r the t e s t to work p r o p e r l y , the a c t u a l d i v e r s i o n must have occurred as a constant amount s t o l e n during each balance p e r i o d . Even i f a l l the d i v e r s i o n took place w i t h i n one balance p e r i o d , the filter will s t i l l c a l c u l a t e the c o r r e c t average per balance over any time i n t e r v a l c o n t a i n i n g the d i v e r s i o n . I m p l i c i t i n the hypotheses statements i s the assumption that the sequence {M(i), i = l,2,...,k} and its associated error variances are equivalent to Z ( k ) , that i s , that knowledge of the separate inventory and t r a n s f e r components of the material balances i s unimportant. This would be true, f o r example, i f the inventory measurement e r r o r s were small compared to those of the t r a n s f e r s . In that case, the Kalman f i l t e r estimate can be shown to be optimal i n the sense that i t i s the l i n e a r , minimumvariance, unbiased estimate whenever the measurement e r r o r proba b i l i t y d e n s i t i e s are symmetric about t h e i r means (16) ; i . e . , the Gaussian e r r o r assumption i s not necessary for calculating the estimate.

The

CUSUM S t a t i s t i c

I f the material-balance e r r o r variances are a l l constant, V (k) = V f o r k = 1,2,..., then s o l u t i o n of Eqs. 10 and 11 results i n M

M

M(k)

= i

Σ

M(i)

,

V(k)

= ^

.

i=l


(13)

46

NUCLEAR

SAFEGUARDS

ANALYSIS

M u l t i p l y i n g both s i d e s o f the f i r s t equation by k y i e l d s a new s t a t i s t i c c a l l e d the CUSUM (cumulative summation) (18,19,20,21): k CUSUM (k) = Σ M(i) i=l

,

(14)


which has v a r i a n c e k-1

V (k) = v (k) c

x

+ v (o> + Σ I

V

( i ) T

( 1 5 )

»

i=0 where V j ( - ) and νχ( · ) are the inventory and t r a n s f e r meas urement e r r o r v a r i a n c e s , r e s p e c t i v e l y . The CUSUM s t a t i s t i c i s i n t e r e s t i n g , even i f the m a t e r i a l - b a l a n c e e r r o r variances are not constant, because i t i s an estimate of the t o t a l amount of missing m a t e r i a l at any time during the p e r i o d o f i n t e r e s t . It i s g e n e r a l l y not optimal i n any sense, but i t has a u s e f u l p h y s i c a l i n t e r p r e t a t i o n and has become q u i t e common. A development analogous to that f o r the one-state Kalman f i l t e r y i e l d s the f o l l o w i n g SPRT: < - / 2 I In T I

, accept H

> + /2|ln Τ J

, accept Η

Q

I f

CUSUM (k)

Q

χ

, ,

(16)

otherwise, take another observation, which i s the same form as Eq. 12 i n that an s i o n i s compared to i t s standard d e v i a t i o n .

The Two-State Kalman F i l t e r

estimate

of

diver

Statistic

I f the assumption that the inventory measurement e r r o r s are small compared to the t r a n s f e r measurement e r r o r s i s not v a l i d , then an approach devised by Pike and h i s coworkers (22,23,24,25) w i l l y i e l d a m a t e r i a l balance estimate having smaller v a r i a n c e than the one-state f i l t e r . The technique i s to estimate both the m a t e r i a l balance and the inventory, which means that the f i l t e r now has two s t a t e v a r i a b l e s r a t h e r than one. In r e c u r sive form, the f i l t e r equations are


4.

Decision

SHIPLEY

Analysis

47

I(k) = l ( k | k - l ) + B ( k ) [ l ( k ) - l ( k | k - l ) ]

,

1

M(k)

= M(k-l) + B ( k ) [ l ( k ) - l ( k | k - l ) ]

î(klk-l) = î(k-l) + T ( k - l ) - M(k-l)


,

2

(17)

,

and I(k) and M(k) are the inventory and material-balance estimates, r e s p e c t i v e l y , at time k based on a l l information through time k. The f i l t e r gains B^(k) and B (k) are given by 2

Vk) V

k

)

V (lc) flGr

v 7 k T

=

·

V

k

)

=

v

7

k

T

(

1

8

)

where Vj(k) and Vfg(k) are r e s p e c t i v e l y the inventory estimate e r r o r v a r i a n c e and the c^variance between the inventory and material-balance estimate ert*yrs. They are given r e c u r s i v e l y by Vj(k|k-l)V (k) I

V

î

(

k

)

=

Vj(k|k-l) +

v (k) x

(19) V-(k|k-l)V_(k) V

ÎM

( k )

" Vj(k|k-l) + V (k) x

'

with

Vj(k|k-1) = V j ( k - l ) - 2 V

îfi[

( k - l ) + V (k-1) + V ( k - l ) ft

T

:

(20) Vj (k|k-1) = V ft

îft

( k - 1 ) - V (k-1)

.

ft

The material-balance e r r o r variance at time k i s

2

V (k|k-1) f t i ) + v (k) fft

tyk) = v (k-D ft

The M(0)

V î ( k

( 2 1 )

x

filter is initiated with 1(0) = 1(0), Vj(0) = ν ( 0 ) , = 0, V ( 0 ) = oo, before. By a development s i m i l a r to that f o r the one-state filter, the SPRT can be shown to reduce to χ

M

a s


NUCLEAR

48

If

fi(k)

v^kl

< - /2 |ln TQ I

, accept H

Q

> + /2 |ln Ί

, accept H

1

χ

I

SAFEGUARDS

ANALYSIS

, ,

(22)


otherwise, take another o b s e r v a t i o n . G e n e r a l l y , t h i s t e s t w i l l be more s e n s i t i v e than Eq. 12 because the estimate e r r o r v a r i a n c e i s smaller. This formulation has two other advantages. First, i t pro v i d e s a b e t t e r estimate of the inventory. Second, the effects of c o r r e l a t e d m a t e r i a l balances caused by the common inventory measurement have disappeared as a r e s u l t of the f i l t e r s t r u c t u r e . However, we have bought these advantages at the expense of com p l e x i t y and information requirements.

Nonparametric Tests A l l t e s t s such as those j u s t discussed are c a l l e d parametric because they depend upon knowledge of the s t a t i s t i c s of the measurement e r r o r s . They a l s o happen to work best when the measurement e r r o r s are Gaussian, a q u i t e common but by no means all-inclusive situation. I f the measurement e r r o r statistics are unknown or non-Gaussian, then nonparametric sufficient sta t i s t i c s ( 2 6 - 3 2 ) may give b e t t e r test results. In a d d i t i o n , nonparametric t e s t s can provide independent support f o r the r e s u l t s of parametric t e s t s even though nonparametric tests are g e n e r a l l y l e s s powerful than parametric ones under conditions for which the l a t t e r are designed. The two most common nonparametric t e s t s are the s i g n test and the Wilcoxon rank sum t e s t . The sufficient s t a t i s t i c for the sign test i s the t o t a l number of positive material balances. That f o r the Wilcoxon t e s t i s the sum of the ranks of the m a t e r i a l balances. The rank of a m a t e r i a l balance i s i t s r e l a t i v e p o s i t i o n index under a r e o r d e r i n g of the m a t e r i a l bal ances according to magnitude. Other, p o s s i b l y more e f f e c t i v e nonparametric t e s t s are being i n v e s t i g a t e d . Further discussion of nonparametric t e s t s i s beyond the scope of t h i s paper.

CORRELATIONS Consider f i r s t the problem of c o r r e l a t e d measurements, i n particular, correlated t r a n s f e r measurements. The following s i m p l i f i e d treatment i s due p r i m a r i l y to F r i e d l a n d (33,34,35). Let the a c t u a l net t r a n s f e r T ( k ) be represented by a

a

T ( k ) = T(k) - v(k) - w(k)

,


(23)

4.

SHIPLEY

Decision

Analysis

49

where T(k) i s the t r a n s f e r measurement, v(k) i s the random meas urement e r r o r ( i . e . , E[v(k)v(k+j) ] = 0 f o r a l l j ψ 0, and Ε [·] i s the expectation o p e r a t o r ) , and w(k) i s the s o - c a l l e d "systema t i c e r r o r . " L e t us assume that w(k) i s a b i a s that a r i s e s from instrument m i s c a l i b r a t i o n , say, and thus i s constant between c a l i b r a t i o n s . Further assume that the (constant) w(k) resulting from any c a l i b r a t i o n i s a Gaussian random v a r i a b l e w i t h mean zero and v a r i a n c e V . Then w(k) can be represented r e c u r s i v e l y by the d i f f e r e n c e equation Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on February 27, 2018 | https://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0079.ch004

w

w(k) = a w(k-l) + ( l - a ) u ( k ) ,

k = 1,2,...,

0

i f a c a l i b r a t i o n j u s t occurred,

1

i f a c a l i b r a t i o n has not j u s t o c c u r r e d ,

(24)

where

and u(k) i s a Gaussian random v a r i a b l e with mean zero and v a r i ance V equal to the covariance between t r a n s f e r measure ments. Equation 24 can be appended to the state equations f o r e i t h e r the one- or two-state Kalman f i l t e r , which w i l l then y i e l d an estimate of the b i a s w(k). Any systematic e r r o r can be t r e a t e d i n t h i s fashion merely by i n c r e a s i n g the order of the f i l t e r , but knowledge of the systematic e r r o r statistics is required. One of F r i e d l a n d s major r e s u l t s (33) i s that the optimum m a t e r i a l balance estimate can be expressed as w

!

M(k) = M(k) + Dw(k)

(25)

where M(k) i s the b i a s - f r e e estimate, computed as i f there were no b i a s , and D i s r e l a t e d to the r a t i o of the covariance o f M(k) and w(k) to the variance of w(k) . Thus, c a l c u l a t i o n o f M(k) can be decoupled from the b i a s estimate u n t i l the f i n a l s t e p . This k i n d of systmatic e r r o r i s an example o f a p o s i t i v e c o r r e l a t i o n , and f a i l u r e to account f o r i t has two d e l e t e r i o u s e f f e c t s . F i r s t , the m a t e r i a l - b a l a n c e estimate i s b i a s e d , pos s i b l y g i v i n g a biased d e c i s i o n . Second, the v a r i a n c e o f the material-balance estimate e r r o r appears to be s m a l l e r than i s a c t u a l l y the case. This may r e s u l t i n a d e c i s i o n that seems to be b e t t e r than i t i s . Now consider m a t e r i a l balances that are c o r r e l a t e d (nega t i v e l y ) through the common inventory measurement, as f o r the one-state Kalman f i l t e r . Write the k t h m a t e r i a l balance as a

a

M(k) = - I ( k ) + I ( k - 1 ) - v_(k) + v ( k - l ) + T ( k ) , T

a

(26)

where I ( k ) i s the k t h a c t u a l inventory and v j ( k ) i s the k t h inventory measurement e r r o r with variance V-j-(k) . Define


50

NUCLEAR

v^k) = + v ( k - l ) ,

v (k) = -v^k)

I

In

2

ANALYSIS

.

(27)

,

(28)

r e c u r s i v e form, these equations are

v (k) = - v ( k - l ) , x


SAFEGUARDS

v (k) = -v^k)

2

2

where v^ and V2 are now considered as s t a t e v a r i a b l e s just as i s the m a t e r i a l balance i n the one-state f i l t e r . In analogy to the treatment of p o s i t i v e c o r r e l a t i o n s , Eq. 28 can be appended to the state equations f o r the one-state f i l t e r ( f o r the twos t a t e f i l t e r there i s no need), which then gives estimates, νχ and v , of the inventory measurement e r r o r s . That i s , this method of t r e a t i n g the negative c o r r e l a t i o n s also generates improved inventory estimates, which are given by 2

I(k-l|k) = K k - l ) - v (k|k) x

, (29)

I(k|k) = I(k) + v (k|k) 2

.

Notice that l(k|k) i s the f i l t e r e d estimate of the inventory at time k and i s based on the f i r s t k inventory measurements. The estimate l(k-l|k) a l s o uses the f i r s t k inventory measurements, but i t i s the lag-one, smoothed estimate of the inventory at time k-1. A negative c o r r e l a t i o n such as t h i s , contrasted to the p o s i t i v e ones treated above, causes no b i a s i n the m a t e r i a l balance estimate. However, i t does r e s u l t i n a m a t e r i a l balance error v a r i a n c e that appears l a r g e r than a c t u a l i f the c o r r e l a t i o n i s ignored. The s e n s i t i v i t y of the corresponding d e c i s i o n test would, t h e r e f o r e , be degraded.

TEST APPLICATION Procedure As d i s c u s s e d above, we seldom w i l l know beforehand when d i v e r s i o n has s t a r t e d or how long i t w i l l l a s t . Therefore, the d e c i s i o n t e s t s must examine a l l p o s s i b l e , contiguous subsequences of the a v a i l a b l e m a t e r i a l s accounting data (1,2,3,18). That i s , i f at some time we have Ν m a t e r i a l balances, then there are Ν s t a r t i n g points f o r Ν p o s s i b l e sequences, a l l ending at the Nth, or c u r r e n t , m a t e r i a l balance, and the sequence lengths range from Ν to 1. Because of the s e q u e n t i a l a p p l i c a t i o n of the t e s t s , sequences with ending p o i n t s less than Ν have already been tested; those with ending p o i n t s g r e a t e r than Ν w i l l be done i f the t e s t s do not terminate before then.


4.

Decision

SHIPLEY

51

Analysis


Another procedure that helps i n i n t e r p r e t i n g the r e s u l t s of tests i s to do the t e s t i n g at several significance levels, or f a l s e - a l a r m p r o b a b i l i t i e s . This i s so because, i n p r a c t i c e , the test thresholds are never e x a c t l y met; thus, the true signifi cance of the data i s obscured. Several thresholds corresponding to d i f f e r e n t f a l s e - a l a r m p r o b a b i l i t i e s give at l e a s t a rough idea of the a c t u a l p r o b a b i l i t y of a f a l s e alarm.

D i s p l a y i n g the

Results

Of course, one of the r e s u l t s of most i n t e r e s t i s the suffi cient s t a t i s t i c . Common p r a c t i c e i s to p l o t the s u f f i c i e n t sta t i s t i c , with symmetric e r r o r bars of length twice the square root of i t s v a r i a n c e , vs the m a t e r i a l balance number. The ini t i a l m a t e r i a l balance and the t o t a l number of m a t e r i a l balances are chosen a r b i t r a r i l y , perhaps to correspond to the shift or campaign s t r u c t u r e of the process. For example, i f balances are drawn hourly, and a day c o n s i s t s of three s h i f t s , then the ini t i a l m a t e r i a l balance might be chosen as the f i r s t of the day, and the t o t a l number of m a t e r i a l balances might be 24, covering three s h i f t s . Remember, however, that t h i s choice is for dis play purposes only; the actual t e s t i n g procedure s e l e c t s a l l p o s s i b l e i n i t i a l p o i n t s and sequence lengths, and any sufficient s t a t i s t i c may be d i layed as seems appropriate. The other important r e s u l t s are the outcomes of the tests, performed at the s e v e r a l s i g n i f i c a n c e l e v e l s . A new t o o l , c a l l e d the alarm-sequence chart (1,2,3,18), has been developed to dis play these r e s u l t s i n compact and readable form. To generate the alarm-sequence c h a r t , each sequence causing an alarm i s assigned (1) a d e s c r i p t o r that c l a s s i f i e s the alarm according to its f a l s e - a l a r m p r o b a b i l i t y , and (2) a pair of integers ^ l > 2 ^ that are, r e s p e c t i v e l y , the indexes of the initial and f i n a l m a t e r i a l balance numbers i n the sequence. I t i s a l s o p o s s i b l e to d e f i n e ( r ^ , ^ ) as the sequence length and the f i n a l m a t e r i a l balance number of the sequence. The two defini tions are e q u i v a l e n t . The alarm-sequence chart i s a point plot of r ^ vs r f o r each sequence that caused an alarm, with the s i g n i f i c a n c e range of each point i n d i c a t e d by the p l o t t i n g sym bol. One p o s s i b l e correspondence of p l o t t i n g symbol to signifi cance i s given i n Table I. The symbol Τ denotes sequences of such low s i g n i f i c a n c e that i t would be fruitless to examine extensions of them; the l e t t e r Τ indicates their termination points. I t i s always true that r\ £ r so that a l l symbols l i e to the r i g h t of the line r^ = r through the origin. Examples of these charts are shown i n the s e c t i o n on r e s u l t s . r

r

2

2

2


52

NUCLEAR

SAFEGUARDS

ANALYSIS

TABLE I ALARM CLASSIFICATION FOR


Classification ( P l o t t i n g Symbol)

THE ALARM-SEQUENCE CHART

False-Alarm P r o b a b i l i t y ΙΟ"

A Β

5 χ

ίο"

to 5

3

to

3

to 5

ίο"

C

2

X

ίο" ίο"

X

3

3

ίο"

D

5 χ ΙΟ"

4

to

ίο"

Ε

ΙΟ"

4

to

ίο"

4

4

5

< ΙΟ"

F

5

0.5

Τ

AN EXAMPLE The

Process

To demonstrate the a p p l i c a t i o n of d e c i s i o n a n a l y s i s , we present r e s u l t s from a study (2) of m a t e r i a l s accounting in a nuclear f u e l r e p r o c e s s i n g p l a n t s i m i l a r to the A l l i e d - G e n e r a l Nuclear S e r v i c e s (AGNS) chemical separations facility at Barnwell, South C a r o l i n a (BNFP). The BNFP (36) i s designed to use the Purex process to recover uranium and plutonium from power-reactor spent fuels c o n t a i n i n g e i t h e r enriched uranium oxide or mixed uranium-plutonium oxide. Nominal c a p a c i t y i s 1500 MT/yr of spent f u e l , which i s approximately equivalent to 50 kg/day of plutonium. For a p l a n t such as BNFP, the most d e s i r a b l e areas f o r mate r i a l s accounting would be those c o n t a i n i n g the l a r g e s t amounts of plutonium i n a form most a t t r a c t i v e to the d i v e r t o r . The plutonium at the head end of the process i s not a t t r a c t i v e be cause i t c o n t a i n s l e t h a l concentrations of f i s s i o n products and i s d i l u t e d approximately 100-fold with uranium. However, a f t e r the IB column, the bulk of the f i s s i o n products have been removed and the uranium/plutonium r a t i o has been reduced to 2/1. From t h i s p o i n t the plutonium becomes i n c r e a s i n g l y a t t r a c t i v e as i t proceeds through the process to the plutonium-nitrate storage tanks. Hence, t h i s area, the plutonium purification process (PPP), was s e l e c t e d f o r design of a dynamic m a t e r i a l s accounting system. A b l o c k diagram of the PPP i s shown i n F i g . 2. Typical values f o r c o n c e n t r a t i o n s and flow r a t e s are given i n Table I I , and Table I I I l i s t s nominal in-process i n v e n t o r i e s .


Decision

Analysis


SHIPLEY

Figure 2. Block diagram of the plutonium purification process


54

NUCLEAR

SAFEGUARDS

ANALYSIS

TABLE I I CONCENTRATIONS AND FLOW RATES IN THE PPP

Stream IBP Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on February 27, 2018 | https://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0079.ch004

Plutonium Concentration (g/L)

Flow (L/h)

3PCP

400

5

8

250

2AW

500

trace

3AW

215

0.1

3PD

32

trace

2BW

150

trace

3BW

105

trace

TABLE I I I IN-PROCESS HOLDUP IN TANKS AND VESSELS OF THE PPP

Identification^

Volume (L)

Plutonium Concentration

(g/D

a

Plutonium Holdup (kg)

4.942

7.4

IBP Tank

1500

2A Column

700

c

4.6

2B Column

500

c

2.8

3A Column

600

c

5.4

c

4.8

58.70

1.2

3Β Column

440

3PS Wash Column

20

3P Concentrator

60

250.

15.

a

These values are not flowsheet values of any existing reprocessing f a c i l i t y but represent t y p i c a l values w i t h i n r e a sonable ranges of a workable flowsheet.

b

See F i g . 2.

c

A model of the c o n c e n t r a t i o n p r o f i l e s and the holdup i n the pulse columns i s described i n Ref. (2^).


4.

Decision

SHIPLEY

55

Analysis


The M a t e r i a l s Accounting

System

To i s o l a t e the PPP as a u n i t process r e q u i r e s flow and con c e n t r a t i o n measurements at the IBP tank (input) and 3P concen t r a t o r (output). In a d d i t i o n , a c i d r e c y l e s (2AW, 3AW, 3PD) and organic r e c y c l e (2BW, 3BW) must be monitored f o r flow and con c e n t r a t i o n , and an estimate o f the in-process inventory must be obtained. Table IV gives the required measurements and some p o s s i b l e measurement methods and a s s o c i a t e d u n c e r t a i n t i e s . The r e l a t i v e p r e c i s i o n o f dynamic volume measurements i s estimated to be 3% (1σ) f o r the IBP tank, t h r e e f o l d more than f o r a conventional p h y s i c a l - i n v e n t o r y measurement because liquid is continuously flowing i n t o and out o f the tank during process i n g . Dynamic estimates o f plutonium c o n c e n t r a t i o n i n the IBP and 3P concentrator tanks can be obtained from d i r e c t , in-line measurements (by absorption-edge densitometry, f o r example), or from combinations o f adjacent a c c o u n t a b i l i t y and p r o c e s s - c o n t r o l measurements.

TABLE IV MATERIALS ACCOUNTING MEASUREMENTS FOR THE PPP

Instrument Precision (1σ, %)

Calibration Error (1σ, %)

Measurement Point

Measurement Type

IBP, 3PCP streams

Flow meter Absorption-edge densitometry

1

0.5

1

0.3

Volume Absorption-edge densitometry

3

2A,2B,3A,3B columns

See text

5-20

2AW,2BW,3AW, 3BW, 3PD streams

Flow meter NDA

IBP surge tank

3

5 10

3PS column

See text

5-20

3P concentrator

Volume (constant) Absorption-edge densitometry

1.5

H i g h - q u a l i t y measurements o f the in-process plutonium tory i n the columns are the most d i f f i c u l t to make.


inven In the

NUCLEAR


56

SAFEGUARDS

ANALYSIS

reference design, the columns are f u l l y instrumented f o r process control, i n c l u d i n g measurements o f the position o f the aqueous-organic i n t e r f a c e and o f the l e v e l and d e n s i t y o f l i q u i d i n the product-disengagement volume. Much of the column holdup i s i n the product-disengagement volume, and a good measurement of t h i s volume i s a v a i l a b l e . However, only a crude estimate of plutonium c o n c e n t r a t i o n can be made without additional instru mentation. R e l a t i v e p r e c i s i o n f o r column-holdup measurements i s estimated to be i n the range o f 5-20% (1σ). The 20% l i m i t appears to be conservative i n terms o f d i s c u s s i o n s with industry and DOE personnel. A p r e c i s i o n o f 10% should be p r a c t i c a b l e using the current p r o c e s s - c o n t r o l instrumentation. Improvements toward the 5% f i g u r e (or b e t t e r ) w i l l require a d d i t i o n a l research and development to i d e n t i f y optimum combinations of a d d i t i o n a l o n - l i n e instrumentation and improved models of column behavior. Waste and r e c y c l e streams from the columns and the concen t r a t o r i n the PPP are monitored by a combination o f flowmeters and NDA-alpha detectors f o r plutonium concentration. The alpha monitors are already used f o r process c o n t r o l i n the reference design and r e q u i r e only modest upgrading (primarily calibration and s e n s i t i v i t y s t u d i e s ) to be used f o r a c c o u n t a b i l i t y as w e l l . Flow measurements i n the waste and r e c y c l e streams can be simple and r e l a t i v e l y crude (5-10%) because the amount o f plutonium i s small. A rough c a l i b r a t i o n o f the a i r l i f t s f o r l i q u i d flow may s u f f i c e , or continuous l e v e l monitors i n the appropriate headpots could provide the r e q u i r e d data. Several measurement s t r a t e g i e s have been i n v e s t i g a t e d , i n c l u d i n g one i n which m a t e r i a l balances are drawn every hour, column inventory measurement p r e c i s i o n i s taken as 5%, and flow meters are r e c a l i b r a t e d every 24 hours. This i s the best o f the s t r a t e g i e s considered and i s the one on which the f o l l o w i n g r e s u l t s are based.

Results D e c i s i o n a n a l y s i s techniques have been used to evaluate the d i v e r s i o n s e n s i t i v i t y o f t h i s m a t e r i a l s accounting s t r a t e g y and others (2^). Part o f the e v a l u a t i o n c o n s i s t s of examining test r e s u l t s , i n the form o f estimate (sufficient s t a t i s t i c ) and alarm-sequence c h a r t s , f o r v a r i o u s time i n t e r v a l s . Examples of t y p i c a l one-day charts f o r the CUSUM and two-state Kalman f i l t e r , both with and without d i v e r s i o n , are given i n F i g s . 3 and 5; the h o r i z o n t a l marks i n d i c a t e the values o f the estimates, and the v e r t i c a l l i n e s are + 1σ e r r o r bars about those estimates. The corresponding alarm-sequence charts are shown i n F i g s 4 and 6. The d i v e r s i o n l e v e l f o r the lower charts i s 0.073 kg Pu/balance p e r i o d , which i s about 0.1 standard d e v i a t i o n o f a s i n g l e mate r i a l balance. Results o f a l a r g e number of t e s t s show that the


Decision


SHIPLEY

Analysis

cn ZD CJ

5

10

15

20

25

MRTERIRL BRLRNCE NUMBER

Q_

2 +

CD

=3

CO

3

0-

"+-

—I— 15


Figure 3.

CUSUM

charts without diversion (upper) and with diversion (lower)



NUCLEAR

ι

1

1

SAFEGUARDS

r

20 +

•

15 +

Q_

az £ 10 4-

1 0

5

1 10

FINRL

5

10

1 15

1 20

25

POINT

15

20

F I N R L POINT

Figure 4. Alarm-sequence charts for the of Figure 3

CUSUMs


ANALYSIS


SHIPLEY

Decision

Analysis

1

1

1

-

ι 1 1 1 1 1 1 1 ι . . . ι

)t'

-

1 0

li

5

1

1

1

10

15

20

25


1

1

!

--

I | t l t n m

m

r

1 -

— h

1

1

10

1 —

15

1


Figure 5. Kalman filter estimates of average missing material with out diversion (upper) and with diversion (lower)


NUCLEAR


60

SAFEGUARDS

Q_ _J CE

25

π

1

1—"

r

20 +

15 + CE

^

10 +

fl R BCDEDCDDCCBflflflC

1 0

5

1 10

FINRL

1 15

1 20

25

POINT

Figure 6. Alarm-sequence charts for the Kalmanfilter estimates of Figure 5


ANALYSIS


4.

SHIPLEY

Decision

Analysis

61

two-state Kalman f i l t e r gives somewhat b e t t e r r e s u l t s than the CUSUM, as expected. In the course of e v a l u a t i o n , many such sets of charts are examined so that the random e f f e c t s of measurement e r r o r s and normal process v a r i a b i l i t y can be assessed; that i s , we perform a Monte Carlo study to estimate the sensitivity to d i v e r s i o n . In applying d e c i s i o n a n a l y s i s to data from a f a c i l i t y operating under a c t u a l c o n d i t i o n s , only one set of data w i l l be a v a i l a b l e f o r making d e c i s i o n s , r a t h e r than the m u l t i p l e data streams gen erated from a s i m u l a t i o n . In p a r t i c u l a r , direct comparison of charts with and without d i v e r s i o n , as shown here, w i l l be impos s i b l e . The decision-maker w i l l have to e x t r a p o l a t e from h i s t o r i c a l information and from c a r e f u l process and measurement analy s i s to determine whether d i v e r s i o n has occurred. The r e s u l t s of the e v a l u a t i o n are given i n Table V. By com p a r i s o n , current r e g u l a t i o n s r e q u i r e that the material-balance u n c e r t a i n t y be l e s s than 1% (2σ) of throughput f o r each six-month accounting p e r i o d , which corresponds to 75 kg of plutonium f o r t h i s process. Such l a r g e improvement in diversion sensitivity i s p o s s i b l e through the combination of timely measurements with the powerful s t a t i s t i c a l methods of d e c i s i o n a n a l y s i s .

TABLE V DIVERSION SENSITIVITY FOR

Average D i v e r s i o n per Balance (kg Pu) 2.6

Detection Time (h)

THE

PPP

T o t a l at Time of Detection (kg

1

2.6

0.075

24

1.8

0.025

168

(1 week)

Pu)

4.2

LITERATURE CITED

1.

Shipley, J . P., Cobb, D. D., Dietz, R. J., Evans, M. L . , Schelonka, E. P., Smith, D. B . , and Walton, R. B., "Coordinated Safeguards for Materials Management in a Mixed-Oxide Fuel Facility," Los Alamos Scientific Laboratory report LA-6536 (February 1977).



62

NUCLEAR SAFEGUARDS ANALYSIS

2.

Hakkila, Ε. Α . , Cobb, D. D . , Dayem, Η. Α . , Dietz, R. J., Kern, Ε. Α . , Schelonka, E . P . , Shipley, J . P . , Smith, D. B . , Augustson, R. H . , and Barnes, J . W., "Coordinated Safeguards for Materials Management in a Fuel Reprocessing Plant," Los Alamos Scientific Laboratory report LA-6881 (September 1977).

3.

Dayem, Η. Α . , Cobb, D. D., Dietz, R. J., Hakkila, Ε. Α . , Kern, Ε. Α . , Shipley, J . P . , Smith, D. B . , and Bowersox, D. F., "Coordinated Safeguards for Materials Management in a Nitrate-to-Oxide Conversion Facility," Los Alamos Scientific Laboratory report LA-7011 (to be published).

4.

Keepin, G. R., and Maraman, W. J., "Nondestructive Assay Technology and In-Plant Dynamic Materials Control--DYMAC," in Safeguarding Nuclear Materials, Proc. Symp., Vienna, Oct. 20-24, 1975 (International Atomic Energy Agency, Vienna, 1976), Paper IAEA-SM-201/32, Vol. 1, pp. 304-320.

5.

Augustson, R. H . , "Development of In-Plant Real-Time Materials Control: The DYMAC Program," Proc. 17th Annual Meeting of the Institute of Nuclear Materials Management, Seattle, Washington, June 22-24, 1976.

6.

Howard, R. Α . , "Decision Analysis: Pespectives on Inference, Decision, and Experimentation," Proc. IEEE, Special Issue on Detection Theory and Applications 58, No. 5, 632-643 (1970).

7.

Ref. (2), Vol. II, App. E .

8.

Shipley, J . P . , "Decision Analysis in Safeguarding Special Nuclear Material," Invited paper, Trans. Am. Nucl. Soc. 27, 178 (1977).

9.

Shipley, J. P . , "Decision Analysis for Dynamic Accounting of Nuclear Material," paper presented at the American Nuclear Society Topical Meeting, Williamsburg, V i r g i n i a , May 15-17, 1978.

10.

Sage, A. P. and Melsa, J . L., Estimation Theory with Applications to Communications and Control (McGraw-Hill, 1971).

11.

Lehmann, Testing S t a t i s t i c a l Sons, Inc., 1959).

12.

Blackwell and Girshick, Μ. Α . , Theory S t a t i s t i c a l Decisions (Wiley, 1954).

Hypotheses

(John of

Wiley Games


and and


4. SHIPLEY

63

Decision Analysis

13.

Wald, Α . , Sequential Analysis (John Wiley 1947).

14.

Kalman, R. E., "A New Approach to Linear F i l t e r i n g and Prediction Problems," Trans. ASME J. Basic Eng. 82D, 34-45 (March 1960).

15.

Kalman, R. E . and Bucy, R. S., "New Results in Linear Filtering and Prediction Theory," Trans. ASME J. Basic Eng. 83D, 95-108 (March 1961).

16.

Meditch, J. S., Stochastic Optimal Control (McGraw-Hill, 1969).

17.

Jazwinski, Α. Η . , Stochastic Processes and F i l t e r i n g (Academic Press, 1970).

18.

Cobb, D. D . , Smith, D. B . , and Shipley, J . P . , Sum Charts in Safeguarding Special Nuclear submitted to Technometrics (December 1976).

19.

Duncan, A. J., Quality Control (R. D. Irwin, Inc., 1965).

20.

Page, E . S., "Cumulative Sum Charts," Technometrics 1, 1-9 (February 1961).

21.

Evans, W. D . , "When and How to Use Cu-Sum Technometrics 5, No. 1, 1-22 (February 1963).

22.

Pike, D. Η . , Morrison, G. W., and Holland, C. W., "Linear F i l t e r i n g Applied to Safeguards of Nuclear Material," Trans. Amer. Nucl. Soc. 22, 143-144 (1975).

23.

Pike, D. Η . , Morrison, G. W., and Holland, C. W., "A Comparison of Several Kalman F i l t e r Models for Establishing MUF," Trans. Amer. Nucl. Soc. 23, 267-268 (1976).

24.

Pike, D. H. and Morrison, G. W., "A New Approach to Safeguards Accounting," Oak Ridge National Laboratory report ORNL/CSD/TM-25 (March 1977).

25.

Pike, D. H. and Morrison, G. W., "A New Approach Safeguards Accounting," Nucl. Mater. Manage. VI, No. 641-658 (1977).

26.

Thomas, J . B . , "Nonparametric Detection," Proc. IEEE, Special Issue on Detection Theory and Applications 58, No. 5, 623-631 (May 1970).

Linear

and Sons,

Inc.,

Estimation

and

Theory

"Cumulative Materials,"

and Industrial

Statistics


3,

No.

Charts,"

to 3,

NUCLEAR SAFEGUARDS


64

Z.,

Theory

of

Rank

Tests

ANALYSIS

27.

Hajek, J. and Sidak, Press, 1967).

(Academic

28.

Carlyle, J. W. and Thomas, J. B . , "On Nonparametric Signal Detectors," IEEE Trans. Info. Theory IT-10, No. 2, 146-152 (1964).

29.

Tantaratana, S. and Thomas, J . B . , Detection of a Constant Signal," IEEE IT-23, No. 3, 304-315 (May 1977).

30.

Capon, J., "A Nonparametric Technique for the Detection of a Constant Signal in Additive Noise," 1959 IRE WESCON Convention Record, Part 4, San Francisco, August 1959.

31.

Gibson, J . D. and Melsa, J. L., Nonparametric Detection with Applications 1975).

32.

Puri, M. L . and Sen, P. Κ., Nonparametric Multivariate Analysis (Wiley, 1971).

33.

Friedland, B . , "Treatment of Bias in Recursive F i l t e r i n g , " IEEE Trans. Autom. Contr. AC-14, No. 4, 359-367 (1969).

34.

Friedland, B . , "Recursive F i l t e r i n g in the Presence of Biases with Irreducible Uncertainty," IEEE Trans. Autom. Contr. AC-21, No. 5, 789-790 (1976).

35.

Friedland, B . , "On the Calibration Problem," Autom. Contr. AC-22, No. 6, 899-905 (1977).

36.

"Barnwell Nuclear Fuel Plant-Separation F a c i l i t y Final Safety Analysis Report," A l l i e d General Nuclear Services, Barnwell, South Carolina (October 1975).

"On Sequential Sign Trans. Info. Theory

Introduction to (Academic Press,

Methods

IEEE

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in

Trans.

Nuclear Safeguards Analysis - American Chemical Society

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