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Annals of Nuclear Energy 112 (2018) 307–321

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Annals of Nuclear Energy

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Predicting correlation coefficients for Monte Carlo eigenvaluesimulations with multitype branching process

https://doi.org/10.1016/j.anucene.2017.10.0140306-4549/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (J. Miao), [email protected] (B. Forget),

[email protected] (K. Smith).

Jilang Miao ⇑, Benoit Forget, Kord SmithMassachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, United States

a r t i c l e i n f o

Article history:Received 20 July 2017Received in revised form 2 October 2017Accepted 6 October 2017

Keywords:Monte CarloAuto-correlation predictionVariance estimateMultitype Branching Processes

a b s t r a c t

This paper provides a prediction method of the generation-to-generation correlations as observed whensolving large scale eigenvalue problems such as full core nuclear reactor simulations. Knowing the cor-relations enables correction of the variance underestimation that occurs when assuming that the activegenerations are independent. The Monte Carlo power iteration is cast in the Multitype Branching Process(MBP) framework by discretizing the neutron phase space which allows calculation of spatial and tem-poral moments. These moments can then provide auto-correlation coefficients between the generationsof MBP and are shown to accurately predict the auto-correlation coefficients of the original Monte Carlosimulation. This prediction capability was demonstrated on the full core 2D PWR BEAVRS benchmark andcompared successfully with variance estimates from independent simulations.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Monte Carlo methods are most often considered as a referencefor neutron transport simulations since very limited approxima-tions are made about nuclear data and system geometry. Uncer-tainty of any tallied quantity is commonly represented by takingthe sample variance over the active generations, which is basedon the assumption that the neutron generations are independent.Correlation effects between neutrons in multiplying systems, par-ticularly when performing power iteration to evaluate eigenvalueshave been observed and quantified in previous work (Brissendenand Garlick, 1986; Dumonteil et al., 2014; Herman et al., 2014;Miao et al., 2016). Neglecting this correlation effect results in anunderestimate of uncertainty reported by Monte Carlo calcula-tions, the magnitude of which depends on the dominance ratio ofthe problem and the size of the phase space being tallied (Uekiet al., 2003). Many studies have quantified this underestimationusing post-processing techniques of the tallied quantities or byde-correlating batches using many generations per batch, both ofwhich can become costly in either memory or runtime (Miaoet al., 2016; Wilson et al., 2012).

Previous work has also proposed methods to predict theunderestimation ratio between the correlated and uncorrelated

estimates. The actively investigated methods are classified intotwo categories. The first performs data fitting on simulation out-puts to capture the correlation. The second directly computecovariance betwen the Monte Carlo generations based on corre-sponding approximations of the Monte Carlo simulation (Ueki,2010). Demaret et al. (1999) fitted AR (auto-regressive) and MA(Moving Average) models to the results of Monte Carlo eigenvaluecalculations and used the AR and MA models to give variance esti-mator of keff . Yamamoto et al. (2014) expanded the fission sourcedistribution with diffusion equation modes, performed numericalsimulation of the AR process of the expansion coefficients and usedthe correlation of the AR process to predict the Monte Carlo eigen-value simulation. Approximating the original neutron transportproblem with a diffusion problem lead to a non-negligible lost inaccuracy. Sutton (2015) applied the discretized phase space(DPS) approach to predict the underestimation ratio but themethod cannot predict the ratio when one neutron generates off-spring in different phase space regions or generates a randomnumber of offspring. Ueki et al. (2016) developed variance estima-tor with orthonormally weighted standardized time series(OWSTS). The estimator is based on the convergence of step-wiseinterpolation of standardized tallies (SIST) to Brownian bridge. SISTweighted by a trigonometric family of weighting functions gives anew statistical estimator for the variance. Asymptotic behavior ofexpectation of the new static leads to a variance estimator that isnot affected by the correlation effect thus converging at a 1/N rate.Numerical results showed that the variance can be accurately

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308 J. Miao et al. / Annals of Nuclear Energy 112 (2018) 307–321

estimated after approximately 5000 � 10000 active generationsfor problemswith non-negligible autocorrelation up to 100 generation lags. Aconvergence diagnosis is also necessary to determine when theasymptotic behavior is reached. However, the convergence diagno-sis cannot be implemented on-the-fly.

This paper presents a method to predict the correlation effectbased on the model of multitype branching processes (MBP)(Mode, 1971).

This method uses tallies from Monte Carlo simulation toconstruct a Multitype Branching Processes model, evaluates thecorrelation effect of the MBP model analytically and can then pre-dict the underestimation ratio for the original Monte Carlo simula-tion. The kernel to construct the MBP model is the transferprobabilities between the multi-types, which are the discretizedphase space regions of the original problem. If phase space regionsare discretized so fine that every region can be viewed as a flatsource region, the transfer probabilities can be tallied from a fixedsource calculation of uniformly sampled neutron sources. Morepractically, phase space regions need to be slightly finer than thetally regions but not as fine as when a flat source approximationwould yield a satisfactory spatial convergence of the flux. Underthis circumstance, transfer probabilities can be tallied from a fewactive generations right after the source becomes stationary.

Section 2 will present the theoretical derivation of auto-correlation of active generation tallies from multitype branchingprocesses. In Section 2.1, a resursive relation betweenmoment gen-erating functions of neutron sources will be derived and used toderive various moments of neutron sources. Section 2.2 approxi-mates the Monte Carlo power iteration by the multitype branchingprocess and uses covariance expansion to relate the derivedmoments of the multitype branching process to the auto-correlation of the power iteration. Section 3 discusses how to con-struct the multitype branching process from unconverged tallies.Numerical results on applying the methods developed in this paperto the BEAVRS benchmark will be found in Section 4. This sectionwill show that auto-correlation and thus variance can be accuratelypredicted from the constructedmultitype branching processmodel.

2. Theory

Suppose generation n yields tally XlðnÞ for tally region l, the sim-

ulation typically reports the average �XlðNÞ ¼PN

n¼1XlðNÞ=N andr2

Xl=N as an approximation of r2

�XlðNÞ. Due to the correlation between

generations in the power iteration process, rXl=ffiffiffiffiN

punderestimates

r �XlðNÞ. The Cov½XlðiÞ;XlðjÞ�, where i; j are the active generationindexes, is required to correctly evaluate the uncertainty.

These correlations across generations result from the fission siteupdate process where the source of generation nþ 1 come fromthe fission sites created during generation n. The correlation ofany tallied quantity can be calculated from the correlation of thefission source distribution Shim, 2015. The auto-correlation coeffi-cient of Xl between generation n and nþ k is defined as

qn;k ¼Cov½XlðnÞ;Xlðnþ kÞ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar½XlðnÞ�Var½Xlðnþ kÞ�

p ð2:1Þ

The theory of MBP model is developed to predict the auto-correlation coefficients (ACC) of the fission source distribution.

2.1. Theory of multitype branching processes

A branching process describes a population of individualswhere each individual produces offsprings independently with

identical distributions. Multitype branching process extends themodel to a population of finite types of individuals, such as spatialposition, energy, angle. An MBP model can approximate the MonteCarlo power iteration if the neutron phase space is discretizedwhere each cell is treated as a unique type.

This subsection first defines the moment generating function(MGF) related to the MBP model. Then the MGF is used to extractthe serial-spatial moments which can then be related to the ACCs.

2.1.1. Moment generating functionsThe model of multitype branching process discretizes the neu-

tron phase space over all independent variables into m discreteregions and denotes the system state at generation n with a vector~ZðnÞ (Eq. (2.2)). The lth component of the vector corresponds to thenumber of neutrons belonging to the discrete phase space l at gen-eration n. A neutron in region l is defined to be of type l.

~ZðnÞ ¼ ðZ1ðnÞ; . . . ; ZlðnÞ; . . . ; ZmðnÞÞ ð2:2ÞThe total number of neutrons of all phase space regions is the

sum of all components of ~ZðnÞ and is denoted as ZðnÞ (Eq. (2.3))

ZðnÞ ¼Xmi¼1

ZiðnÞ: ð2:3Þ

The state vector at generation n is related to generation n� 1through

~ZðnÞ ¼Xmi¼1

XZiðn�1Þ

j¼1

~Yij; ð2:4Þ

where Yij

!is the state vector generated by the jth neutron of type i at

generation n� 1.

The moment generating function of ~ZðnÞ is defined as

Fnð~r0;~sÞ � EYmi¼1

sZiðnÞi

��~Zð0Þ ¼~r0

" #

¼X~r

P ~ZðnÞ ¼~r��~Zð0Þ ¼~r0

� �Ymi¼1

srii ; ð2:5Þ

where ~r0 denotes the initial configuration of neutrons in the dis-cretized phase space and~s is the argument of the moment generat-ing function. If ~r0 is a point source of type i ð~r0 �~ei; ðr0Þj ¼ dijÞ, we

denote Fnð~r0;~sÞ as Fnði;~sÞ. The random vector ~ZðnÞ initiated by ~Zð0Þis the sum of the random vectors initiated by the Zð0Þ neutrons rep-resented by the vector ~Zð0Þð¼ ~r0Þ

~ZðnÞ��~Zð0Þ¼~r0

¼Xmi¼1

Xri0j¼1

~ZðnÞ��~Zð0Þ¼~ei

ð2:6Þ

Eq. (2.6) expresses ~ZðnÞ as sum of contributions from all

neutrons in the 0th generation. Since the Zð0Þ random vectors areindependent, the moment generating function of the state vector~ZðnÞ

��~Zð0Þ¼~r0

is written as the product of the moment generating

function of the state vectors of the Zð0Þ components ~ZðnÞ��~Zð0Þ¼~ei

.

Therefore, Fnð~r0;~sÞ and Fnði;~sÞ are related by

Fnð~r0;~sÞ ¼Ymi¼1

Fnði;~sÞri0 : ð2:7Þ

Fnði;~sÞ can be evaluated from F1ði;~sÞ recursively according to itsdefinition in Eq. (2.5),

J. Miao et al. / Annals of Nuclear Energy 112 (2018) 307–321 309

Fnði;~sÞ ¼ EYml¼1

sZlðnÞl

��~Zð0Þ ¼~ei

" #

¼ E EYml¼1

sZlðnÞl

��~Zðn� 1Þ" #��~Zð0Þ ¼~ei

" #: ð2:8Þ

where the second equality in Eq. (2.8) is based on the Markov prop-

erty of process ~ZðnÞ. Substituting the relation between ~ZðnÞ and~Zðn� 1Þ (Eq. (2.4)) into the conditional expectation yields

EYml¼1

sZlðnÞl j~Zðn� 1Þ" #

¼ EYml¼1

Ymk¼1

YZkðn�1Þ

j¼1

sðYk;j

!Þll

��~Zðn� 1Þ" #

¼ EYmk¼1

YZkðn�1Þ

j¼1

Yml¼1

sðYk;j

!Þll

��~Zðn� 1Þ" #

¼Ymk¼1

YZkðn�1Þ

j¼1

EYml¼1

sðYk;j

!Þll

" #

¼Ymk¼1

EYml¼1

sðYk;j

!Þll

" # !Zkðn�1Þ

ð2:9Þ

Because the neutrons at generation n� 1 reproduce indepen-dently, the product over k and j on the second line of Eq. (2.9)can be pulled out of the expectation operator. By the definition

of Yk;j

!,

EYml¼1

sðYk;j

!Þll

" #¼ F1ðk;~sÞ: ð2:10Þ

Combining the conditional expectation and Eq. (2.8) gives

Fnði;~sÞ ¼ EYmk¼1

F1ðk;~sÞð ÞZkðn�1Þ��~Zð0Þ ¼~ei

" #ð2:11Þ

which is the moment generating function of ~Zðn� 1Þ with~f ð1;~sÞ asthe argument, where the vector~f ð1;~sÞ is defined with F1ðk;~sÞ as itskth component:

~f ð1;~sÞ ¼ ðF1ð1;~sÞ; . . . ; F1ðm;~sÞÞ ð2:12ÞTherefore the recursive relation between Fnði;~sÞ and Fn�1ði;~sÞ

reads

Fnði;~sÞ ¼ Fn�1ði;~f ð1;~sÞÞ ð2:13ÞThis equivalence can be demonstrated by induction on the

above equation which gives

Fnði;~sÞ ¼ Fn�1 i;~f ð1;~sÞ� �

¼ Fn�2 i;~f 1;~f ð1;~sÞ� ��

� Fn�2 i;~f ð2;~sÞ� �

¼ � � �¼ F1ði;~f ðn� 1;~sÞÞ

ð2:14Þ

where~f ðn;~sÞ is naturally defined as:

~f ðn;~sÞ ¼~f ðn� 1;~f ð1;~sÞÞ; n P 2 ð2:15Þ

Combining the definitions of ~f ðn;~sÞ (Eq. (2.12), Eq. (2.15)) andthe last recursive evaluation of Fnði;~sÞ (Eq. (2.14)) yields

Fnði;~sÞ ¼ ð~f ðn;~sÞÞi ð2:16Þ

which means Fnði;~sÞ is the ith element of the vector function~f ðn;~sÞ.

2.1.2. Spatial MomentsThe spatial moments of ZlðnÞ defined in Eq. (2.17) can be evalu-

ated by taking derivatives of Fnð~r0;~sÞ.

llðnÞ � E½ZlðnÞ�Cl;jðnÞ � E½ZlðnÞZjðnÞ�Tl;j;kðnÞ � E½ZlðnÞZjðnÞZkðnÞ�

ð2:17Þ

This would involve commuting the derivatives into the recur-

sive definition of~f ðn;~sÞ (Eq. (2.15)) which can be quite lengthy. Asimpler way is to evaluate the expectations at generation nþ 1

conditional on ~ZðnÞ. For simplicity, E½XjY � is denoted as E½X�jY inthe following. The moment generating function at generationnþ 1 can be written as

Fnþ1 ~r0;~sð Þj~ZðnÞ ¼ F1ð~ZðnÞ;~sÞ

¼Ymk¼1

F1ðk;~sÞZkðnÞ:ð2:18Þ

From the moment generating function in Eq. (2.18), liðnþ 1Þ isfirst evaluated as a conditional expectation on ~ZðnÞ and thenexpressed as a function of liðnÞ.

liðnþ 1Þ��~ZðnÞ ¼ @

@si

Ymk¼1

F1ðk;~sÞZkðnÞ��~s¼~1

¼Xml¼1

F1ðl;~sÞZlðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ

Yk–l


¼Xml¼1

ZlðnÞ @

@siF1ðl;~sÞ

��~s¼~1

¼Xml¼1

ZlðnÞMli

ð2:19Þwhere in the second to last line, the identity F1ðl;~sÞj~s¼~1 ¼ 1 is used.

Mli is used to denote @

@siF1ðl;~sÞ

��~s¼~1

, which by definition is the

expected number of neutrons found of type i after one generationgiven a source neutron of type l. It will be referred as the first spatialmoment response and denoted as:

Mli ¼ E½Zið1Þj~r0 ¼~el�: ð2:20ÞEvaluating the expected value of Eq. (2.19) removes the depen-

dence on ~ZðnÞ and yields

liðnþ 1Þ ¼ llðnÞMli: ð2:21Þ

where the Einstein tensor notation is used (the sum is taken over allvalues of the index whenever the same symbol appears as a sub-script and superscript in the same term) and will be used through-out this paper.

A similar procedure will provide the second order factorial

moments, eCl;jðnÞ, which can eventually be related to Cl;jðnÞ (Eq.(2.17)) through Eq. (2.22).

~Cl;jðnÞ ¼ @

@sl

@

@sjFnð~r0;~sÞ ¼ E½ZlðnÞZjðnÞ � dl;jZlðnÞ� ð2:22Þ

~Ci;jðnþ 1Þ is first evaluated conditional on ~ZðnÞ and then

expressed as a function of ~Ci;jðnÞ,


~Cj;iðnþ 1Þ��~ZðnÞ

¼ @

@sj

@

@si

Ymk¼1


¼ @

@sj

Xml¼1

F1ðl;~sÞZlðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ

Yk–l

F1ðk;~sÞZkðnÞ !��

~s¼~1

¼Xml¼1

@

@sjF1ðl;~sÞZlðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ

� �Yk–l

F1ðk;~sÞZkðnÞ

þF1ðl;~sÞZlðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ @

@sj

Yk–l


~s¼~1

¼Xml¼1

ZlðnÞ ðZlðnÞ � 1Þ @

@sjF1ðl;~sÞ @

@siF1ðl;~sÞ

�

þF1ðl;~sÞZlðnÞ�1 @2

@sj@siF1ðl;~sÞ

!

þZlðnÞMli

Xh–l

ZhðnÞF1ðh;~sÞZhðnÞ�1 @

@sjF1ðh;~sÞ

�Y

g–h;g–l

F1ðg;~sÞZg ðnÞ !!��

~s¼~1

¼Xml¼1

ZlðnÞ ðZlðnÞ � 1ÞMljM

li þ ~Cl

j;ið1Þ� �

þZlðnÞMli

Xh–l

ZhðnÞMhj

¼Xml¼1

ZlðnÞ ZlðnÞMljM

li

� �þ ZlðnÞMl

i

Xh–l

ZhðnÞMhj

þZlðnÞð~Clj;ið1Þ �Ml

jMliÞ ¼

Xml¼1

ZlðnÞMli

Xmh¼1

ZhðnÞMhj

þZlðnÞð~Clj;ið1Þ �Ml

jMliÞ

¼Xml¼1

Xmh¼1

ZlðnÞZhðnÞMliM

hj þ ZlðnÞð~Cl

j;ið1Þ �MljM

liÞ:

ð2:23Þ

Taking the expected value of ~Cj;iðnþ 1Þj~ZðnÞ (Eq. (2.23)) removes

the dependence of ~ZðnÞ and yields

~Cj;iðnþ 1Þ ¼ Mhj M

liCh;lðnÞ þ llðnÞ ~Cl

j;ið1Þ �MljM

li

� �ð2:24Þ

Substituting the relation between ~Cj;i and Cj;i (Eq. (2.22)) simpli-fies Eq. (2.24) to

Cj;iðnþ 1Þ ¼ Mhj M

liCh;lðnÞ þ llðnÞVl

j;i ð2:25Þwhere

Vli;j ¼ E½ðZið1Þ � lið1ÞÞðZjð1Þ � ljð1ÞÞj~r0 ¼~el�; ð2:26Þ

with Vli;j being the covariance between the number of new neutrons

of type i and type j after one generation given a neutron born of typel. It will be referred as the second spatial (central) momentresponse.

Following a similar but more lengthy derivation (Appendix A),the third order spatial moments can be given as

Tk;j;iðnþ 1Þ ¼ Tg;h;lðnÞMgkM

hj M

li þ Vl

j;iMakCl;aðnÞ

þ Vlk;iM

aj Cl;aðnÞ þ Vl

j;kMai Cl;aðnÞ þWl

i;j;kllðnÞ ð2:27Þ

whereWli;j;k is the third spatial (central) moment response to a point

source of type l:

Wli;j;k ¼ E ðZið1Þ � lið1ÞÞðZjð1Þ � ljð1ÞÞðZkð1Þ � lkð1ÞÞ

��~r0 ¼~elh i

:

ð2:28Þ

2.1.3. Serial momentsTo calculate the generation-to-generation correlation, the spa-

tial moments in the form of E½ZiðnÞ�, E½ZiðnÞZjðnÞ�,E½ZiðnÞZjðnÞZkðnÞ�, are not sufficient, the serial-spatial moments inthe form of E½ZiðnÞZjðnþ kÞ�, are also required. Therefore, the joint

moment generating function of~ZðnÞ and~Zðnþ kÞ is needed (wherek is the generation lag):

Fn;nþk ~r0;~s;~t� ¼ E ~ZðnÞ~Zðnþ kÞ

��~Zð0Þ ¼ ~r0h i

¼X~r;~q

P ~ZðnÞ ¼~r;~Zðnþ kÞ ¼~q��~Zð0Þ ¼~r0

� �Ymi¼1

srii tqii

ð2:29ÞThe relation between the moment generating function (Fnð~r0;~sÞ)

of ~ZðnÞ and the joint moment generating function (Fn;nþkð~r0;~s;~tÞ) of~ZðnÞ and ~Zðnþ kÞ can be found:

Fn;nþk ~r0;~s;~t� ¼X

~r;~q

P ~ZðnÞ ¼~r;~Zðnþ kÞ ¼~q��~Zð0Þ ¼~r0

� �Ymi¼1

srii tqii

¼X~r;~q

P ~Zðnþ kÞ ¼~q��~ZðnÞ ¼~r

� �P ~ZðnÞ ¼~r

��~Zð0Þ ¼~r0� �Ym

i¼1

srii tqii

¼X~r


� �

�X~q

P ~Zðnþ kÞ ¼~q��~ZðnÞ ¼~r

� �Ymj¼1

tqjj

24 35Ymi¼1

srii

¼X~r


� �Fkð~r;~tÞ �Ym

i¼1

srii

¼X~r


� � Ymj¼1

Fkðj;~tÞrj" #Ym

i¼1

srii

¼X~r


� � Ymj¼1

Fkðj;~tÞrj srjj" #

ð2:30ÞIt shows that the joint moment generating function

Fn;nþkð~r0;~s;~tÞ satisfies the functional equation

Fn;nþk ~r0;~s;~t� ¼ Fnð~r0;~uðkÞÞ

uiðkÞ ¼ siFkði;~tÞð2:31Þ

2.2. Approximating the ACCs

Section 2.1 derived various moments to evaluate the ACCs of anMBP model. This subsection shows the modifications of an MBPmodel to fit the Monte Carlo power iteration and explicitly writesACC in term of the spatial and serial moments.

2.2.1. Expansion of fission source distributionBefore specifying the serial-spatial moments to be evaluated

from the moment generating functions Fnð~r0;~sÞ and Fn;nþkð~r0;~s;~tÞ,ACCs must be expressed in the form of such moments. Since theMultitype Branching Processes does not include neutron popula-tion normalization, the ACC’s are defined over a normalized fissionsource distribution XlðnÞ (defined in Eq. (2.32)) rather than theunnormalized fission source ZlðnÞ.

XlðnÞ � ZlðnÞZðnÞ ¼ ZlðnÞPm

i¼1ZiðnÞð2:32Þ

The ACC of Xl between generation n and nþ k is defined in Eq.(2.1).

Taking derivatives of moment generating functions in Eqs. (2.5)and (2.29) gives expectation of products of ZlðnÞ such asE½ZlðnÞ�; E½ZlðnÞZjðnÞ� and E½ZlðnÞZlðnþ kÞ�. However, to evaluate


the variance and covariance terms in Eq. (2.1), expectations in theform of E½XlðnÞ� and E½XlðnÞXlðnþ kÞ� are required.

To perform the transformation, the definition of XlðnÞ(Eq. (2.32)) is viewed as a function of ~ZðnÞ and expanded around

E½~ZðnÞ� (� ~lðnÞ):XlðnÞ � glð~ZðnÞÞ

¼ glð~lðnÞÞ þXmi¼1

@glð~ZðnÞÞ@ZiðnÞ

��~ZðnÞ¼~lðnÞ

ðZiðnÞ � liðnÞÞ

þXmi;j¼1

@2glð~ZðnÞÞ@ZiðnÞ@ZjðnÞ


ðZiðnÞ � liðnÞÞðZjðnÞ

� ljðnÞÞ þ � � � ð2:33Þ

where the derivatives of glð~ZðnÞÞ can be calculated explicitly fromEq. (2.32) as

@glð~ZðnÞÞ@ZiðnÞ


� g0l iðnÞ ¼

di;llðnÞ �

llðnÞlðnÞ2

ð2:34Þ

@2glð~ZðnÞÞ@ZiðnÞ@ZjðnÞ


� g00l i;jðnÞ ¼ � di;l

lðnÞ2� dj;l

lðnÞ2þ 2

llðnÞlðnÞ3

ð2:35Þ

where

lðnÞ �Xmi¼1

liðnÞ: ð2:36Þ

The covariance terms required to evaluate qn;kðXlÞ (Eq. (2.1)) areof the form E ðXl � EXlÞðXl � EXlÞ½ �. Therefore Xl and EXl must pre-

serve terms of the same order following an expansion. glð~ZðnÞÞ(corresponding to Xl) and Eglð~ZðnÞÞ (corresponding to EXl) will beexpressed explicitly below to obtain consistent expansion orders.Also for simplicity, the dependence on generation n and neutrontype l is suppressed and the first and second order derivatives of

gð~ZðnÞÞ evaluated at ~lðnÞ are denoted as g0i and g00

i;j respectively.

Eqs. (2.34) and (2.35) show that g0i is of order 1

lðnÞ, and g00i;j is of

order 1lðnÞ2. And g0

i and g00i;j are coefficients of Zi � li � DZi and

ðZi � liÞðZj � ljÞ � DZiDZj respectively. Therefore, as long as thesystem has a large number ðlðnÞÞ of expected neutrons, the expan-sion in Eq. (2.33) is justified. Eq. (2.33) can be conceptually writtenas Eq. (2.37):

gðZÞ ¼ gðlÞ þ DZl

þ DZl

� �2

þ o �2� ð2:37Þ

where � denotes DZl . It is worthwhile to note that if a function of~ZðnÞ

is defined as

h ~ZðnÞ� �

¼Pm

i¼1ZiðnÞPmi¼1Ziðn� 1Þ ð2:38Þ

MBP methodology can be applied to analyze correlation andvariance underestimate of keff eigenvalue tallies.

2.2.2. Variance termsFrom the expansion of fission source distribution

gð~ZÞ ¼ gð~lÞ þ g0iðZi � liÞ þ 1

2g00i;jðZi � liÞðZ j � l jÞ þ o �2

� ð2:39Þ

its expectation is given by

Egð~ZÞ ¼ gð~lÞ þ 12g00i;jCov ½Zi; Z j� þ o E�2

� : ð2:40Þ

From Eqs. (2.39) and (2.40), an expression for Var½gð~ZÞ� can beobtained

Var½gð~ZÞ� ¼E g0iðZi � liÞ þ 1

2g00i;jðZi � liÞðZ j � l jÞ � 1

2g00i;jCov ½Zi; Z j� þ oð�2Þ

� �2

¼g0ig

0i0EðZi � liÞðZi0 � li0 Þ

þ g0i0g

00i;jEðZi0 � li0 ÞðZi � liÞðZ j � l jÞ þ oðE�3Þ:

ð2:41Þ

Since the lowest order of gð~ZÞ � Egð~ZÞ is Oð�Þ, a second order

expansion of gð~ZÞ is sufficient to provide the third order expansion

of Var½gð~ZÞ�. Otherwise, third order expansion is required to com-

prise all third order terms in Var½gð~ZÞ� with the leading constant

term in gð~ZÞ. The third order terms of the form

g0ig

00i0 ;j0 ðZi � liÞCov ½Zi0 ; Zj0 � vanish because EZi ¼ li.

Inserting the expansion coefficients (Eqs. (2.34), (2.35)) andspatial moments, (Eq. (2.17)) and restoring the neutron typeand generation dependence, the variance can be explicitlyexpressed as

Var½glð~ZðnÞÞ�¼dil

lðnÞ�llðnÞlðnÞ2

!Ci;jðnÞ�liðnÞljðnÞ� � d j

l

lðnÞ�llðnÞlðnÞ2

!

þ dhllðnÞ�

llðnÞlðnÞ2

!� dillðnÞ2

� d jl

lðnÞ2þ2

llðnÞlðnÞ3

!� Th;i;jðnÞ�liðnÞCj;hðnÞ�ljðnÞCh;iðnÞ�lhðnÞCi;jðnÞ�

þ2liðnÞljðnÞlhðnÞ�

ð2:42Þ

2.2.3. Covariance terms

The expansion of gð~ZðnÞÞ (Eq. (2.39)) and Egð~ZðnÞÞ (Eq. (2.40))can also be used to evaluate the covariance terms. Since thedependence on generation is suppressed, g1 and g2 are used to

denote gð~ZðnÞÞ and gð~Zðnþ kÞÞ. l1;l2; Z1 and Z2 are introducedsimilarly.

Cov ½gð~ZðnÞ;gð~ZðnþkÞÞ� ¼ E g01 iðZi

1�li1Þþ

12g001 i;jðZi

1�li1ÞðZj

1�l j1Þ

��12g00i;jCov½Zi

1;Zj1�þoð�2Þ

�� g0

2 i0 ðZi02�li0

2Þþ12g002 i0 ;j0 ðZi0

2�li02ÞðZj0

2�lj0

2Þ�

�12g00i0 ;j0Cov ½Zi0

2;Zj0

2�þoð�2Þ�

¼ g01 ig

02 i0EðZi

1�li1ÞðZi0

2�li02Þ

þ12g01 ig

002 i0 ;j0EðZi

1�li1ÞðZi0

2�li02ÞðZj0

2�lj0

2Þ

þ12g02 i0g

001 i;jEðZi0

2�li02ÞðZi

1�li1ÞðZj

1�l j1Þ

þoðE�3Þ ð2:43Þ

Inserting the expansion coefficients (Eqs. (2.34), (2.35)) andreintroducing the neutron type and generation dependence, thecovariance can be explicitly written as


Cov ½glð~ZðnÞÞ;glð~Zðnþ kÞÞ�

¼ dillðnÞ�

llðnÞlðnÞ2

!EZiðnÞZjðnþ kÞ �liðnÞljðnþ kÞ� �

� d jl

lðnþ kÞ �llðnþ kÞlðnþ kÞ2

!þ12

dhllðnÞ�

llðnÞlðnÞ2

!

� � dillðnþ kÞ2

� d jl

lðnþ kÞ2þ 2

llðnþ kÞlðnþ kÞ3

!� EðZhðnÞ �lhðnÞÞðZiðnþ kÞ �liðnþ kÞÞðZjðnþ kÞ�ljðnþ kÞÞ

þ12

dhllðnþ kÞ �

llðnþ kÞlðnþ kÞ2

!� dillðnÞ2

� d jl

lðnÞ2þ 2

llðnÞlðnÞ3

!� EðZhðnþ kÞ �lhðnþ kÞÞðZiðnÞ �liðnÞÞðZjðnÞ �ljðnÞÞ ð2:44ÞEq. (2.44) indicates the moments that need be evaluated. The

first term is of the lowest expansion order ðoð�2ÞÞ. It requires thevalue of serial-spatial moments EZiðnÞZjðnþ kÞ. The second andthird terms are of the higher expansion order ðoð�3ÞÞ. They requirethe value of serial-spatial moments EZhðnÞZiðnþ kÞZjðnþ kÞ andEZhðnþ kÞZiðnÞZjðnÞ. The moments in Eq. (2.44) are spatialmoments between different generations. Therefore, the spatialmoments evaluated in Eq. (2.17) are not sufficient. The jointmoment generating function for two generations defined inEq. (2.29) and the functional equation it satisfies (Eq. (2.31)) areneeded to evaluate additional serial-spatial moments. Derivativeof the joint moment generating function gives fractional serial-spatial moments, which can be easily converted to the serial-spatial moments needed in Eq. (2.44). Denote the expectationoperator for factorial moments with :

where the derivative can be evaluated explicitly using chain-ruleand repeatedly replacing in Eq. 2.31

@

@sl

@

@tjFnð~r0;~uðkÞÞ ¼ @

@sl

@Fnð~r0;~uðkÞÞ@uhðkÞ

@uh

@tj

¼ @

@sl

@Fnð~r0;~uðkÞÞ@uhðkÞ sh

@Fkðh;~tÞ@tj

¼ @Fkðh;~tÞ@tj

@

@sl

@Fnð~r0;~uðkÞÞ@uhðkÞ sh

� �¼ @Fkðh;~tÞ

@tj

@

@sl

@Fnð~r0;~uðkÞÞ@uhðkÞ sh þ @Fnð~r0;~uðkÞÞ

@uhðkÞ dlh

� �¼ @Fkðh;~tÞ

@tj

@2Fnð~r0;~uðkÞÞ@ugðkÞ@uhðkÞ

@ugðkÞ@sl

sh þ @Fnð~r0;~uðkÞÞ@uhðkÞ dlh

!

¼ @Fkðh;~tÞ@tj

@2Fnð~r0;~uðkÞÞ@ugðkÞ@uhðkÞ Fkðg;~tÞdlgsh þ

@Fnð~r0;~uðkÞÞ@uhðkÞ dlh

!

¼ @Fkðh;~tÞ@tj

@2Fnð~r0;~uðkÞÞ@ulðkÞ@uhðkÞ Fkðl;~tÞsh þ @Fkðl;~tÞ

@tj

@Fnð~r0;~uðkÞÞ@ulðkÞ

ð2:46Þ

Evaluated at ~1, the above equation leads to.

where lljðnÞ is defined the same way as ljðnÞ except that ll

jðnÞ is aconditional expectation on ~r0 ¼~el. When k P 1; ZlðnÞ and Zjðnþ kÞcannot be the same variable, factorial moments of ZlðnÞ andZjðnþ kÞ are identical to the ordinary moments,

Eq. (2.46) reveals much of the origin of the auto-correlation.Given the number of neutrons in one cell at one generation, thenumber of neutrons in this cell at future generations are correlateddue to two parts: (1) neutrons from a given cell remain or re-enter

a given cell at a later generation, and (2) neutrons initially inanother cell enters a given cell at a later generation. The first partis determined by the diagonal elements of the spatial covariancematrix. Contributions from neighboring cells is determined bythe off-diagonal elements of the spatial covariance matrix.

After a more lengthy but similar procedure, the third orderserial-spatial moments are found as.

Similarly, ~Chi;jðnÞ is defined the same way as ~Ci;jðnÞ except that

~Chi;jðnÞ is a conditional expectation on ~r0 ¼~eh. It should be noted

of an exception to Einstein notation in the above equation. The lefthand side depends on index l and l is not dummy (summed over)on the right hand side.

2.2.4. Auto-correlation qn;k of coarse tally regionsIn realistic simulations, auto-correlation might be needed on

discretized regions of different size (e.g. pin vs assembly size talliesin a reaction simulation). To avoid recomputing multiple momentson varying mesh sizes, a condensation process is proposed allow-ing to evaluate ACCs on coarser meshes. Suppose tally region I con-tains phase space cells I1; . . . ; Ii; . . ., the number of neutrons inregion I (denoted by YIðnÞ) is the sum of neutrons in the phasespace cells. YIðnÞ is defined in Eq. (2.51).

YIðnÞ �XIi2I

ZIi ðnÞ ð2:51Þ

The covariance terms of tally region I can be directly written as

Var½YIðnÞ� ¼XIi ;Ij2I

Cov ZIi ðnÞ; ZIj ðnÞh i

ð2:52Þ

Cov ½YIðnÞ;YIðnþ kÞ� ¼XIi ;Ij2I

Cov ZIi ðnÞ; ZIj ðnþ kÞh i

ð2:53Þ

All the terms on the right hand side of equations in Eq. (2.53)have already been explicitly written in Eq. (2.44). Though Eq.(2.44) only evaluates Cov ½ZlðnÞ; Zlðnþ kÞ�, it is straight to substitutesubscript l of terms at generation n with Ii and substitute subscriptl at generation nþ 1 with Ij. Therefore, the condensation processcan be easily done.

2.3. Procedure to calculate qn;k

In summary, knowing the spatial moments, serial-spatialmoments, evolution equations and expansions of XlðnÞ,generation-to-generation correlation coefficients, qn;kðXlÞ, can becalculated following the steps below:

1. Calculate the spatial moment responses Mli;V

li;j and Wl

i;j;k givenin Eqs. (2.20), (2.26) and (2.28)

2. Evolve the spatial moments liðnÞ;Ci;jðnÞ and Ti;j;kðnÞ according toEqs. (2.21), (2.25) and (2.27)

3. Combine the spatial moments of different generations to evalu-ate the serial-spatial moments according to Eqs. (2.48), (2.49)and (2.50)

4. Condense the covariances of discretized phase space cells intoconcerned tally regions with Eq. (2.53)

5. Substitute the moments into expansion of Var½YIðnÞ� andCov½YIðnÞ;YIðnþ kÞ� to evaluate qn;kðYIÞ

The above procedure yields qn;k after evolving the MBP for ngenerations.


It should be noted that the above procedure starts beforeperforming a full converged simulation. If the moment responses

Mlj;V

li;j are affordable with all phase space cells being finely dis-

cretized into flat source regions, they can be tallied from fixedsource calculation of uniformly sampled neutron sources. Sincesuch a fine mesh can be costly, the moment responses can betallied from a stationary source on a mesh slightly finer than thetallied quantities. Generation-to-generation correlation of a tallyregion is captured by the generation-to-generation and spatialcorrelation among the comprising phase space cells. Predictionshould be accurate if the migration distance of neutrons betweengenerations is on the same order or larger than the MBP phasespace cells for a given tally region.

Following the above procedure, evolving the constructed MBPmodel for k generations gives ACC of generation lag k and thus cor-rection for variance estimator after k active generations. In order topredict termination of Monte Carlo simulation to a given userdetermined variance criterion, very long calculation of MBP isrequired to evaluate ACC’s up to arbitrary generation lag k. Alterna-tively, the method of fitting ACC to a sum of exponentially decayingterms can be applied on the ACC’s found from MBP instead of thenoisy estimated value as in Miao et al. (2016). With data fitting,the MBP model can be evolved for a fixed number of generationsand the fit can be used to predict ACC and variance underestima-tion to any arbitrary number of generations. Results from MBP alsoverifies the data fitting method where ACC’s are fitted to a sum ofexponentially decaying terms. If ACC’s predicted from MBP arewritten explicitly, a quadratic form of the response matricesappears, whose dominant eigenvalues are equivalent to the expo-nential terms used in data fitting.

3. Application

3.1. Evaluate spatial moment responses

With the procedure given in Section 2.3, the spatial momentresponses are needed to initialize the calculation of ACC’s. Basedon their definition, spatial moment responses are just spatialmoments except that they specify the starting phase space regionand are one generation away from the source. Therefore, these spa-tial moment responses can be tallied from a single generation of aMonte Carlo simulation. Due to computational cost, it is unreason-able to tally directly higher order spatial moments. For example,

the moment response Vli;j of second order spatial moments is a

third order tensor that depends on three phase space regions andwould require many neutrons per generation to accurately resolve.We propose instead to tally the fission-to-absorption probabilityfor each region pair and moments of new neutrons (from fission)for each region and use these to construct the spatial momentresponses. The feasibility of constructing the spatial momentresponses is limited to cases where only spatial (excluding energy,angle) correlation effects are of concern.

Assuming that the probability of absorption and the number ofneutrons born from fission are independent, we can denote theprobability that a neutron born at phase space region l is absorbed

at phase space region i as Pli. Defining the random variable mi as the

number of new neutrons out of absorption at phase space region i,

then by definition of Mli:

Mli ¼ Pl

iEmi: ð3:1ÞAs for the second order spatial moment responses, Eq. (2.26)

can be rearranged as

Vli;j ¼ EðZið1ÞZjð1Þj~r0¼~el Þ �Ml

iMlj ð3:2Þ

Since energy and angle are not considered, new neutrons out ofabsorption cannot appear at different phase space regions,

Zið1ÞZjð1Þ ¼ m2i di;j ð3:3ÞTherefore the second order spatial moment responses Vl

i;j can beconstructed from fission-to-absorption probabilities and momentsof new neutrons from fission as

Vli;j ¼ Pl

iEm2i di;j � Pl

iPljEmiEmj ð3:4Þ

Although Vli;j is a third order tensor, it can be constructed from

a matrix Pli and a vector Emi. The matrix Pl

i and vectors Emi and Em2ican be either computed from the unconverged or converged fis-sion source. If unconverged fission source is used, a fine meshis needed such that the flat source approximation yields a suit-able approximation of the Monte Carlo eigenvalue problem.Under this circumstance, the distribution of fission source does

not alter Pli; Emi and Em2i . If starting from a converged fission

source, Pli; Emi and Em2i can be tallied directly on a mesh as coarse

as the tally regions to be analyzed and used to construct an MBPmodel.

The above construction of spatial moment responses Mli and Vl

i;j

are based on the reasonable assumption that the occurrence ofbranching defines the concept of generation. However, in MonteCarlo simulations, other branching processes like ðn; xnÞ are treatedwithin a generation (Romano and Forget, 2013).

To describe such processes exactly, a more detailed structure oftransfer is required which can be shown to require 3 additionalmatrices and 2 additional vectors. The detailed derivation is givenin Appendix B, but results were shown unnecessary for capturingthe bulk correlation behavior in nuclear systems and were thusomitted.

3.2. Approximating source normalization

Another difference between the Multitype Branching Processes(MBP) presented previously and realistic Monte Carlo simulationsis the normalization of neutron population. This problem can bemitigated by making the corresponding MBP critical. In OpenMC(Romano and Forget, 2013), the number of new neutrons for eachfission event is normalized by the estimated keff to maintain theneutron population near constant with subsequent normalizationsteps to enforce a constant fission bank. Correspondingly, withthe tallied fission-to-absorption matrix P and the expected numberof new neutrons for each cell Em, a fission matrix can be con-structed and the vector Em normalized by the estimated eigenvalueof the fission matrix.

4. Results and analysis

The MBP model was implemented in OpenMC and its predictivecapability was compared on assembly size tallies using the 2DBEAVRS benchmark (Horelik et al., 2013) at Hot Zero Power condi-tions. The 2D model is represented by a 10 cm axial slice at mid-height with reflective top and bottom boundary conditions. Toserve as a reference for the variance and prediction of ACCs, 300independent simulations were performed. Each simulation con-tains 100 inactive generations and 1000 active generations. Eachgeneration tracks 4� 106 neutrons. These independent simula-tions provide a true measure of variance at each generation.

To limit the amount of tally data while still exploring the spatialdependency of the predicted quantities, three representativeassemblies were selected as shown in Fig. 1. Among the three fuelassemblies selected, one is at the edge, one is at the center and the

Fig. 1. Layout of BEAVRS Benchmark and selected assemblies.


last one is in the middle. U235 enrichment of Assembly 1, 2, 3 are3.1%, 1.6% and 1:6% respectively. The correlation level decreasesfrom the boundary to the center. The MBP model’s transfer matrix

Pli was computed on a quarter-assembly mesh using the first 8

active generations.

4.1. Application of the predicted ACC’s

The predictive capability of the MBP model is shown on threequantities:

� ACC’s as function of generation lag: qk

� Bias of the estimator of sample variance as function of activegenerations: bðNÞ

� Underestimation ratio of the variance of the estimator (tallyaveraged over active generations) as function of active genera-tions: rðNÞ

The underestimation ratio, rðNÞ, is the ratio between the realvariance of the estimator accumulated up to Nth active generationand the variance of the estimator at any generation divided by N.

rðNÞ �r2

�Xl

r2Xl=N

: ð4:1Þ

The relation between the underestimation ratio ðrðNÞÞ and theACC’s is shown in Eq. (4.2) (Miao et al., 2016).

rðNÞ ¼ 1þ 2XN�1

k¼1

1� kN

� �qk ð4:2Þ

q̂kðlÞ ¼ 1N � k

XN�k

n¼1

sPs

a¼1XðaÞl ðnÞXðaÞ

l ðnþ kÞ �Psa¼1X

ðaÞl ðnÞPs

a¼1XðaÞl ðnþ kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sPs

a¼1XðaÞl ðnÞ2 � Ps

a¼1XðaÞl ðnÞ

� �2r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisPs

a¼1XðaÞl ðnþ kÞ2 � Ps

a¼1XðaÞl ðnþ kÞ

� �2r ; ð4:6Þ

However, finding rðNÞ is only part of the problem in estimatingr2

�Xl. The ratio rðNÞ corrects the convergence rate from 1=N to

rðNÞ=N, but the leading term r2Xl

is biased from the correlation.

Instead of estimating r2Xl

with

dr2Xl

¼ 1N � 1

XNi¼1

XlðiÞ � �XlðNÞ� 2 ð4:3Þ

the unbiased estimator is defined as (Miao et al., 2016)

dr2Xl

¼ 1N � rðNÞ

XNi¼1

XlðiÞ � XlðNÞ� 2 ð4:4Þ

where rðNÞ < N unless qðkÞ ¼ 1 for all k. Therefore, a correctionterm can be defined as

bðNÞ ¼ N � 1N � rðNÞ : ð4:5Þ

4.2. Comparing to reference

The above three quantities, qk, rðNÞ and bðNÞ can be solved ana-lytically using the MBP model and thus used as a prediction for the2D BEAVRS benchmark. These three quantities can be also esti-mated directly from the s independent simulations using the fol-lowing definitions.

The ACC, qk, can be estimated by first calculating qn;k as thePearson correlation coefficient of the two samples,

fXð1Þl ðnÞ; . . . ;XðsÞ

l ðnÞg and fXð1Þl ðnþ kÞ; . . . ;XðsÞ

l ðnþ kÞg and then tak-

ing the average of all n’s using the fact that qn;k is stationary. XðaÞl

uses the same notation as Section 2 with a superscript ðaÞ to iden-tify the independent simulations from a ¼ 1 to a ¼ s.

Eq. (4.6) is equivalent to the definition in Eq. (2.1) by simplyreplacing the expectation operator in Eq. (2.1) with average overthe independent simulations:

where l denotes the location, s denotes the number of simulations,n denotes the number of generations in each of the s simulations.The outermost sum exploits the stationary feature of correlationcoefficients and averages over the N � k active generations.


The reference of rðNÞ can be obtained by calculating Var XlðNÞ �

directly. For any generation N, Each of the s independent simula-tions gives an estimate of XlðNÞ. The variance of the s of them is

an estimate of Var½XlðNÞ� and is denoted as dVar½XlðNÞ�.

dVar½XlðNÞ� ¼Ps

a¼1 XðaÞl ðNÞ

� �2� Ps

a¼1XðaÞl ðNÞ

� �2=s

s� 1ð4:7Þ

Eq. (4.7) forms a sample of XlðNÞ from each independent simu-lation a and then calculates the variance over the s independentsimulations. Since the s simulations are independent, Var XlðNÞ

�estimated by Eq. (4.7) is not underestimated. To view the conver-

gence rate, predicted rlðNÞN will be compared with drlðNÞ

N calculated

from Var XlðNÞ �

.

drlðNÞN

¼dVar XlðNÞ �dVar Xlð1Þ � ð4:8Þ

Similarly, the reference of bðNÞ can be obtained by estimatingVar Xl½ � directly.

dVar½Xl� ¼ 1N

XNn¼1

Psa¼1ðXðaÞ

l ðnÞÞ2 � Psa¼1X

ðaÞl ðnÞ

� �2�s

s� 1ð4:9Þ

The ratio between the variance calculated from Eq. (4.9) and thevariance calculated from Eq. (4.3) is the reference expression forbðNÞ

b̂ðNÞ ¼1N

XNn¼1

Ps

a¼1ðXðaÞ

lðnÞÞ2�

Ps

a¼1XðaÞl

ðnÞ� 2.

s

s�1

1s

Psa¼1

1N�1

PNn¼1 XðaÞ

l ðnÞ � XðaÞl ðnÞ

� �2 ð4:10Þ

4.2.1. ACC predictionThe predicted and reference ACC’s for the three assemblies as a

function of the generation lag are shown in Fig. 2. The fitted to pre-diction ACC’s are also given. To show the stability of the MBPmethodology, the prediction is performed for each of the 300 inde-pendent simulations. In Fig. 2, the black curves correspond to aver-age of the 300 independent predictions, the black shaded regionscorrespond to one standard deviation of the 300 predictions. thegreen curves correspond to average of the 300 independent ACC fit-tings, the green shaded regions correspond to one standard devia-tion of the 300 ACC fittings. The fitted ACC curve agrees well withthe predicted values as expected from the analysis in Section 2.3.The deviation of the predicted curves is negligible, which implies

that estimation of the moments Mlj and Vl

i;j from first 8 active gen-erations is accurate enough to capture the ACC of the system. The

Fig. 2. Predicted and Reference ACC for the three selected assemblies. The black curves coone standard deviation of the 300 predictions. the green curves correspond to average ofthe 300 fittings. (For interpretation of the references to colour in this figure caption, the

greater deviation of the fitted curves is due to the unstability ofdata fitting to sum of exponential terms (Acton, 1990).

4.2.2. Underestimation ratio predictionThe ACC’s are then used to correct the underestimation ratio.

The real convergence rates from independent simulations accord-ing to Eq. (4.7) are shown in Fig. 3 with the convergence rates pre-dicted from the MBP model and from the fitted ACC of MBP. As

mentioned in Section 4.2, dVar½XlðNÞ� is renormalized to giverðNÞ=N (Eq. (4.8)). In the peripheral assembly with the highest cor-relation level among the three, real variance deviates from 1=N bya factor of 7. In the central assembly with lowest correlation level,this factor is around 4.

4.2.3. Variance bias predictionThe impact of the correlation coefficients on bias of the estima-

tor of Var½XlðNÞ� is shown in Fig. 4. In the peripheral assembly withhigh correlation, the variance for each generation is underesti-mated by a factor of 2 if 10 active generations are simulated forstatistics. In the central assembly with lower correlation, the factoris around 1:6. In comparison with the deviation from 1=N, whichremains constant asymptotically, the underestimation of leadingvariance becomes negligible as more active generations aresimulated.

Fig. 5 plots the variance after incorporating both correctionfrom bias ratio and underestimation of 1=N.

4.3. Cost of the prediction method

Theoretical predictions above are based a MBP model (resultingfrom discretizing the BEAVRS 2D benchmark intom ¼ 34� 34 ¼ 1156 cells) and the second order expansion of fis-sion source distributions (Section 2.2.1). The 1156 cells are actuallyquarter-assembly cells because a 17 discretization exactly matchesthe boundary of assemblies (Fig. 1). To perform the second order

calculation, moment responses Mlj and Vl

i;j and moments

llðnÞ;lilðkÞ and Ci;jðnÞ are needed. Ml

j and Vli;j can be constructed

from the fission to absorption matrix (Pli) and first ðEmiÞ and second

ðEm2i Þ order moments of mi, where mi is the number of neutrons born

from absorption in phase space region i. The quantities Pli; Emi and

Em2i are tallied during the first few active generations. These corre-spond to one m�m matrix and two length m vectors, respectively.

Mlj and Vl

i;j are used to find the stationary distribution of llðnÞ andspatial covariance Ci;jðnÞ.

ThenMlj is used to evolve the first order point response moment

lhj ðkÞ with lh

j ð0Þ ¼ dhj . According to Eqs. (2.53) and (2.44),

llðnÞ;Ci;jðnÞ and lhj ðkÞ are sufficient to evaluate qn;kðXlÞ.

rrespond to average of the 300 predictions. The black shaded regions correspond tothe 300 fittings. The green shaded regions correspond to one standard deviation ofreader is referred to the web version of this article.)

Fig. 3. Predicted and Reference underestimation ratio rðNÞ at three selected assemblies. The black curves correspond to average of the 300 independent predictions. The blackshaded regions correspond to one standard deviation of the 300 independent predictions, which is too small to be visible. the green curves correspond to average of the 300independent fittings. The green shaded regions correspond to one standard deviation of the 300 independent fittings. (For interpretation of the references to colour in thisfigure caption, the reader is referred to the web version of this article.)

Fig. 4. Predicted and Reference bias of variance at three selected assemblies. The black curves correspond to average of the 300 independent predictions. The black shadedregions correspond to one standard deviation of the 300 independent predictions. the green curves correspond to average of the 300 independent fittings. The green shadedregions correspond to one standard deviation of the 300 independent fittings. (For interpretation of the references to colour in this figure caption, the reader is referred to theweb version of this article.)

Fig. 5. Predicted (Fully Corrected) and Reference Variance at three selected assemblies. The black curves correspond to average of the 300 independent predictions. The blackshaded regions correspond to one standard deviation of the 300 independent predictions, which is too small to be visible. the green curves correspond to average of the 300independent fittings. The green shaded regions correspond to one standard deviation of the 300 independent fittings. (For interpretation of the references to colour in thisfigure caption, the reader is referred to the web version of this article.)


To provide a better variance estimate, rðNÞ rather than qðkÞ isrequired for each tally region. According to Eq. (4.2), twoaccumulated summation of ACC’s for each tally region should be

maintained:PN�1

k¼1qk andPN�1

k¼1 kqk, thus needing to store twovectors of length m. In total, the minimal memory requirementof a second order MBP calculation is four m length vectors

Emi; Em2i ;PN�1

k¼1 kqkðXiÞ;PN�1

k¼1qkðXiÞ� ��

and three m�m matrices

(Mlj;Ci;jðnÞ;lh

j ðkÞ). For the BEAVRS 2D benchmark discretized to34� 34 ¼ 1156 cells with all matrix element represented asdouble precision numbers, the memory requirements are approxi-mately 30 MB.

However, once the ACCs for each tally region have been calcu-lated for enough MBP generations ð� 100Þ to perform exponentialfitting, only 2 � 3 coefficients per tally are needed to provide a bet-ter variance estimate.

5. Conclusion

This work proposes an approach for predicting the underes-timation of variance often seen in full core nuclear reactorsimulations prior to performing a detailed simulation. Knowingthis underestimation will allow for more efficient simulationswith truly quantifiable error estimates and stopping criteria.By discretizing the phase space, the Multitype Branching Pro-cess was used to explicitly write the spatial and temporalmoments which were then related to the auto-correlation coef-ficients. The discretization process introduces an approximationof the true transport process that can be used to efficientlyapproximate the variance. The discretization must be finer thanthe tally regions of interest. The more stationary the fissionsource used to compute the moments, the coarser the MBPmesh can be.


Numerical results compared to 300 independent simulationshave shown to accurately predict assembly auto-correlation coeffi-cients and variance estimates in various spatial location of the 2Dfull core PWR BEAVRS benchmark. This paper used a mesh 4-timesfiner than the assembly-sized tally regions needed for the MBPprocess with moments computed in the first 8 actives batches.MBP tallies were performed to compute the fission to absorptionmatrix and moments of mi (new neutrons produced from anabsorption site at each phase space cells). Evolution of the MBPmodel by moment responses M and V can be related to the ACCsfor any coarser tally region at any generation lag. For efficiency,the ACCs can be exponentially fitted using 2 � 3 parameters whichcan then be used in calculating the variance underestimation. Thispredictive capability can help us calculate the number of activegenerations needed to achieve a target variance.

Presently, the prediction method requires memory / m2, wherem measures the fineness of the discretized mesh on which MBP isapplied. Future work will investigate further approximations thatcan be made to reduce memory requirement and identify waysof using the moment estimates to de-correlate the fission bankand reduce the variance.

Acknowledgments

This research was supported in part by the Consortium forAdvanced Simulation of Light Water Reactors (CASL), an Energy

~Tk;j;iðnþ 1Þ��~ZðnÞ

¼ @3

@sk@sj@si

Ymk¼1


¼ @2

@sk@sj

Xml¼1

F1ðl;~sÞZl ðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ

Yk–l


~s¼~1

¼ @

@sk

Xml¼1

@

@sjF1ðl;~sÞZlðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ

� �Yk–l

F1ðk;~sÞZkðnÞþF1ðl;~sÞZlðnÞ�1Z

¼Xml¼1

@

@skZlðnÞðZlðnÞ � 1ÞF1ðl;~sÞZl ðnÞ�2 @

@sjF1ðl;~sÞ @

@siF1ðl;~sÞ

� �Yk–l

F1ðk;~sÞZ

þ F1ðl;~sÞZlðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ

Xh–l

ZhðnÞF1ðh;~sÞZh ðnÞ�1 @

@sjF1ðh;~sÞ

Yg–hg–l

F1

0B@¼Xml¼1

ZlðnÞðZlðnÞ � 1ÞðZlðnÞ � 2ÞF1ðl;~sÞZl ðnÞ�3 @

@skF1ðl;~sÞ @

@sjF1ðl;~sÞ @

@si

�� @2

@sk@sjF1ðl;~sÞ @

@siF1ðl;~sÞ þ ZlðnÞðZlðnÞ � 1ÞF1ðl;~sÞZl ðnÞ�2 @

@sjF1ðl;~sÞ @

@sk

� @

@skF1ðl;~sÞ @2

@sj@siF1ðl;~sÞ þ F1ðl;~sÞZlðnÞ�1ZlðnÞ @3

@sk@sj@siF1ðl;~sÞ

!Yk–l

F1ð

þ ZlðnÞðZlðnÞ � 1ÞF1ðl;~sÞZl ðnÞ�2 @

@sjF1ðl;~sÞ @

@siF1ðl;~sÞ þ F1ðl;~sÞZlðnÞ�1Zlð

�X

a–lZaðnÞF1ða;~sÞZaðnÞ�1 @

@skF1ða;~sÞ

Yb–ab–l

F1ðb;~sÞZbðnÞ0B@

1CAþ ZlðnÞðZlðnÞ � 1ÞF1ðl;~sÞZl ðnÞ�2 @

@skF1ðl;~sÞ @

@siF1ðl;~sÞ þ F1ðl;~sÞZlðnÞ�1Zl

�Xh–l

ZhðnÞF1ðh;~sÞZhðnÞ�1 @

@sjF1ðh;~sÞ

Yg–hg–l

F1ðg;~sÞZh ðnÞ0B@

1CAþ F1ðl;~sÞZl ðnÞ�1ZlðnÞ @

@siF1ðl;~sÞ �

Xh–l

ZhðnÞðZhðnÞ � 1ÞF1ðh;~sÞZh ðnÞ�2

0B@264

þ ZhðnÞF1ðh;~sÞZhðnÞ�1 @2

@sk@sjF1ðh;~sÞ

Yg–hg–l

F1ðg;~sÞZg ðnÞ0B@

1CAþ ZhðnÞF1ðh;~sÞZh ðn

0BBB@

Innovation Hub for Modeling and Simulation of Nuclear Reactorsunder U.S. Department of Energy Contract No. DE-AC05-00OR22725.

Appendix A. Derivation of third order spatial moments

This section derives Eq. (2.27). Third order derivatives of themoment generating function will provide the third order factorialmoments, ~Tk;j;iðnÞ, which can eventually be related to Ti;j;iðnÞthrough Eq. (A.1).

~Tk;j;iðnÞ ¼ @3

@sk@sj@siFnð~r0;~sÞ

��~s¼~1

¼ E ZkðnÞZjðnÞZiðnÞ � ZiðnÞZkðnÞdi;j � ZiðnÞZjðnÞdi;k

�ZjðnÞZiðnÞdj;k þ 2ZiðnÞdi;jdi;k�

¼ Tk;j;iðnÞ � Ci;kðnÞdi;j � Ci;jðnÞdi;k � Cj;iðnÞdj;k þ 2liðnÞdi;jdi;kðA:1Þ

~Tk;j;iðnþ 1Þ is first evaluated conditional on ~ZðnÞ and then

expressed as a function of ~Tk;j;iðnÞ. The conditional expectation isevaluated as,

lðnÞ @

@siF1ðl;~sÞ @

@sj

Yk–l


~s¼~1

kðnÞ þ F1ðl;~sÞZlðnÞ�1ZlðnÞ @2

@sj@siF1ðl;~sÞ

!Yk–l

F1ðk;~sÞZkðnÞ

ðg;~sÞZg ðnÞ1CA��~s¼~1

F1ðl;~sÞ þ ZlðnÞðZlðnÞ � 1ÞF1ðl;~sÞZlðnÞ�2

2

@siF1ðl;~sÞ þ ZlðnÞðZlðnÞ � 1ÞF1ðl;~sÞZlðnÞ�2

k;~sÞZkðnÞ

nÞ @2

@sj@siF1ðl;~sÞ

!

ðnÞ @2

@sk@siF1ðl;~sÞ

!

@

@skF1ðh;~sÞ @

@sjF1ðh;~sÞ

Yg–hg–l

F1ðg;~sÞZg ðnÞ1CA

Þ�1 @

@sjF1ðh;~sÞ

Xg–hg–l

ZgðnÞF1ðg;~sÞZg ðnÞ�1 @

@skF1ðg;~sÞ

Yb–gb–hb–l

F1ðb;~sÞZbðnÞ

1CCCA377759>>>=>>>;��~s¼~1

ðA:2Þ


Evaluating Eq. (A.2) at~s ¼ ~1, the moment generating functionsbecome 1, the derivatives become factorial moments.


¼Xml¼1

ZlðnÞðZlðnÞ � 1ÞðZlðnÞ � 2ÞMlkM

ljM

li þ ZlðnÞðZlðnÞ � 1Þ ~Cl

k;jð1ÞMli þMl

j~Clk;ið1Þ þMl

k~Clj;ið1Þ

� �þ ZlðnÞ~Tl

k;j;ið1Þ� �

þXml¼1

ZlðnÞðZlðnÞ � 1ÞMljM

li þ ZlðnÞ~Cl

j;ið1Þ� �

�Xa–l

ZaðnÞMak þ

Xml¼1

ZlðnÞðZlðnÞ � 1ÞMlkM

li þ ZlðnÞ~Cl

k;ið1Þ� �

�Xh–l

ZhðnÞMhj

þXml¼1

ZlðnÞMli �Xh–l

ZhðnÞðZhðnÞ � 1ÞMhkM

hj

� �þ ZhðnÞ~Ch

k;jð1Þ þ ZhðnÞMhj

Xg–hg–l

ZgðnÞMgk

2666664

3777775¼Xml¼1

ZlðnÞðZlðnÞ � 1ÞðZlðnÞ � 2ÞMlkM

ljM

li þ ZlðnÞðZlðnÞ � 1Þ ~Cl

k;jð1ÞMli þMl

j~Clk;ið1Þ þMl

k~Clj;ið1Þ

� �þ ZlðnÞ~Tl

k;j;ið1Þ� �

þXml¼1

ZlðnÞðZlðnÞ � 1ÞMljM

li þ ZlðnÞ~Cl

j;ið1Þ� �

�Xa–l

ZaðnÞMak þ

Xml¼1


li þ ZlðnÞ~Cl

k;ið1Þ� �

�Xh–l

ZhðnÞMhj

þXml¼1


lj þ ZlðnÞ~Cl

k;jð1Þ� �

�Xh–l

ZhðnÞMhi þ

Xml¼1

ZlðnÞMli �Xh–l

ZhðnÞMhj

Xg–hg–l

ZgðnÞMgk

2666664

3777775 ðA:3Þ

Denote the 1st term, the 2nd to 4th terms and the last term inEq. (A.3) as T1, T2 and T3 respectively for simplification in the fol-lowing derivations. T3 can be simplified as Eq. (A.4).

T3 ¼Xml;h;g¼1

l–h;h–g;g–l

ZlðnÞMliZhðnÞMh

j ZgðnÞMgk ðA:4Þ

T1 can be simplified as Eq. (A.5).

T1 ¼Xml¼1

ZlðnÞ � ðZlðnÞ � 1ÞðZlðnÞ � 2ÞMlkM

ljM

li þ ðZlðnÞ � 1Þ

h� ~Cl

k;jð1ÞMli þMl

j~Clk;ið1Þ þMl

k~Clj;ið1Þ

� �þ ~Tl

k;j;ið1Þi

¼Xml¼1

ZlðnÞ � ðZlðnÞ2 � 3ZlðnÞÞMlkM

ljM

li

hþZlðnÞ ~Cl

k;jð1ÞMli þMl

j~Clk;ið1Þ þMl

k~Clj;ið1Þ

� �þ 2Ml

kMljM

li

�~Clk;jð1ÞMl

i �Mlj~Clk;ið1Þ �Ml

k~Clj;ið1Þ þ ~Tl

k;j;ið1Þi

ðA:5Þ

The third spatial (central) moment response to a point source oftype l (defined in Eq. (2.28)) can be expanded as Eq. (A.6)

Wlk;j;i ¼ EðZkð1ÞZjð1ÞZið1Þj~r0 ¼~elÞ þ 2EðZkð1Þj~r0 ¼~elÞEðZjð1Þj~r0 ¼~elÞ

� EðZið1Þj~r0 ¼~elÞ � EðZið1ÞZjð1Þj~r0 ¼~elÞEðZkð1Þj~r0 ¼~elÞ� EðZið1ÞZkð1Þj~r0 ¼~elÞEðZjð1Þj~r0 ¼~elÞ� EðZkð1ÞZjð1Þj~r0 ¼~elÞEðZið1Þj~r0 ¼~elÞ

¼ Tlk;j;ið1Þ þ 2Ml

kMljM

li � Cl

i;jð1ÞMlk � Cl

k;ið1ÞMlj � Cl

k;jð1ÞMli ðA:6Þ

The third factorial moment response to a point source of type lcan be defined by replacing the moments in Eq. (A.6) with factorialmoments.

~Wlk;j;i ¼ ~Tl

k;j;ið1Þ þ 2MlkM

ljM

li � ~Cl

i;jð1ÞMlk � ~Cl

k;ið1ÞMlj � ~Cl

k;jð1ÞMli

ðA:7Þ

~Wlk;j;i defined in Eq. (A.7) simplifies T1 to

T1 ¼Xml¼1

ZlðnÞ ZlðnÞ ðZlðnÞ � 3ÞMlkM

ljM

li þ ~Cl

k;jð1ÞMli þMl

j~Clk;ið1Þ

hnþMl

k~Clj;ið1Þ

iþ ~Wl

k;j;i

oðA:8Þ

For simplification, the three terms in T2 are defined asT2;1; T2;2; T2;3. T2;1 is simplified in Eq. (A.9).

T2;1 ¼Xml¼1

ZlðnÞ ZlðnÞ ~Clj;ið1Þ �Ml

jMli þ ZlðnÞMl

jMli

� �h i�Xa–l

ZaðnÞMak

¼Xml¼1

ZlðnÞ ~Vlj;i þ ZlðnÞMl

jMli

� ��Xa–l

ZaðnÞMak

¼Xml¼1;a¼1a–l

ZlðnÞ ~Vlj;i þ ZlðnÞMl

jMli

� �ZaðnÞMa

k

¼Xml¼1;a¼1a–l

ZlðnÞZaðnÞ~Vlj;iM

ak þ ZlðnÞ2ZaðnÞMl

jMliM

ak

ðA:9Þ

Simplifying T2;2 and T2;3 in the same way and combining themto T2 lead to

T2 ¼Xml¼1;a¼1a–l

ZlðnÞZaðnÞ ~Vlj;iM

ak þ ~Vl

k;iMaj þ ~Vl

j;kMai

� �þ ZlðnÞ2ZaðnÞ Ml

jMliM

ak þMl

kMliM

aj þMl

jMlkM

ai

� �ðA:10Þ

where ~Vlj;i is defined in a similar way to ~Wl

k;j;i by replacing momentsin Eq. (2.26) with factorial moments.

~Vlj;i ¼ ~Cl

j;ið1Þ �MljM

li ðA:11Þ


Combining T1 (Eq. (A.8)), T2 (Eq. (A.10)) and T3 (Eq. (A.4)) sim-plifies ~Tk;j;iðnþ 1Þ (Eq. (A.3)) to Eq. (A.12)


¼Xml¼1

ZlðnÞ ~Wlk;j;i þ ZlðnÞ3Ml

kMljM

li

� 3ZlðnÞ2MlkM

ljM

li

þ ZlðnÞ2 ~Clk;jð1ÞMl

i þMlj~Clk;ið1Þ þMl

k~Clj;ið1Þ

� �þXml¼1;a¼1a–l


ak þ ~Vl

k;iMaj þ ~Vl

j;kMai

� �þ ZlðnÞ2ZaðnÞ Ml

jMliM

ak þMl

kMliM

aj þMl

jMlkM

ai

� �þ

Xml;h;g¼1

l–h;h–g;g–l

ZlðnÞMliZhðnÞMh

j ZgðnÞMgk

ðA:12Þ

Denote the sum of terms in the form of Z3M3 in Eq. (A.12) asTZM .

TZM ¼Xml¼1

ZlðnÞ3MlkM

ljM

li þ

Xml;h;g¼1

l–h;h–g;g–l

ZlðnÞMliZhðnÞMh

j ZgðnÞMgk

þXml¼1;a¼1a–l

ZlðnÞ2ZaðnÞ MljM

liM

ak þMl

kMliM

aj þMl

jMlkM

ai

� �

¼Xml;h;g¼1l¼h¼g

ZlðnÞZhðnÞZgðnÞMgkM

hj M

li

þXml;h;g¼1

l–h;h–g;g–l

ZlðnÞMliZhðnÞMh

j ZgðnÞMgk

þXml;h;g¼1l¼h;g–l

ZlðnÞZhðnÞZgðnÞMhj M

liM

gk

þXml;h;g¼1l¼g;g–h

ZlðnÞZhðnÞZgðnÞMgkM

liM

hj

þXml;h;g¼1g¼h;g–l

ZlðnÞZhðnÞZgðnÞMhj M

gkM

li

¼Xm

l;h;g¼1

ZlðnÞZhðnÞZgðnÞMliM

hj M

gk ðA:13Þ

Denote the sum of terms in the form of Z2M3 or Z2M~C in Eq.(A.12) as TZC .

TZC ¼Xml¼1

ZlðnÞ2 ~Clk;jð1ÞMl

i þMlj~Clk;ið1Þ þMl

k~Clj;ið1Þ

� �� 3ZlðnÞ2Ml

kMljM

li

¼Xml¼1

ZlðnÞ2 ~Clk;jð1ÞMl

i �MlkM

ljM

li þMl

j~Clk;ið1Þ

��Ml

kMljM

li þMl

k~Clj;ið1Þ �Ml

kMljM

li

�¼Xml¼1

ZlðnÞ2 Mli~Vlk;j þMl

j~Vlk;i þMl

k~Vli;j

� �ðA:14Þ

where the last line uses the definition of ~Vlj;i in Eq. (A.11). Denote the

sum of TZC (Eq. (A.14)) and terms in the form of Z2 ~VM in Eq. (A.12)as TZV .

TZV ¼Xml¼1

ZlðnÞ2 Mli~Vlk;j þMl

j~Vlk;i þMl

k~Vli;j

� �þXml¼1;a¼1a–l


ak þ ~Vl

k;iMaj þ ~Vl

j;kMai

� �¼ m

m

l;a¼1ZlðnÞZaðnÞ ~Vl

j;iMak þ ~Vl

k;iMaj þ ~Vl

j;kMai

� �ðA:15Þ

TZM; TZV simplify ~Tk;j;iðnþ 1Þ��~ZðnÞ

to


¼Xm

l;h;g¼1

ZlðnÞZhðnÞZgðnÞMliM

hj M

gk

þXml;a¼1


ak þ ~Vl

k;iMaj þ ~Vl

j;kMai

� �þXml¼1

ZlðnÞ ~Wlk;j;i

ðA:16ÞThen taking the expected value of ~Tk;j;iðnþ 1Þj~ZðnÞ removes the

dependence of ~ZðnÞ and yields

~Tk;j;iðnþ 1Þ ¼ Tl;h;gðnÞMliM

hj M

gk

þ Cl;aðnÞ ~Vlj;iM

ak þ ~Vl

k;iMaj þ ~Vl

j;kMai

� �þ llðnÞ ~Wl

k;j;i ðA:17Þ

Substituting the relation between ~Tk;j;i and Tk;j;i (Eq. (A.1)) sim-plifies Eqs. (A.17) to (2.27).

Appendix B. Branching processes within generations

To treat the ðn; xnÞ processes accurately, instead of one fission-to-absorption probability matrix, four matrices are needed.

1. Plf i: the probability of a neutron born from fission at region l to

being absorbed at region i

2. Qlf i: the probability of a neutron born from fission at region l to

inducing a ðn; xnÞ reaction at region i

3. Plsi: the probability of a neutron born from ðn; xnÞ at region l to

being absorbed at region i

4. Qlsi: the probability of a neutron born from ðn; xnÞ at region l to

inducing a ðn; xnÞ reaction at region i

Similarly, rather than one random variable m, moments of tworandom variables are needed

1. mi: number of fissioned neutrons from absorption at phase spaceregion i, for simplicity, the moments of mi are denoted as below

Emi ¼ ki

Em2i ¼ r2i

ðB:1Þ

2. ni: number of new neutrons out of ðn; xnÞ at phase spaceregion i, ni P 2, for simplicity, the moments of ni are denotedas below

Eni ¼ vi

Eniðni � 1Þ ¼ s2iðB:2Þ

Similarly to the case without ðn; xnÞ processes discussed above,all the transport processes Pf ;Qf ; Ps;Qs and the branching pro-cesses m; n are assumed to be independent of each other. In other


words, mi and ni is assumed to be function of phase space region ionly (not dependent on whether the incoming neutron is bornfrom fission or ðn; xnÞ). Therefore, the expectation of the compositeprocesses can be expressed as the product of the expectation ofeach process.

1st order moment response Mli

The lowest order component of Mli corresponds to the process

where a neutron is born at cell l and finally fissioned at cell i with-out any ðn; xnÞ processes in between. The expected number of newneutrons would be

Mlð0Þi ¼ Pl

f iki ðB:3Þ

The next order component would come from the process wherea neutron is born at cell l, then encounter ðn; xnÞ reaction at cell j,and finally fissioned at cell i. Since it is assumed branching andtransport are independent and the neutrons out of ðn; xnÞ behaveindependently, the expected number of new neutrons of this pro-cess would be

Mlð1Þi ¼ Ql

f jvjP

js iki ðB:4Þ

The above process continues with infinitely higher order termsand can be conveniently written in matrix forms. Denote themoments of new neutrons from absorption and ðn; xnÞ in the diag-onal of matrices K and X respectively.

Ki;j ¼ kidi;j ðB:5ÞXi;j ¼ vidi;j ðB:6ÞK0

i;j ¼ r2i di;j ðB:7Þ

Then, the moment response M can be expressed as the series

M ¼ Mð0Þ þMð1Þ þMð2Þ þ � � �¼ PfKþ QfXPsKþ Qf XQsXPsKþ � � �¼ PfKþ QfXðPsKþ QsXðPsKþ QsXð� � �ÞÞÞ¼ PfKþ QfXð1þ QsX þ ðQsXÞ2 þ � � �ÞPsK

ðB:8Þ

The series of QsX must converge and thus sums to ð1� QsXÞ�1.Therefore the infinite contributions of M sum to

M ¼ PfKþ QfXð1� QsXÞ�1PsK ðB:9Þ

2nd order moment response Vli;j

By Eq. (3.2), the undetermined part of Vli;j so far is EZl

ið1ÞZljð1Þ.

The lowest order contribution of Vli;j come from the neutrons that

were born at phase space cell l and fissioned into i; j withoutðn; xnÞ processes in between.

Vlð0Þi;j ¼ Pl

f ir2

i di;j ðB:10ÞHigher order terms result from ðn; xnÞ neutrons. Define the

number of fissioned neutrons at phase space cell i induced by thebth neutron among the nk neutrons out of the ðn; xnÞ reaction atphase space cell k as qk

b;i. Then contribution from ðn; xnÞ neutronsin Vl

i;j can be written as

Vlð1Þi;j ¼ Ql

f kEXnkb¼1

qkb;i

Xnkb¼1

qkb;j ðB:11Þ

The expectation part can be solved following the processesbelow

Dki;j � E

Xnkb¼1

qkb;iq

kb;j þ

Xnkb–c

qkb;iq

kc;j

!¼ E Eðnkqk

biqk

bjjnkÞ þ Eðnkðnk � 1Þqk

biqkcjjnkÞ

� �¼ EnkEqk

biqk

bjþ Eðnkðnk � 1ÞÞEqk

biqkcj

� �¼ vkEqk

biqk

bjþ s2kEqk

biEqk

cj

� �ðB:12Þ

where the nk neutrons out of a ðn; xnÞ reaction are assumed tobehave independently. For simplicity of below equations, expecta-tions in Eq. (B.12) are denoted as

Rki � Eqk

biðB:13Þ

Fki;j � Eqk

biqk

bjðB:14Þ

Gki;j � vkFk

i;j ðB:15ÞSki;j � s2kRk

i Rkj ðB:16Þ

Dki;j ¼ Gk

i;j þ Ski;j ðB:17ÞThe first order moment response of ðn; xnÞ reaction denoted as

Rki (Eq. (B.13)) can be solved the same way as the above analysis

for Mli and turned out to be in a very similar form.

R ¼ ð1� QsXÞ�1PsK ðB:18ÞThe second order moment response of ðn; xnÞ reaction denoted

as Fki;j (Eq. (B.14)) can be solved recursively. Similarly to Vl

i;j, decom-

pose Fki;j into zeroth order contributions (neutrons out of ðn; xnÞ

were absorbed before any ðn; xnÞ reactions) and high order terms(neutrons out of ðn; xnÞwere absorbed after some ðn; xnÞ reactions).

Fki;j ¼ Pk

s i;jr2i di;j þ Qk

s k0EXnk0b¼1

qk0b;i

Xnk0b¼1

qk0b;j

¼ Pks i;jr

2i di;j þ Qk

s k0Dk0i;j

ðB:19Þ

Plugging the solution of Dki;j in the form of Fk

i;j (Eq. (B.12)) into

the above equation gives another equation where Fki;j is the only

unknown term.

Fki;j ¼ Pk

s ir2i di;j þ Qk

s k0 ðvk0Fk0i;j þ Sk

0

i;jÞ ðB:20Þ

which can be easily converted to an equation of Dki;j

Dki;j ¼ Pk

s ivkr2

i di;j þ Ski;j þ Qks k0v

kDk0i;j ðB:21Þ

and Dki;j is solved as

Dki;j ¼ ð1� QsXÞ�1

� �kk0ðPk0

sivk0r2

i di;j þ Sk0

i;jÞ ðB:22Þ

In summary, the 2nd order moment response Vli;j is found as

Vli;j ¼ Vlð0Þ

i;j þ Vlð1Þi;j �Ml

iMlj

V ¼ Pf K0 þ Qf ð1� QsXÞ�1 ðPsX K0 þ SÞ �M �MðB:23Þ

where four types of tensor production are implied

ðABÞij � AikB

kj ðB:24Þ

ðA BÞki;j � Akk0B

k0i;j ðB:25Þ

ðA BÞki;j � Aki Bi;j ðB:26Þ

ðA � BÞki;j � Aki B

kj ðB:27Þ


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https://doi.org/10.1016/0306-4549(86)90095-2

https://doi.org/10.1016/0306-4549(86)90095-2

http://www.sciencedirect.com/science/article/pii/0306454986900952

http://www.sciencedirect.com/science/article/pii/0306454986900952


http://linkinghub.elsevier.com/retrieve/pii/S0306454913004787

http://refhub.elsevier.com/S0306-4549(17)30343-2/h0025








https://doi.org/10.1080/00223131.2015.1039620

https://doi.org/10.1080/00223131.2015.1039620

http://cat.inist.fr/?aModele=afficheN&cpsidt=15256039




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