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The stepwise time dependent backward Kolmogorov equations with delayed neutrons in counting statistics II. Covariance
Nuclear Instruments and Methods 205 (1983) 505-510 North-Holland Publishing Company
505
THE STEPWISE TIME DEPENDENT BACKWARD KOLMOGOROV EQUATIONS WITH D E L A Y E D N E U T R O N S IN C O U N T I N G S T A T I S T I C S II. Covariance F. C A R L O N I * a n d M. M A R S E G U E R R A lstituto di lngegneria Nucleare - C E S N E F , Politecnico di Milano, Italy
Received 5 July 1982 This paper completes work dealing with the neutron counting statistics in a zero power reactor subject to an arbitrary sequence of stepwise variations of all the parameters. Using the one-velocity, point reactor model with delayed neutrons and following the backwards Kolmogorov formalism, expressions for the mean value, variance and covariance of the count distribution have been obtained. Knowledge of these moments allows the estimation of kinetic parameters of a reactor together with their statistical errors under any time variable condition concerning criticality, source intensity and detector efficiency. Numerical examples relating to a rod drop and to a sinusoidal pile oscillator experiments are reported for illustration.
1. Introduction This paper is the second of two [1] which aim at o b t a i n i n g same experimentally useful statistics of the n e u t r o n c o u n t d i s t r i b u t i o n in an arbitrarily time varying zero power reactor. The variation is modeled by discretizing the time in a sequence of intervals in which all the system parameters assume c o n s t a n t values, possibly different from interval to interval. The system parameters are those p e r t a i n i n g to the reactor itself, n a m e l y the reactivity a n d the p r o m p t n e u t r o n life-time, and, in addition, the external n e u t r o n source strength a n d the efficiencies of the n e u t r o n detectors. All these parameters m a y vary at will, since n o use is m a d e of p e r t u r b a t i o n theory; in particular the system may be supercritical, critical or subcritical a n d it may make transitions a m o n g these states. U s i n g the one-velocity, p o i n t reactor model with delayed neutrons, the b r a n c h i n g processes occurring in the system have been analyzed in the realm of the b a c k w a r d K o l m o g o r o v equations [2] (BKE) for the p r o b a b i l i t y generating f u n c t i o n (pgf) of the c o u n t i n g distribution. In the first part of this work we c o m p u t e d the m e a n value a n d the variance; here we complete the analysis giving expressions for the covariance. N u m e r i c a l results o b t a i n e d by m e a n s of a F o r t r a n IV p r o g r a m [3] are reported for illustration a n d to help in the conclusions.
2. The backward Kolmogorov equations ** As in the first part of this work, we consider a m u l t i p l y i n g system whose kinetic parameters, n a m e l y the reactivity 0 a n d the n e u t r o n life-time l, may be subject to stepwise variations at given instants of time. The system also c o n t a i n s n e u t r o n detectors, operating one at a time in s u b s e q u e n t time intervals. Possible waiting times, considered as c o u n t i n g intervals with zero detector efficiency, are also allowed. W o r k i n g with the reversed time, defined in ref. 1 a n d illustrated there by fig. I - l , the intervals are ~, - (t~_ ], t,), (i = 1, 2,..., M; t M = ~ ) . We recall that these intervals are chosen so that in each of them the c o u n t i n g efficiency, the reactor parameters a n d the external source intensity which will be considered in section 4, are all constant. * Universith degli Studi, Facoltfi di Ingegneria, Bologna, Italy. ** In the following,the symbols indicate the same quantities as defined in the first part of this work. The equations and the figures appearing in the earlier work will be referred to with their numbers preceded by I. 0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
F. Carloni, M. Marseguerra / The backward Kolmogorov equations 11
506
We consider the branching processes originated by the injection of a single initial neutron into the system, taking into account the neutrons, the six groups of delayed neutrons precursors and the neutron counts (detectrons). The state of the system at any time is characterized by the numbers of all these particles at that time. However since the detectrons are the unique observable quantities we are only interested in their distribution, whatever the numbers of neutrons and precursors m a y be. Let F.,(x; t) and F~.(x; t) be the pgf's of the numbers of the detectrons at the initial (reversed) time t 0 = 0, when a single initial neutron or a precursor of t h e j t h group respectively is injected in the system at time t ~ 6: with t >/0; then the B K E are (I-1) ~ - F , , ( x ; t) 0 ~Fp.(x;
Y( F,,, F p i ) - 1 +
t)=X/(F.,-
Fp/,)
j:
(1 - F,,) + , , ( x , -
1)
1.2 ..... 6,
(I)
where A ~, k~, (~, indicate respectively the mean generation time, the effective multiplication factor and the counting efficiency (count/fission) of the system during the i th time interval. These equations are to be integrated separately for the different time intervals, starting from the first one with the b o u n d a r y conditions (I-2) F,,,(1; t ) = F,,,(x; 0 ) = Fp/i(1; t ) = Fv:,(x; O)= 1 and assuming continuity of all functions at times t~, tz ..... tM ~.
3. Covariance of the counts originated by a single initial neutron (kernel covariance) We are interested in the expression for the covariance between the counts in the rnth and the lth intervals, with m < l, due to the branching processes originated by a single initial neutron injected in the system at the (reversed) time t. For simplicity's sake, in the following we determine the covariance between the counts in the first and the /th intervals; that relating to the ruth and the lth intervals can be readily obtained by suitably treating the subscripts. The kernel mean value of the joint distribution of the counts in the first and in the lth intervals, when the initial neutron is injected at time t ~ 8~ is defined as:
q ~ ! : ' ( t ) = t Ox,Ox:
i>~l; X=
/ = 1 , 2 ..... M .
I
The corresponding kernel m o m e n t ~:(.~::I ) (t). relating to the injection of one delayed neutron precursor of t h e j t h group is analogously defined. Differentiating system (1) with respect to x I and x: and then setting x = 1 yields *" 1 X-'R .t,(n.I)
A7t "j - ' ' ' "
=
~:ll,")(t ) =
+ g/,(t)
~.j (qll "l) ,L (1"1, ) --
'~3h
:
where /:0
g/,(t) = ~ i
PoX
(1):,(I) ~_
tc~ { ~(1).1 (1)
:.(l)A (I)
~ Ul,'t/, T E h"/tttlt %1i + qti ~t//li J
and a, is the Rossi-a of the reactor during 8,. * T h e dots d e n o t e differentiation with respect to time.
i = l , l + 1 ..... M
F. Carloni, M. Marseguerra / The backward Kolmogorov equations I1
507
These equations are to be integrated in the different intervals, starting from t h e / t h one with zero initial values of the functions at t~ i and assuming continuity at times t/, t~+ l ..... tM I" Using the Laplace transform technique, after a lot of tedious manipulations, the following recursive expressions are obtained: ~l
q~l,,)(t) _-- .~t~ y '"
AplAq/Arl
pqr
,,
OOrl
bqrl'lrql(I) ( tl 1)Cpq,/(t-t/
At,, ~rpl (ti l) e
,-)=
l)
AqiAribqri,lrql (ti_,)%z (ti_ I
",'*('-
qr Cl ~pqrXi h/oap I A p l A q l A r l +)//l'(t)=7]Tvt~
,/,(I,l)/, viii ~ ' ) = ~p
(1) (tz 1)Ceq~,(t-tz u,lrr,},
1)
(2)
~] [ - ' A qi--ri--q~i"q,~ A h -(b~'t, l) )~jS~ipiApi ~rpt(l'l)c" ~'i I)~ e ~'~'('-" '~ + 3,.., qr
X rr~rll)(ti l)Bpqri( t-- ti_l)] , I where q.g(1,1 , '(t,
l)
= ~,(1,1) ( t, ttl,i-I"
e '%'~ --
1)+l--Y" Ai j
g ~k.]-
°°pi
,(t, ,),
~'2X + bqr,= ~
e -~aqlr
~o(2°~l - ~,1l-- 02rl)' e -weir -- e -('~qi+w',)r
OOql -- OOpl
(aOqi -[- 02ri -- ¢.Op l
Notice that q~))(t)= (OFnJOxl)x: n represents the kernel mean value of the counts in the lth interval, when the initial neutron is injected at time t E 8,. The expressions for q}])(t) and q}))(t), (i > 1) are given by the rhs of qll)(t) and q}l)(t) of eqs. (I-5) provided the index 1 in the rhs of qll)(t) be changed to l and ~r¢l'(t,) be defined in terms of q}))(ts) and +)tl,)(t~) in place of q}l'(t~) and q,),l)(t,) respectively. The same holds true for the expression of +)//(t) representing the kernel mean value of the counts in the lth interval when the initial j t h precursor is injected at time t ~ 8,.The Ap~ and - % , values ( p = 0,1 ..... 6) are the residues and the poles of the reactor transfer function during 8,; if in this interval the reactor is critical, one pole vanishes and some terms in eqs. (2) assume indefinite form which are to be suitably c o m p u t e d as detailed in ref. (3).
4. Covariance of the counts in a system with an external neutron source
Let us now assume that the multiplying system is fed with the neutrons emitted by an external Poisson-type source with stepwise time varying mean emission rate and let S i be the source intensity (n/s), possibly equal to zero, in the ith interval. The/:;,i(x; t) represent the Green functions for the pgf G(x) of the n u m b e r of counts collected in the various time intervals at time t o = 0, which reads:
G(x)=exp
~ S~ t, [ F~(x; r ) _ l] d r . i=1
,-t
The coveriance between counts in 8~ and 8.,, (l > 1) is then:
coy,, ,,=[
,
=
F. Carloni, M. Marseguerra/ The backwardKolrnogorovequations I1
508
Substitution of eqs. (2) yields:
El
AplAqlArl lxlv pqr 02rl
Coy(1 l) = S/T-=_ Y'~
~ ~qrl ~r"'l', ql ~,'1
I ]~ ~ p:q r l
71- EM St [V'p ~ ~pimPiTr(IA p, )(/'i--l)( 1 --e-wp,B,) i=l+1
Ap~AqiAri + Z
p q r O'~qi
+
OOri
_
cp ,)<,"(,,
03P i
] ,
where
%,qr:= f",Ceqr:(r)d*
and
_f',
39pqr,6~
B p q , ( r ) dr.
II i
To reduce the burden of formulae, the evaluation of these simple integrals is left to the reader. Again, if during an interval the reactor is critical, some of these expressions assume indefinite forms and proper limits are to be computed, as detailed in ref. (3).
5. Results Following the line presented in the first part of this work, we shall now illustrate with numerical examples the expression for the covariance obtained here. Indeed it is evident that explicit analytical expressions for the general case would be too intricate, while the recursive formulae given here do not lend themselves to an immediate interpretation. The examples refer to the experiments described in the first part of this work to which we refer for details. Let us first consider the rod drop experiment performed in a system which has been mantained critical for 300 s. The system has a prompt neutron life-time l = 1.6 × 10-Ss and contains detectors with efficiencies E = 10-3 count/fission. For the analysis of the data measured with the integral count technique [4] we computed the mean value and variance of the counts taken in an interval of one second preceding the drop and in the (infinite) time interval following the drop and the covariance between the counts collected in these two intervals. The mean value and the variance in the second preceding the drop are = 689;
02 = 5.02 X 10 4.
This variance appears to be very large, in agreement with the fact that in a critical system the variance diverges exponentially with time. The corresponding values relating to the total counts following the drop are ~
= = 5828;
o~ = 2.05 X 10 6.
Since both the above variances are very large in comparison with the mean values, assumption of independent count measurements would lead to largely scattered values with an unreliable ratio. Actually the counts are highly correlated as indicated by the computed covariance Cov(m, m~o ) = 2.58 x 105, corresponding to a correlation coefficient of 0.81. This means that the count before the drop and the total count after the drop, both deviate from their respective mean values in the same direction and by almost the same relative amounts. Indeed assuming for these two random variables a bivariate normal distribution with parameters as given above, the relative standard deviation of their ratio is -- 20%, to be compared with the value of = 50% obtained assuming no correlation. Assuming Poissonian counts the same ratio has a small relative standard deviation of 4%. The conclusion is that it is worthwhile to take properly into
F Carloni, M. Marseguerra / The backward Kolmogorov equations 1I
509
account both the second order moments, i.e. variance and covariance, thus justifying the burden of their computation. Analogous results have been obtained when investigating the detailed rod drop and the sinusoidal pile
10s
t0.
+ E
o> 1 03
]0
t
I
I0
20
J
10~Io
[
10
I
j
20
30
Fig. 1. Detailed rod drop numerical experiment, as illustrated in ref. 1. Covariance between pairs of channels versus channel separation. Fig. 2. Pile oscillator numerical experiment, as illustrated in ref. 1. Covariancebetween pairs of channels versus channel separation.
oscillator experiments. Here we limit ourselves to report in figs. 1 and 2 the time behaviours of some computed covariances. Concerning the computation problems, the expressions for the moments obtained in this work are very involved so that they have been given in a recursive form well suited for programming. A F O R T R A N IV program has been written for a DEC PDP 11/23 microcomputer and the listing is available from the authors. Since the formulae contain up to quadruple sums and are recursive in nature, it was almost mandatory to use double precision (8-byte words) which however lengthens the running time. Concerning this time, most of it is spent calculating the covariances whose number quadratically increases with the number of intervals. In addition, the computing time linearly increases with the number of intervals following (in reversed time) the single interval in the case of the mean value and the variance and following the second interval in the case of the covariance. The conclusion is that the running time increases very fast with the number of intervals. In our numerical examples the times required for working out the integral count, the detailed rod drop and the sinusiodal pile oscillator experiments were about one minute, one hour and one night respectively,
6. Conclusions
This paper completes work dealing with the moments up to the second order of the count distribution taken in a multiplying system subject to a sequence of arbitrary stepwise variations of the reactor, source and detector parameters. In the usual one-velocity, point-reactor model with delayed neutrons, and following the Kolmogorov and D'Mitriev theory of the branching stochastic processes, we obtained rigorous expressions for the mean value and variance of the counts taken in a single interval and for the covariance of the counts taken in a pair of non-intersecting intervals.
F. Carloni, M. Marseguerra / The backward Kolmogorov equations 11
510
In the first part of this work we reported numerical examples relating to the rod drop and to the pile oscillator experiments. It turned out that the variances were much greater than the corresponding mean values, thus supporting the conclusion that parameters obtained from weighted minimization procedures based upon Poissonian counts could be largely in error. In the present paper~ working out the same numerical experiments, we obtained also large covariances characterized by correlation coefficients near to unity. This result confirms that proper variance-covariance matrices should be used for the correct estimation of the parameters and their errors. In other words the assumption of independent Poissonian counts may lead to the estimation of biased parameters affected by unrealistic statistical errors. The computation problems also deserve some comments. The time required for computing the moments obviously increases with the number of intervals, but it is important to stress that this increase is very fast. In addition the moments are computed by adding up a great many of terms in recursive form so that the truncation errors would propagate unacceptably unless double precision is adopted. The upshot is that experiments, such as the rod drop, characterized by few rapid variations of the parameters and therefore suitable to be modeled by a few time intervals, are readily worked out. On the contrary, experiments with slowly and continuously varying parameters, such as the sinusoidal pile oscillator, which can be closely followed only by discretizing the time in a large number of intervals, require large amounts of computer time. A practical limit for our PDP 11/23 seems to range from about twenty to thirty intervals.
References [l] [2] [3] [4]
F. Carloni and M. Marseguerra, Nucl. Instr. and Meth. 190 (1981) 583-591. A.N. Kolmogorov and N.A. D'Mitriev, Dokl. Akad. Nauk SSSR 56, (1947) 7. F. Carloni and M. Marseguerra, CNEN R T / F I (81) 26. G.R. Keepin, Physics of nuclear kinetics (Addison Wesley, Reading, Mass., (1965) p. 200.
505
THE STEPWISE TIME DEPENDENT BACKWARD KOLMOGOROV EQUATIONS WITH D E L A Y E D N E U T R O N S IN C O U N T I N G S T A T I S T I C S II. Covariance F. C A R L O N I * a n d M. M A R S E G U E R R A lstituto di lngegneria Nucleare - C E S N E F , Politecnico di Milano, Italy
Received 5 July 1982 This paper completes work dealing with the neutron counting statistics in a zero power reactor subject to an arbitrary sequence of stepwise variations of all the parameters. Using the one-velocity, point reactor model with delayed neutrons and following the backwards Kolmogorov formalism, expressions for the mean value, variance and covariance of the count distribution have been obtained. Knowledge of these moments allows the estimation of kinetic parameters of a reactor together with their statistical errors under any time variable condition concerning criticality, source intensity and detector efficiency. Numerical examples relating to a rod drop and to a sinusoidal pile oscillator experiments are reported for illustration.
1. Introduction This paper is the second of two [1] which aim at o b t a i n i n g same experimentally useful statistics of the n e u t r o n c o u n t d i s t r i b u t i o n in an arbitrarily time varying zero power reactor. The variation is modeled by discretizing the time in a sequence of intervals in which all the system parameters assume c o n s t a n t values, possibly different from interval to interval. The system parameters are those p e r t a i n i n g to the reactor itself, n a m e l y the reactivity a n d the p r o m p t n e u t r o n life-time, and, in addition, the external n e u t r o n source strength a n d the efficiencies of the n e u t r o n detectors. All these parameters m a y vary at will, since n o use is m a d e of p e r t u r b a t i o n theory; in particular the system may be supercritical, critical or subcritical a n d it may make transitions a m o n g these states. U s i n g the one-velocity, p o i n t reactor model with delayed neutrons, the b r a n c h i n g processes occurring in the system have been analyzed in the realm of the b a c k w a r d K o l m o g o r o v equations [2] (BKE) for the p r o b a b i l i t y generating f u n c t i o n (pgf) of the c o u n t i n g distribution. In the first part of this work we c o m p u t e d the m e a n value a n d the variance; here we complete the analysis giving expressions for the covariance. N u m e r i c a l results o b t a i n e d by m e a n s of a F o r t r a n IV p r o g r a m [3] are reported for illustration a n d to help in the conclusions.
2. The backward Kolmogorov equations ** As in the first part of this work, we consider a m u l t i p l y i n g system whose kinetic parameters, n a m e l y the reactivity 0 a n d the n e u t r o n life-time l, may be subject to stepwise variations at given instants of time. The system also c o n t a i n s n e u t r o n detectors, operating one at a time in s u b s e q u e n t time intervals. Possible waiting times, considered as c o u n t i n g intervals with zero detector efficiency, are also allowed. W o r k i n g with the reversed time, defined in ref. 1 a n d illustrated there by fig. I - l , the intervals are ~, - (t~_ ], t,), (i = 1, 2,..., M; t M = ~ ) . We recall that these intervals are chosen so that in each of them the c o u n t i n g efficiency, the reactor parameters a n d the external source intensity which will be considered in section 4, are all constant. * Universith degli Studi, Facoltfi di Ingegneria, Bologna, Italy. ** In the following,the symbols indicate the same quantities as defined in the first part of this work. The equations and the figures appearing in the earlier work will be referred to with their numbers preceded by I. 0 1 6 7 - 5 0 8 7 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
F. Carloni, M. Marseguerra / The backward Kolmogorov equations 11
506
We consider the branching processes originated by the injection of a single initial neutron into the system, taking into account the neutrons, the six groups of delayed neutrons precursors and the neutron counts (detectrons). The state of the system at any time is characterized by the numbers of all these particles at that time. However since the detectrons are the unique observable quantities we are only interested in their distribution, whatever the numbers of neutrons and precursors m a y be. Let F.,(x; t) and F~.(x; t) be the pgf's of the numbers of the detectrons at the initial (reversed) time t 0 = 0, when a single initial neutron or a precursor of t h e j t h group respectively is injected in the system at time t ~ 6: with t >/0; then the B K E are (I-1) ~ - F , , ( x ; t) 0 ~Fp.(x;
Y( F,,, F p i ) - 1 +
t)=X/(F.,-
Fp/,)
j:
(1 - F,,) + , , ( x , -
1)
1.2 ..... 6,
(I)
where A ~, k~, (~, indicate respectively the mean generation time, the effective multiplication factor and the counting efficiency (count/fission) of the system during the i th time interval. These equations are to be integrated separately for the different time intervals, starting from the first one with the b o u n d a r y conditions (I-2) F,,,(1; t ) = F,,,(x; 0 ) = Fp/i(1; t ) = Fv:,(x; O)= 1 and assuming continuity of all functions at times t~, tz ..... tM ~.
3. Covariance of the counts originated by a single initial neutron (kernel covariance) We are interested in the expression for the covariance between the counts in the rnth and the lth intervals, with m < l, due to the branching processes originated by a single initial neutron injected in the system at the (reversed) time t. For simplicity's sake, in the following we determine the covariance between the counts in the first and the /th intervals; that relating to the ruth and the lth intervals can be readily obtained by suitably treating the subscripts. The kernel mean value of the joint distribution of the counts in the first and in the lth intervals, when the initial neutron is injected at time t ~ 8~ is defined as:
q ~ ! : ' ( t ) = t Ox,Ox:
i>~l; X=
/ = 1 , 2 ..... M .
I
The corresponding kernel m o m e n t ~:(.~::I ) (t). relating to the injection of one delayed neutron precursor of t h e j t h group is analogously defined. Differentiating system (1) with respect to x I and x: and then setting x = 1 yields *" 1 X-'R .t,(n.I)
A7t "j - ' ' ' "
=
~:ll,")(t ) =
+ g/,(t)
~.j (qll "l) ,L (1"1, ) --
'~3h
:
where /:0
g/,(t) = ~ i
PoX
(1):,(I) ~_
tc~ { ~(1).1 (1)
:.(l)A (I)
~ Ul,'t/, T E h"/tttlt %1i + qti ~t//li J
and a, is the Rossi-a of the reactor during 8,. * T h e dots d e n o t e differentiation with respect to time.
i = l , l + 1 ..... M
F. Carloni, M. Marseguerra / The backward Kolmogorov equations I1
507
These equations are to be integrated in the different intervals, starting from t h e / t h one with zero initial values of the functions at t~ i and assuming continuity at times t/, t~+ l ..... tM I" Using the Laplace transform technique, after a lot of tedious manipulations, the following recursive expressions are obtained: ~l
q~l,,)(t) _-- .~t~ y '"
AplAq/Arl
pqr
,,
OOrl
bqrl'lrql(I) ( tl 1)Cpq,/(t-t/
At,, ~rpl (ti l) e
,-)=
l)
AqiAribqri,lrql (ti_,)%z (ti_ I
",'*('-
qr Cl ~pqrXi h/oap I A p l A q l A r l +)//l'(t)=7]Tvt~
,/,(I,l)/, viii ~ ' ) = ~p
(1) (tz 1)Ceq~,(t-tz u,lrr,},
1)
(2)
~] [ - ' A qi--ri--q~i"q,~ A h -(b~'t, l) )~jS~ipiApi ~rpt(l'l)c" ~'i I)~ e ~'~'('-" '~ + 3,.., qr
X rr~rll)(ti l)Bpqri( t-- ti_l)] , I where q.g(1,1 , '(t,
l)
= ~,(1,1) ( t, ttl,i-I"
e '%'~ --
1)+l--Y" Ai j
g ~k.]-
°°pi
,(t, ,),
~'2X + bqr,= ~
e -~aqlr
~o(2°~l - ~,1l-- 02rl)' e -weir -- e -('~qi+w',)r
OOql -- OOpl
(aOqi -[- 02ri -- ¢.Op l
Notice that q~))(t)= (OFnJOxl)x: n represents the kernel mean value of the counts in the lth interval, when the initial neutron is injected at time t E 8,. The expressions for q}])(t) and q}))(t), (i > 1) are given by the rhs of qll)(t) and q}l)(t) of eqs. (I-5) provided the index 1 in the rhs of qll)(t) be changed to l and ~r¢l'(t,) be defined in terms of q}))(ts) and +)tl,)(t~) in place of q}l'(t~) and q,),l)(t,) respectively. The same holds true for the expression of +)//(t) representing the kernel mean value of the counts in the lth interval when the initial j t h precursor is injected at time t ~ 8,.The Ap~ and - % , values ( p = 0,1 ..... 6) are the residues and the poles of the reactor transfer function during 8,; if in this interval the reactor is critical, one pole vanishes and some terms in eqs. (2) assume indefinite form which are to be suitably c o m p u t e d as detailed in ref. (3).
4. Covariance of the counts in a system with an external neutron source
Let us now assume that the multiplying system is fed with the neutrons emitted by an external Poisson-type source with stepwise time varying mean emission rate and let S i be the source intensity (n/s), possibly equal to zero, in the ith interval. The/:;,i(x; t) represent the Green functions for the pgf G(x) of the n u m b e r of counts collected in the various time intervals at time t o = 0, which reads:
G(x)=exp
~ S~ t, [ F~(x; r ) _ l] d r . i=1
,-t
The coveriance between counts in 8~ and 8.,, (l > 1) is then:
coy,, ,,=[
,
=
F. Carloni, M. Marseguerra/ The backwardKolrnogorovequations I1
508
Substitution of eqs. (2) yields:
El
AplAqlArl lxlv pqr 02rl
Coy(1 l) = S/T-=_ Y'~
~ ~qrl ~r"'l', ql ~,'1
I ]~ ~ p:q r l
71- EM St [V'p ~ ~pimPiTr(IA p, )(/'i--l)( 1 --e-wp,B,) i=l+1
Ap~AqiAri + Z
p q r O'~qi
+
OOri
_
cp ,)<,"(,,
03P i
] ,
where
%,qr:= f",Ceqr:(r)d*
and
_f',
39pqr,6~
B p q , ( r ) dr.
II i
To reduce the burden of formulae, the evaluation of these simple integrals is left to the reader. Again, if during an interval the reactor is critical, some of these expressions assume indefinite forms and proper limits are to be computed, as detailed in ref. (3).
5. Results Following the line presented in the first part of this work, we shall now illustrate with numerical examples the expression for the covariance obtained here. Indeed it is evident that explicit analytical expressions for the general case would be too intricate, while the recursive formulae given here do not lend themselves to an immediate interpretation. The examples refer to the experiments described in the first part of this work to which we refer for details. Let us first consider the rod drop experiment performed in a system which has been mantained critical for 300 s. The system has a prompt neutron life-time l = 1.6 × 10-Ss and contains detectors with efficiencies E = 10-3 count/fission. For the analysis of the data measured with the integral count technique [4] we computed the mean value and variance of the counts taken in an interval of one second preceding the drop and in the (infinite) time interval following the drop and the covariance between the counts collected in these two intervals. The mean value and the variance in the second preceding the drop are = 689;
02 = 5.02 X 10 4.
This variance appears to be very large, in agreement with the fact that in a critical system the variance diverges exponentially with time. The corresponding values relating to the total counts following the drop are ~
= = 5828;
o~ = 2.05 X 10 6.
Since both the above variances are very large in comparison with the mean values, assumption of independent count measurements would lead to largely scattered values with an unreliable ratio. Actually the counts are highly correlated as indicated by the computed covariance Cov(m, m~o ) = 2.58 x 105, corresponding to a correlation coefficient of 0.81. This means that the count before the drop and the total count after the drop, both deviate from their respective mean values in the same direction and by almost the same relative amounts. Indeed assuming for these two random variables a bivariate normal distribution with parameters as given above, the relative standard deviation of their ratio is -- 20%, to be compared with the value of = 50% obtained assuming no correlation. Assuming Poissonian counts the same ratio has a small relative standard deviation of 4%. The conclusion is that it is worthwhile to take properly into
F Carloni, M. Marseguerra / The backward Kolmogorov equations 1I
509
account both the second order moments, i.e. variance and covariance, thus justifying the burden of their computation. Analogous results have been obtained when investigating the detailed rod drop and the sinusoidal pile
10s
t0.
+ E
o> 1 03
]0
t
I
I0
20
J
10~Io
[
10
I
j
20
30
Fig. 1. Detailed rod drop numerical experiment, as illustrated in ref. 1. Covariance between pairs of channels versus channel separation. Fig. 2. Pile oscillator numerical experiment, as illustrated in ref. 1. Covariancebetween pairs of channels versus channel separation.
oscillator experiments. Here we limit ourselves to report in figs. 1 and 2 the time behaviours of some computed covariances. Concerning the computation problems, the expressions for the moments obtained in this work are very involved so that they have been given in a recursive form well suited for programming. A F O R T R A N IV program has been written for a DEC PDP 11/23 microcomputer and the listing is available from the authors. Since the formulae contain up to quadruple sums and are recursive in nature, it was almost mandatory to use double precision (8-byte words) which however lengthens the running time. Concerning this time, most of it is spent calculating the covariances whose number quadratically increases with the number of intervals. In addition, the computing time linearly increases with the number of intervals following (in reversed time) the single interval in the case of the mean value and the variance and following the second interval in the case of the covariance. The conclusion is that the running time increases very fast with the number of intervals. In our numerical examples the times required for working out the integral count, the detailed rod drop and the sinusiodal pile oscillator experiments were about one minute, one hour and one night respectively,
6. Conclusions
This paper completes work dealing with the moments up to the second order of the count distribution taken in a multiplying system subject to a sequence of arbitrary stepwise variations of the reactor, source and detector parameters. In the usual one-velocity, point-reactor model with delayed neutrons, and following the Kolmogorov and D'Mitriev theory of the branching stochastic processes, we obtained rigorous expressions for the mean value and variance of the counts taken in a single interval and for the covariance of the counts taken in a pair of non-intersecting intervals.
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In the first part of this work we reported numerical examples relating to the rod drop and to the pile oscillator experiments. It turned out that the variances were much greater than the corresponding mean values, thus supporting the conclusion that parameters obtained from weighted minimization procedures based upon Poissonian counts could be largely in error. In the present paper~ working out the same numerical experiments, we obtained also large covariances characterized by correlation coefficients near to unity. This result confirms that proper variance-covariance matrices should be used for the correct estimation of the parameters and their errors. In other words the assumption of independent Poissonian counts may lead to the estimation of biased parameters affected by unrealistic statistical errors. The computation problems also deserve some comments. The time required for computing the moments obviously increases with the number of intervals, but it is important to stress that this increase is very fast. In addition the moments are computed by adding up a great many of terms in recursive form so that the truncation errors would propagate unacceptably unless double precision is adopted. The upshot is that experiments, such as the rod drop, characterized by few rapid variations of the parameters and therefore suitable to be modeled by a few time intervals, are readily worked out. On the contrary, experiments with slowly and continuously varying parameters, such as the sinusoidal pile oscillator, which can be closely followed only by discretizing the time in a large number of intervals, require large amounts of computer time. A practical limit for our PDP 11/23 seems to range from about twenty to thirty intervals.
References [l] [2] [3] [4]
F. Carloni and M. Marseguerra, Nucl. Instr. and Meth. 190 (1981) 583-591. A.N. Kolmogorov and N.A. D'Mitriev, Dokl. Akad. Nauk SSSR 56, (1947) 7. F. Carloni and M. Marseguerra, CNEN R T / F I (81) 26. G.R. Keepin, Physics of nuclear kinetics (Addison Wesley, Reading, Mass., (1965) p. 200.