propagation of uncertainties in unfolding procedures

Nuclear Instruments and Methods in Physics Research A 476 (2002) 230–241

Propagation of uncertainties in unfolding procedures

Manfred Matzke*

Physikalisch-Technische Bundesanstal, Section 6.42, Bundesallee 100, D-38116 Braunschweig, Germany

Abstract

A review of methods currently used to unfold particle spectra from measured pulse height distributions or other

detector readings is given. It is pointed out that most of the measurements in particle spectrometry reveal ill-conditioned

or ill-posed problems. The presentation which is given here for examples of such inverse problems is focussed on the

algorithms used in the HEPRO unfolding program package of PTB. The question of uncertainty propagation is

discussed for least-squares algorithms as well as for those based on maximum entropy. A first attempt has been made to

quantify generally the ‘‘ambiguity’’ in ill-posed unfolding problems. The maximum entropy algorithm realized in the

MIEKE code allows a clear distinction to be made between two parts of uncertainty, one part coming from ambiguity

and one part coming from the usual uncertainty propagation. The resulting uncertainty matrix of the MIEKE code

provides these two parts. r 2002 Elsevier Science B.V. All rights reserved.

PACS: 29.30.Hs; 02.50.Ng; 29.85.+c

Keywords: Inverse problems; Spectrum unfolding; Maximum entropy; Unfolding codes; Neutron spectrometry; Uncertainty

propagation

1. Introduction

The determination of the spectral particle

fluence via unfolding of measured detector read-

ings has been investigated by many authors e.g.

[1–8]. Many papers are available, which refer to

the so-called inverse problem, where the existence

of solutions for various numbers of metrological

examples is investigated. The paper presented here

expands on the overview given in reference [1]. The

spectrometry performed by means of so-called

‘multi-channel’ or ‘few-channel’ measurements is

investigated and attention is given to the question

of uncertainty analysis and of propagation of

uncertainties in under- and overdetermined inverse

problems.

The evaluation of the spectral particle fluence

FEðEÞ from integrating measurements involves

solving the basic system of linear integral equa-

tions

z0i ¼ZRiðEÞFEðEÞ dE i ¼ 1;y;M ð1Þ

which represent the model of the measurement.

The vector zT0 ¼ ðz01;yz0i;yz0MÞ denotes the

expectation values of the (measured) readings of

the detector system, where the actual readings are

z00 ¼ z0 þ e with the statistically fluctuating quan-

tity e (T meaning transposition). It is assumed that

the uncertainty matrix (covariance matrix) Sz0 of

the vector z0 is known. The kernels RiðEÞ are the*Tel.: +49-531-592-6420; fax: +49 531-592-7015.

E-mail address: [email protected] (M. Matzke).

0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 4 3 8 - 3

response functions for the energy E of the Mdetector channels of the measuring system.

Mathematically, Eq. (1) is a degenerate case of

the Fredholm integral equation of the first kind. It

has no unique solution since a finite number of

discrete measurements cannot define a continuous

function FEðEÞ: Eq. (1) can be transformed at least

approximately into the discretised linear matrix

equation

z0 ¼ RU ð2Þ

with the group fluence vector UT¼ðF1; :::;Fn; :::;FNÞ: The N components Fn can be regarded as

average fluence values in the intervals between the

energies En and Enþ1: Eq. (2), too, has no unique

solution for N >M:In the overdetermined case of NoM (or for

N ¼M), the solution spectrum and its uncertainty

matrix is often calculated by the least-squares

method [see 1,2,5,7–10], where the quantity

w2 ¼ ðz0 � R � UÞT � S�1z0 � ðz0 � R � UÞ ð3Þ

is minimized, provided Sz0 is non-singular. The

solution of Eq. (3) is obtained from the so-called

normal equations

RT � S�1z0 � z0 ¼ RT � S�1

z0 � R � U ¼ B � U; ð4Þ

which have to be solved for U: Eq. (4) has a uniquesolution if the rank of the matrix

B ¼ RT � S�1z0 � R ð5Þ

is maximum, i.e. if it equals the number of fluence

groups N: Matrix B is sometimes called the

structure matrix [11].

In practice, it has been found that in many cases

the normal Eq. (4) are very close to singular

[1,3,9,12], even for a maximum rank of R: Thesituation is then similar to the underdetermined case

ðN >MÞ; and it must be concluded that, in practice,there is probably a manifold of possible solutions of

Eq. (4). This ambiguity may be extremely hard to

notice in complicated problems [12].

There are some possibilities of overcoming these

difficulties. First, all the information available on

the spectrum must be used within the unfolding

algorithm. Sometimes a pre-calculated fluence

vector U0 with uncertainty matrix SF0 is known,

sometimes a smoothing condition for the fluence

can be formulated. This a priori knowledge can be

used with Eq. (3) to construct a solvable system of

normal equations. However, if SF0 contains large

uncertainty components, non-physical results such

as negative elements of the fluence vector may be

obtained.

It is a conditio sine qua non for the resulting

fluence vector to be non-negative for all particle

energies. The discussion must therefore be ex-

tended to algorithms which include this condition,

such as the logarithmic least-squares method

[8,13–15] or, more general, to algorithms based

on the principle of maximum entropy with

constraints based on given information.

Computer programs based on this principle

have been developed at the PTB and are contained

in the HEPRO program system together with

other programs [1,2,4,13]. It is the aim of this

paper to review some algorithms used in particle

spectra unfolding (mainly those of the HEPRO

package) and to investigate how the propagation

of uncertainties can be performed.

2. Methods, algorithms and codes

A number of methods have been developed for

solving the inverse problem; details may be found

elsewhere in textbooks. Only a small overview can

be given here, concentrated on the algorithms used

in the HEPRO program system [13]. During the

recent years, however, important progress has been

achieved in the development of other methods and

codes not included in the HEPRO package. Some

of these methods are mentioned here, too.

In reactor dosimetry for ‘few channel’ unfolding

the codes based on linear least-squares methods

were been successfully applied during the REAL-

84 exercise [16]. STAY’SL [7], LEPRICON [17],

LSL [14], DIFBAS [18] and MSITER and

MINCHI [19] are representative of a large number

of available least-squares adjustment codes using a

priori information on the fluence. These codes are

also successfully applied to the unfolding of the

measured data obtained by Bonner spheres. In a

second group, the codes excluding negative fluence

values, e. g. SAND-II [15], GRAVEL [13], LSL-

M2 [14], LOUHI [20], BUNKI [21], have to be

M. Matzke / Nuclear Instruments and Methods in Physics Research A 476 (2002) 230–241 231

noticed. These codes perform in principle a non-

linear least-square adjustment, with the constraint

of non-negative particle fluence. In this context the

approach of Schmittroth [8] has to be mentioned,

where a log-normal distribution of the variables is

used to avoid negative fluence values. Secondly

Schmittroth uses correlations in the a priori

spectrum as a means of ensuring smoothness.

Some of the least-squares codes use other smooth-

ing or a regularizing procedures to construct a

non-singular matrix B in Eq. (5), e.g. a Tikhonov

regularization [22] or a condition of small curva-

ture or small fluctuation, e. g. the FERDOR code

[23,24]. In this context, the singular value decom-

position [3,12] should be mentioned here.

Recently, genetic algorithms (e. g. those of

Freeman et al. [25] and Mukherje [26]) have been

applied in particle spectrometry, where an evolu-

tion model is used together with a ‘‘survival’’

strategy in order to solve complex mathematical

equations. Here, the optimisation criterion (defini-

tion of the fittest) is an important model para-

meter. Some of the codes mentioned here allow an

uncertainty analysis to be performed, namely

those codes based on the least-squares method.

Uptill now it has not been quite clear how to

propagate the uncertainties for the evolution

models or the models referring to neural networks.

Some progress has been made in this field by

introduction of Bayesian methods into neural

network theories [27–29].

Bayesian methods or methods based on the

maximum entropy method have been realized in

the MIEKE and UNFANA programs of the

HEPRO package [13] and in the MAXED code,

recently developed [38,39]. These codes make

consistent uncertainty propagation possible, how-

ever, it is not yet completely understood how the

uncertainty resulting from the ‘‘ambiguity’’ of the

solution (singularity of matrix B) can be taken into

account; a first trial is contained in the MIEKE

code (see below).

3. Uncertainties in least-squares adjustment

It is assumed that a pulse height spectrum or a

number of detector readings z0 with the uncer-

tainty matrix Sz0 have been determined. For Sz0;only a diagonal form is sometimes used with the

variances of z0 (often determined by Poisson

statistics). It is further assumed that pre-informa-

tion R0 on the response matrix is available (or that

some parameters p0 of the response matrix are

known before) with the uncertainty matrix SR0 or

Sp0: If, in addition, the particle fluence U0 was

previously calculated or measured by another

independent ‘‘measurement’’ (e.g. by a transport

code calculation) with the uncertainty matrix SF0;the general w2 expression:

w2 ¼ðz0 � zÞT � S�1z0 � ðz0 � zÞ

þ ðR0 � RÞT � S�1R0 � ðR0 � RÞ

þ ðU0 � UÞT � S�1F0 � ðU0 � UÞ ð6Þ

has to be minimized under constraint z ¼ RðpÞ � Uwith respect to z, R (or p) and U. Eq. (6) is the

general w2 expression for an adjustment procedure,when the parameters or the fluence are already

known to a certain extent and the new measure-

ment z0 is used for adjustment only. The various

algorithms used to find the minimum of Eq. (6)

make a distinction in the modelling of the last two

terms. There are only some codes, mainly used in

reactor dosimetry, where the a priori information

on the response matrix can be taken into account

(e.g. STAY’SL [7], LSL [14], LEPRICON [17],

MSITER and MINCHI of PTB [19]); in all other

cases, the second term in Eq. (6) is missing. If

nothing is known a priori on the fluence, the last

term in Eq. (6) may be replaced by a smoothing or

shape condition (see e.g. [20,23,24,30]) in order to

obtain a non-singular system of normal equations

for the solution as mentioned in the introduction.

With the constraint z ¼ R � U the minimization

of Eq. (6) leads to non-linear normal equations.

Since it can be assumed that the adjusted values

will not be too far from the a priori information

values, this constraint equation is replaced by a

Taylor approximation in the vicinity of R0 and U0

[13], leading to the linear expression

z � R � U ) z � z1 � zR � ðR � R1Þ � zF � ðU � U1Þ;

ð7Þ

where z1 ¼ R0 � U0; and zR and zF are matrix

derivatives of z at R0 and U0; where zF ¼ R0 and

M. Matzke / Nuclear Instruments and Methods in Physics Research A 476 (2002) 230–241232

zR ¼ U0: With Eq. (7), the linear least-squares

adjustment is performed. The solution for the

fluence vector and its uncertainty matrix is [13]:

U ¼ U0 þ SF0 � zF � W�1 � ðz0 � R0 � U0Þ;

SF ¼ SF0 � SF0 � zF � W�1 � zF � SF0:ð8Þ

with the weighting matrix

W ¼ Sz0 þ zR � SR0 � zR þ zF � SF0 � zF ð9Þ

(transposition not indicated here and in the

following).

From Eq. (8) it is apparent that the introduction

of a priori information into the least-squares

formalism leads to normal equations, where only

an inversion of the M M matrix W instead of

the N N matrix B of Eq. (5) is required. Due to

the minus sign in the uncertainty equation, the

uncertainty is reduced after adjustment. From

Eq. (8) it is also apparent that the solutions and

their uncertainty matrices strongly depend on the a

priori information values U0 and the a priori

uncertainty matrix SF0: It is also seen that negativefluence values may appear here.

In practice, the a priori spectrum is often not

known on an absolute scale. In this case, the

scaling factor of the fluence has also to be

determined by the unfolding code. A non-linear

evaluation of this factor has been included in the

MSITER(MINCHI) program of PTB. It is not

included in the STAY’SL code.

From Eq. (6) it can be concluded that the a

priori information on both, the response matrix R0

and the fluence U0 is considered in the same way, i.

e. it is considered as if both quantities are intended

to be adjusted. This is the philosophy of the ISO

Guide [31], the German DIN Standard DIN1319

[32] and of the LEPRICON [17] methodology: all

measured quantities, those currently measured and

data quantified before have to be included into the

adjustment procedure, i.e. the ‘‘properties of the

entire world’’ are influenced by the current

measurement.

There is another way of adjustment not yet

established in particle fluence unfolding, namely to

consider the response function as a so-called

‘‘parameter’’ and applying the adjustment proce-

dure with this ‘‘parameter’’ being constant during

the unfolding procedure, and performing the

uncertainty propagation afterwards. In this case,

the second term of the right-hand side of Eq. (6) is

missing. The solution of this model is similar to

Eq. (8), but the term with SR0 is missing in the

expression for matrix W. From the uncertainty

propagation one finds (with new matrix W0):

SF ¼SF0 � SF0 � zF � W0�1 � zF � SF0

þ SF0 �zF �W0�1 �zR �SR0 � zR � W0�1 � zF � SF0:

ð10Þ

This model is different from that usually taken

in least-squares adjustment, and it has obviously

not been used to date in the standard software like

STAY’SL [7]; It offers advantages in cases where

the response functions are known with small

uncertainties.

Some remarks have to be made on the

construction of the matrices SF0 and SR0; whichare to date, in general, not available from the

transport code calculation used to derive U0 and

R0: In many cases, one has a rough idea of the

slope of the particle spectrum. In a reactor

moderator, for instance, a moderated fission

spectrum is expected, expressed as a sum of

parameterised subspectra with a number of kparameters (e. g. the temperature parameter of a

maxwell spectrum or the slope of a 1=E spectrum).

If the subspectra and the uncertainties of their

parameters are used to construct the matrix SF0; itturns out that SF0 has deficient rank rEk:Generally, S�1

F0 then is non-existent, however, a

solution vector U may be compiled from Eq. (8). It

can be shown that the possible solution spectra are

restricted to a very small subspace spanned by the

rEk eigenvectors of SF0; which means in principle

that the solution spectrum is a superposition of the

k subspectra defined a priori, thus putting severe

constraints on the solution spectrum by the special

choice of the a priori covariance matrix [37]. It is,

therefore, preferable to use a diagonal SF0 instead

in order that the entire solution space be available.

This in compliance with the German standard

DIN 1319 [32]: ‘‘If nothing is known on the

correlations, a diagonal uncertainty matrix must

be used’’.

In practical examples the uncertainty matrix SR0should be constructed in a way already outlined in


the ENDF uncertainty file [33], and it is recom-

mended that the format used there should also be

used, for instance, for establishing the uncertainty

matrix of the Bonner spheres response function.

Apart from reactor dosimetry, where the use of a

full uncertainty matrix SR0 compiled from ENDF

data is considered to be the state of the art, only a

few attempts have been made in the past to include

response matrix uncertainties into the unfolding

process for other problems [34]. The HEPRO

package [13] provides the possibility of including

uncertainties of energy calibration and resolution

parameters in multichannel unfolding problems.

4. Non-linear least-squares methods

The linear least-squares method, well estab-

lished in the STAY’SL [7] code and the LEPRI-

CON methodology [17], is the unfolding algorithm

recommended when good a priori information and

consistent measurements are available. The dis-

advantage is that negative fluence values cannot be

excluded. To take into account the condition of

non-negative fluence, a method first realized in the

SAND-II code [15], later used in the LSL-M2 [14]

and GRAVEL [13] codes, can be applied. In this

case, not the group fluence elements Fn but their

logarithms ln ðFnÞ are determined by a special

iteration procedure which minimizes an expression

similar to Eq. (3) with the logarithms of the zoiinstead of zoi: A similar method is used in the

LOUHI code [20] where the unknown quantities

Fn are expressed as squares of real numbers.

The SAND-II (GRAVEL) solution is deter-

mined by a special gradient method [13]: iteration

is started with a spectrum Fð1Þn : Weights w

ð1Þin ¼

Rin � Fð1Þn =zð1Þi with z

ð1Þi ¼

Pn Rin � expðlnF

ð1Þn Þ are

calculated. In each iteration step the current

solution ðkþ 1Þ is obtained from the previous

solution ðkÞ by

lnFðkþ1Þm � lnFðkÞ

m ¼ lðkÞm �Xi

ðln z0i � ln zðkÞi Þ �

wðkÞim

r2i;

where lðkÞm ¼Xi

wðkÞim

r2i

!�1

: ð11Þ

The ri are the relative standard deviations of the

z0i; namely ri ¼ s0i=z0i which were not included in

the original SAND-II code.

For the iteration procedure in the SAND-II

(GRAVEL) code, a first input spectrum is needed

when the iteration is started. A solution always

exists [13], but the solution spectrum depends on

this input spectrum in a way which is not quite

transparent so that an uncertainty propagation

cannot be easily performed. It has been found [13]

that the matrix B of Eq. (5) controls the variety of

solutions; for non-singular B a unique solution

exists, for ill-conditioned B a manifold of solutions

may exist, depending on the rank of B.

For the uncertainty analysis, the method of

randomly chosen start spectra has been tried

[35,42], with final averaging over all solutions.

This gives a (model-dependent) plausible uncer-

tainty of the solution spectrum, including correla-

tions. This uncertainty represents the variety

(ambiguity) of the solutions and cannot take into

account the normal uncertainty propagation of the

uncertainty matrices Sz0 and SR0:The other non-linear least-squares methods

available may be considered as special unfolding

models which are used to regularize the matrix B

(e.g. [22–24]). Additional parameters (e.g. for

smoothing) are introduced to get a unique solu-

tion. Uncertainty propagation can be performed

by these algorithms but has obviously not yet been

established in the codes available. The uncertain-

ties of the solutions depend on the uncertainties of

the input data and, also, of the parameters

introduced.

5. Maximum entropy, MIEKE, UNFANA,

MAXED

The properties of the least-squares codes are less

satisfactory, when there is a lack of pre-informa-

tion on the particle spectrum. In this case, the

singularity or quasi-singularity of the matrix B of

Eq. (4) may lead to an ambiguity of the solutions,

even if the constraint of non-negative fluence is

included into the least-squares algorithm. For the

equivalent maximum likelihood method it follows

that a unique, most probable particle spectrum


might not exist. However, similar to the maximum

likelihood method, it is possible to construct a

posteriori probability density of the fluence using

Bayes’ theorem [36]. The solution spectrum can

then be defined as the expectation value of U;denoted by Uh i; which can be calculated from the

probability density. This method was used in Ref.

[9]; later, a more modern interpretation was given

in Refs. [2,4] using the principle of maximum

entropy.

From statistics and information theory it is

known that a statistical system of a multidimen-

sional random variable x prefers a probability

density PðxÞ; for which the expectation value of theso-called entropy [2,36]

S ¼ �ZPðxÞlogðPðxÞÞ dx ð12Þ

has a maximum value. When the constraining

conditions to be imposed are included, it is

possible to formulate an extremum principle for

the determination of the fluence probability

density PðUÞ: For the case realized in the MIEKE

code [1,13], two constraints are assumed to exist:

(a) all group fluence values should be non-

negative, and (b) the expectation value w2� �

of w2

(see Eq. (3)), should be equal to the number of

degrees of freedom involved, which is equal to the

number M of elements in z0: By means of the

extremalisation procedure, the probability density

PðUÞ ¼ C1 � exp �b2� w2ðUÞ

� for all FnX0 ð13Þ

is obtained, which is defined for all FnX0 and

vanishes for negative values due to condition (a).

C1 is the normalization constant. b is a

‘‘temperature’’ parameter to be determined from

condition (b). The probability density PðUÞrepresents a multivariate normal distribution with

a w2-exponent, truncated because of condition (a).

Although the exponent is degenerate in the case

N >M (or for ill-conditioned matrix B), the

distribution can be normalized and the expectation

value Uh i with its uncertainty matrix

SF ¼ UUh i � Uh i Uh i ð14Þ

can be calculated. For b ¼ 0:5; PðUÞ is equivalentto the likelihood expression used in Bayes’

theorem if a normal distribution is assumed for

the measured data.

The probability density of Eq. (13) is introduced

into the MIEKE code of the HEPRO package [13].

A Monte Carlo code with importance sampling is

used to calculate expectation values. For the

MIEKE code, the computing time turned out to

be very long, in particular for M5N: Weise [4]

therefore proposed an analytical approach to the

Monte Carlo results to reduce the computing time.

Weise replaced the distribution of Eq. (13) by the

simpler exponential ansatz [2,4]:

PðUÞ ¼ C1 � expð�bT � R � UÞ ð15Þ

and used the same constraints for the extremum

principle of maximum entropy. The parameters b

for maximum entropy can be obtained by a simple

solution of a non-linear matrix equation. Here, the

expectation values can easily be calculated. The

covariance matrix SF of the distribution turns out

to be diagonal and must be properly interpreted.

To obtain the uncertainty associated with the

expectation value Uh i due to the uncertainty

matrix Szo; the Gaussian law of uncertainty

propagation is used. The subroutine UNFANA

[4] of the SPECAN code uses Eq. (15) and

performs this analysis.

Weise [4] has shown that for N-N the

expectation values Uh i obtained are identical for

both distributions of Eqs. (13) and (15). It has

been found from a variety of examples that there is

good agreement for MEN; too.It should be noted that, in principle, a priori

information could be included in both distribu-

tions (Eqs. (13) and (15)) by using for w2 an

expression similar to Eq. (6) instead of Eq. (3). For

the algorithm used in the MAXED code [38], this a

priori information is already included into the

maximum entropy formulas. In MAXED, the

(normalized) spectral fluence is taken as a prob-

ability density, leading to very interesting equa-

tions. The uncertainty propagation [39] has been

implemented recently, the code runs within the

HEPRO package.

For the uncertainty analysis it must be taken

into account that, in principle, two fluctuation

components exist, one resulting from the uncer-

tainties of the input quantities x0 (readings,


response functions, a priori information on the

fluence), one resulting from the ambiguity of the

solution (compare, for example with the micro-

scopic and macroscopic fluctuations in particle

statistics). The fluctuation of the resulting fluence

can be considered as being composed of both

contributions:

dU ¼ oqUqx0

> dx0 þ dU1 ð16Þ

perpendicular to each other, i.e. dx0dU1h i ¼ 0:The resulting covariance matrix is thus the sum

of two components: ðq Uh i=qx0ÞSx0ðq Uh i=qx0Þ þSF1: For the MIEKE algorithm, the corresponding

formulas are quite simple to obtain from the

probability distribution. Considering only the

uncertainties Sz0 of z0, it can be shown

thatq Uh i=qz0 ¼ �bSF � R � S�1z0 ; and with the de-

finition of B (Eq. (5)) one finds SF1 ¼ SF � b2SF �B � SF: In linear least-squares methods b2SF ¼B�1; which holds for non-singular B and for a

Gaussian distribution. In this case, the solution is

unique and SF1 vanishes. The ambiguity part SF1

cannot be obtained so easily for the other

maximum entropy models. For UNFANA and

MAXED, only the conventional uncertainty pro-

pagation corresponding to z0 and (in principle to

R0) is included. It has been found that the part

from the uncertainty propagation of z0 for the

MIEKE code ðb2SF � B � SFÞ agrees well with the

corresponding result of UNFANA.

For the MIEKE algorithm one obtains in

obvious matrix notation (indices have to be chosen

carefully):

q Uh iqR

¼ bfSF � S�1z0 � z0

� UUUh i � Uh i UUh ið Þ � R � S�1z0 g

As is usual in uncertainty propagation, the

triple correlation function can be approximated

in Gaussian approximation: F1 � F1h ið ÞhF2 � F2h ið Þ F1 � F1h ið Þi ¼ 0: This means that

the usual propagation of uncertainties even for

the response matrix, can be compiled after an

unfolding run, provided the resulting covariance

matrix SF given by Eq. (14) is compiled by the

code.

It should be mentioned that the maximum

entropy algorithms allow other probability dis-

tributions to be constructed [40]. In the MIEKE

code the evaluation of PðH�Þ for the dose-

equivalent H� ¼P

n hFnFn ¼ hF � U is performed

(hF: fluence-to-dose conversion factor):

PðH�Þ ¼ dðH� �hF � UÞh i ð17Þ

The variance H�H�h i � H�h i H�h i ¼ hFSFhFof the distribution again includes contributions

from the uncertainty propagation and the ambi-

guity.

6. Results obtained from practical examples

Two groups of unfolding tasks are important in

neutron spectrometry: ‘‘few channel’’ and ‘‘multi

channel’’ measurements. As examples representa-

tive of ‘‘few channel’’ spectrometry, the measure-

ments in reactor dosimetry have to be mentioned

(e.g. [16]), where activities or reaction rates of

different irradiation probes are determined. It was

shown in the past that exact uncertainty propaga-

tion is possible in this field [1]. The other

important examples are from neutron dosimetry,

namely the superheated drop (bubble) detectors

(e.g. [41]) and the Bonner sphere detectors (e.g.

[42,43]).

6.1. ‘‘Few channel’’ analysis

The determination of neutron spectra from the

readings z0 of a Bonner sphere system is a

representative example of ‘‘few channel’’ unfold-

ing. Count rates forM detectors are measured (Mbeing of the order of 10), and it is intended to

evaluate the spectral neutron fluence or the dose

equivalent in 50–100 energy groups. Obviously,

the a priori information on the fluence is more

important here than in the multichannel case.

Adding of a priori information during unfolding is

equivalent to increasing the number of measure-

ments fromM toM þN: This means that a prioriinformation controls the results more than the

measured values. The results obtained in various

examples confirm this conjecture.


In the present version of the MIEKE and

UNFANA codes, a priori information cannot be

taken into account. If real a priori information is

available (no guess!), the only codes which should

be used are those based on Eq. (6), e.g. STAY’SL,

LEPRICON, DIFBAS, MSITER(MINCHI),

LSL. However, there is a multitude of examples

(e.g. Refs. [11,44]), where neutron spectra have

also been successfully unfolded using only guess

functions together with the SAND–II (GRAVEL)

program. Careful investigation of these methods

allows the conclusion to be drawn that a priori

information is introduced as well, although in an

interactive procedure, which is not easy to

quantify mathematically.

A lot of experience is needed for a successful

unfolding of neutron spectra employing the

SAND-II or the GRAVEL code. For instance,

the use of a thermal or a fission spectrum as the

starting guess spectrum in SAND-II or GRAVEL

means quantifying a priori information. On the

other hand, when codes are used which need no

pre-information at all, like MIEKE and UNFA-

NA, it is hardly possible to unfold a reasonable

neutron spectrum. However, due to negative

correlations between adjacent neutron groups of

the solution spectrum, resulting from the matrix

SF1; it has been shown that integrals over the

fluence, such as the total dose equivalent, can be

calculated with much smaller uncertainties [1,13].

The following example presents the results of

measurements with Bonner spheres performed in a

moderated 252Cf reference field [45]. Twelve

Bonner spheres were irradiated; the relative

uncertainties of the readings (uncertainty of a

common scaling factor of the response functions

included) were between 1.2% and 4%. A priori

information was available from MCNP calcula-

tions [46], the relative uncertainties of the MCNP

calculations were estimated to be 15% below

100 keV and 10% above this value.

From Fig. 1 it is apparent that the MINCHI

result is nearly the a priori spectrum with the

exception of the thermal part, where the MCNP

results are uncertain because of the not well-

known components of reflecting concrete. The

SAND-II result [45] was obtained by an educated

guess using various combinations of thermal,

intermediate and fission spectra. From the UN-

FANA result the spectrum can hardly be recog-

nized, since available a priori information was not

included. Nevertheless, it was found that integral

values such as the total fluence or the total dose

equivalent agree within the uncertainties.

This also holds for the integral values obtained

with the MIEKE code. The uncertainty of the dose

equivalent is the sum of two nearly equal parts

(from ambiguity matrix SF1 and from uncertainty

propagation) which can be seen in Fig. 2, where

the probability distribution of H� (as obtained by

MIEKE) is given.

Considering the other examples of Bonner

spheres unfolding given in the literature, it must

be concluded that in moderated fission spectra like

reactor surroundings the fluence can in general be

Fig. 1. Spectral fluence of a moderated 252Cf neutron field

unfolded from the readings of 12 Bonner spheres.

Fig. 2. Probability density of H� in a moderated 252Cf neutron

field due to Eq. (17). There are two contributions to the

uncertainty coming from ambiguity and from the usual

propagation of uncertainties. The peak is asymmetric.


determined with a relative uncertainty of about

5% and the total dose equivalent with an

uncertainty of less than 15%.

6.2. ‘‘Multichannel’’ analysis

Successful unfolding of a measured pulse height

spectrum requires that the particle response func-

tion for each particle energy and each channel be

known. The progress made over the past years in

Monte Carlo calculations of response functions

has induced progress in unfolding techniques, too.

As regards gas-filled neutron detectors, the GNSR

code [47] and the SPHERE code [54] have to be

mentioned which calculate the neutron response

function in various neutron beam configurations

for cylindrical and spherical detectors. Included

are recoil detectors filled with H2,3He, 4He, and

mixtures with quenching gases commonly used.

For organic scintillators, the NRESP4 [48], EGS4

[49], and MCNP [46] codes have been used to

calculate neutron and photon response functions

for NE-213 [50]. In practice, only the so-called

ideal response function for an energy En of an

incoming particle is calculated by determining the

distribution of energy En deposited within the

detector. The other parameters, referring to energy

calibration and resolution, have to be determined

separately by the experimenter [13].

For practical unfolding work the question

arises, how many multichannel and fluence groups

should reasonably be chosen. It has been found in

Ref. [1] that the bin width of the pulse height

channels should be of the order of 1/5 of the

FWHM; the same holds for the corresponding

fluence groups.

The organic liquid scintillator NE-213 is very

well suited for measurements in mixed neutron-

photon fields because of its excellent neutron–

photon discrimination capabilities. Photons

produce scintillations by the compton and pair-

production interactions, which, together with wall

effects, lead to a broad response function for

monoenergetic incoming photons [51]. In the

following, the photon unfolding example of a

cylindrical NE-213 detector (5 cm in radius, 5 cm

in height) irradiated in the high-energy reference

photon field of the PTB [50] is considered. Details

of the experiment can be found in Refs. [50–52].

The response functions for monoenergetic

photons were calculated by Novotny [51] using

the EGS4 code. There is a non-linear relationship

between the measured pulse height and the

corresponding calculated light output. This rela-

tionship was experimentally determined together

with the resolution parameters from irradiations in

reference photon fields (see also Ref. [52]). In the

photon spectrum, apart from a peak at 0.511MeV,

three lines were expected between 1.2 and 1.5MeV

and, in addition, lines at 6.1, 6.9 and 7.1MeV.

The upper part of Fig. 3 shows the measured

pulse height distribution. The resolution for a

pulse height at the COMPTON edge amounts to

about 400 keV (6%) [51] (full width at half

maximum). The unfolded results are shown in

the lower part of Fig 3, where it can be seen that

lines of FWHM of about 50 keV (0.8%) could be

resolved in the unfolded photon spectrum.

The resolution of the unfolded peaks of Fig. 3

depends strongly on the uncertainties of the

measured pulse height spectrum and on the

resolution of the detector used. The spectra (a)

and (b) of Fig 4 were obtained from the same pulse

height spectrum of Fig. 3 but with an uncertainty

of the multichannel counts assumed to be two or

four times higher. In the lower curve (c) the results

Fig. 3. Experimental pulse height spectrum (a) with properly

scaled abscissa and unfolded photon spectrum (b) in the high-

energy photon field of the PTB.


of a simulation are shown, where the resolution of

the detector was assumed to be twice the resolu-

tion of the detector from Fig. 3.

There was good agreement between the un-

folded photon spectra and the spectra determined

previously with a germanium detector [50]. In-

tegrals over peak areas were reproduced within the

range of uncertainties [51].

The uncertainty analysis with the MIEKE code

results in large uncertainties of the fluence in the

individual energy groups, but the results of

adjacent groups are strongly anti-correlated, as a

result of the ambiguity part SF1; this leads to muchsmaller uncertainties of fluence integrals over

broader energy groups.

7. Eigenvalues and limits of unfolding

It has been shown in previous papers [1,53] that

there is a criterion to quantify the amount of

information available in an unfolding process. The

number of non-vanishing eigenvalues of the matrix

B (Eq. (5)) determines the amount of ambiguity,

or, which is equivalent, the number of independent

parameters available in the solution spectrum.

Values near zero of the ratio r (ratio of an

individual eigenvalue to the largest one) cause

high uncertainties in the solution spectrum and are

related to the ambiguity involved. Non-negative

eigenvalues with low ratio r near the floating pointprecision of the computer may in addition lead to

numerical errors in unfolding.

To quantify the resulting precision of an

unfolding process, the number nl of eigenvalues,

with a ratio r above a certain limit (e. g. r > 10�4),

should be compiled. nl might be compared with

the maximum possible rank nr (number of neutrongroups) of the matrix B. For the Bonner sphere

example of Fig. 1, nl ¼ 5 ðnr ¼ 52Þ has been

obtained, which means that in principle only five

‘‘parameters’’ of the solution spectrum can be

determined and that a priori information is

required for successful unfolding.

For the NE-213 example, strong dependency of

nl on the resolution function was found. The data

used for Figs. 3 and 4 lead to nl ¼ 70 ðnr ¼ 278Þfor unfolding with the ‘‘ideal’’ response function

(ideal resolution), to nl ¼ 45 ðnr ¼ 278Þ for the

experimental resolution available (used in Fig. 3)

and to nl ¼ 26 ðnr ¼ 278Þ for the resolution used inFig 4c. nl does not depend on the uncertainties of

the measured values z0: The loss of resolution in anunfolding process due to ‘‘poor statistics’’ is a

consequence of the principle of maximum entropy

(from all possible solutions, those of minimum

curvature are taken).

The ambiguity part SF1 always leads to large

uncertainties but strong anti-correlations of the

fluence values in adjacent energy groups so that

integral values can be obtained with much lower

uncertainty. Recently attempts have been made to

unfold the full energy- and angle-dependent

differential neutron fluence from the measure-

ments of six electronic dosimeters mounted on a

sphere [53]. High uncertainties were obtained for

the unfolded fluence, since nl ¼ 13 ðnr ¼ 450Þ wasfound. [53]. However, integral values like the

personal dose equivalent could be compiled after

unfolding with a reasonably small uncertainty [55].

8. Summary

In the present paper, it has been tried to review

some of the unfolding algorithms used in neutron

Fig. 4. (a) and (b): Broadening of the lines due to higher

uncertainties, (c): due to increased resolution parameters.

Results obtained with the MIEKE code.


physics and to quantify the amount of information

involved in an unfolding process. As a first attempt

the ambiguity part due to the underdetermination

was analysed by means of the MIEKE code. On

the basis of the results of the discussion of the

algorithms and the examples presented here, the

following statements can be made:

Unfolding of measured detector readings in

particle spectrometry often constitutes a system of

ill-conditioned normal equations, even in the case

M > N: The matrix to be inverted might be

numerically unstable, in particular for Gaussian

broadened response functions.

Unfolding codes should not be used as ‘‘black

boxes’’. Some experience is required, and it is

recommended to use more than one of the codes

mentioned here. In the HEPRO package [13], the

GRAVEL, MIEKE and UNFANA codes may be

used in succession.

References

[1] M. Matzke, G. Vourvopoulos (Ed.), International Con-

ference: Neutrons in Research and Industry, Crete, Greece,

1996, Proc. of SPIE 2867 (1997), 598–607, ISBN 0-8194-

2263-0.

[2] K. Weise, M. Matzke, Nucl. Instr. and Meth. A 280 (1989)

103–112.

[3] J.T. Routti, J.V. Sandberg, Rad. Prot. Dosim. 10 (1985)

103–110.

[4] K. Weise, Report PTB-N-24, Physikalisch-Technische

Bundesanstalt, Braunschweig, 1995.

[5] J. Pulp!an, M. Kr!al!ık, Nucl. Instr. and Meth. A 325 (1993)

314–318.

[6] V.B. Anykeyev, A.A. Spiridonov, P. Zhigunov, Nucl.

Instr. and Meth. A 325 (1993) 314–318.

[7] F.G. Perey, Report ORNL/TM-6062, Oak Ridge National

Laboratory, Oak Ridge, 1978.

[8] F. Schmittroth, Nucl. Sci. Eng. 72 (1979) 19–34.

[9] M. Matzke, K. Weise, Nucl. Instr. and Meth. A 234 (1985)

324–330.

[10] M. Tichy, Nucl. Instr. and Meth. A 267 (1988) 502–508.

[11] A.V. Alevra, B.R.L. Siebert, A. Aroua, M. Buxerolle, M.

Grecescu, M. Matzke, C.A. Perks, H. Schraube, D.J.

Thomas, H.L. Zaborowski, Report PTB-7.22-90-1, Physi-

kalisch-Technische Bundesanstalt, Braunschweig, 1990.

[12] W.H. Press, B.P. Flannery, W.T. Vetterling, Numerical

Recipes, Cambridge University, Cambridge, 1986.

[13] M. Matzke, Report PTB-N-19, Physikalisch-Technische


[14] F.W. Stallmann, Report NUREG/CR-4349, ORNL/TM-

9933, Oak Ridge National Laboratory, Oak Ridge, 1986.

[15] W.N. McElroy, S. Berg, T. Crockett, R.G. Hawkins,

Report AFWL-TR-67-41, US Air Force Weapons La-

boratory, 1967.

[16] W.L. Zijp, E.M. Zsolnay, H.J. Nolthenius, E.J. Szondi,

Report ECN-212; IAEA-NDS-212, Netherlands Energy

Research Foundation, Petten, 1988.

[17] R.E. Maerker, B.L. Broadhead, J.J. Wagschal, Nucl. Sci.

Eng. 91 (2) (1986) 137–170.

[18] M. Tichy, Report PTB-7.2-93-1, Physikalisch-Technische


[19] N. Kocherov, Neutron metrology file NMF-90, Report

IAEA-NDS-171, International Nuclear Energy Agency,

Vienna, 1996 (the MINCHI code is a revised version of

MSITER).

[20] J.T. Routti, J.V. Sandberg, Comput. Phys. Commun. 21

(1980) 119–144.

[21] K.A. Lowry, T.L. Johnson, NRL Report 5340, Naval

Research Laboratory, Washington DC, 1984.

[22] J. Weese, Comput. Phys. Commun. 69 (1992) 99–111.

[23] W.R. Burrus, J.D. Drischler, Report ORNL-4154, Oak

Ridge, 1965.

[24] R.H. Johnson, B.W. Wehring, Report ORNL/RSIC-40,

Oak Ridge, 1976, pp. 33–39.

[25] D.W. Freeman, D.R. Edwards, A.E. Bolon, Nucl. Instr.

and Meth. A 425 (1999) 549–576.

[26] B. Mukherjee, Nucl. Instr. and Meth. A 432 (1999) 305–

312.

[27] D.J.C. MacKay, Nucl. Instr. and Meth. A 354 (1995)

73–80.

[28] E. Cordes, G. Fehrenbacher, R. Schuetz, M. Sprumck, K.

Hahn, R. Hofmann, J.P. Biersack, W. Wahl, IEE Trans.

Nucl. Sci. 45 (1998) 1464–1469.

[29] C.C. Braga, M. S. Dias, Nucl. Instr. and Meth. A 476

(2002) 252, these proceedings.

[30] C.A. Oster, in: Current Status of Neutron Spectrum

Unfolding, Proc. of a Technical Committee Meeting,

Oak Ridge 1977, Report IAEA-TECDOC-221, Vienna

1979, pp. 29–44.

[31] ISO Guide, Guide to expression of uncertainty in

measurement, International Organisation for Standardiza-

tion, Geneva, 1995.

[32] DIN 1319-4 Grundlagen der Messtechnik, Beuth Verlag,

Berlin, 1999.

[33] P.F. Rose, C.L. Dunford, ENDF-102, Brookhaven Na-

tional Laboratory, 1990.

[34] M. Matzke, W.G. Alberts, E. Dietz, in: H. Farrar, E.P.

Lippincott, J.G. Williams, D.W. Vehar (Eds.), Symposium

on Reactor Dosimetry, ASTMSTP1228, American

Society for Testing and Materials, Philadelphia, 1994,

pp. 310–319.

[35] R. Sanna, K. O’Brien, Nucl. Instr. and Meth. 91 (1971)

573–576.

[36] H. Jeffreys, Theory of Probability, Clarendon Press,

Oxford, 1967.

[37] M. Matzke, Report PTB-FMRB-108, Physikalisch-Tech-

nische Bundesanstalt, Braunschweig, 1986.


[38] M. Reginatto, P. Goldhagen, Report EML-595, Environ-

mental Measurements Laboratory, New York, 1998.

[39] M. Reginatto, P. Goldhagen, S. Neumann, Nucl. Instr.

and Meth. A 476 (2002) 242, these proceedings.

[40] M. Matzke, Rad. Prot. Dosim. 23 (1988) 297–300.

[41] F. d’Errico, Rad. Prot. Dosim. 84 (1999) 55–62.

[42] A.V. Alevra, Radioprotection 34 (1999) 305–333.

[43] B. Wiegel, A. V. Alevra, Nucl. Instr. and Meth. A 476

(2002) 36, these proceedings.

[44] H. Klein, L. Lindborg (Eds.), SSI-Rapport 95-15, Swedish

Rad. Protection Institute, Stockholm, 1995.

[45] H. Kluge, A.V. Alevra, S. Jetzke, K. Knauf, M. Matzke,

K. Weise, J. Wittstock, Rad. Prot. Dosim. 70 (1997) 327–

330.

[46] J. F. Briesmeister (Ed.), Report LA-12625-M, Los Alamos,

1993.

[47] M. Weyrauch, A. Casnati, P. Schillebeecks, Nucl. Instr.

and Meth. A 403 (1998) 442.

[48] G. Dietze, H. Klein, NRESP4 and NEFF4 Monte Carlo

Codes for the Calculation of Neutron Response Functions

and Detection Efficiencies for NE-213 Scintillation Detec-

tors Report PTB-ND-22, PTB Braunschweig, 1982.

[49] W.R. Nelson, H. Hirayama, D.W.O. Rogers, SLAC

Report No 256, Stanford, 1985.

[50] L. B .uermann, S. Ding, S. Guldbakke, H. Klein, T.

Novotny, M. Tichy, Nucl. Instr. and Methods A 332

(1993) 483–492.

[51] T. Novotny, S. Guldbakke, H. Klein, in: G. Vourvopoulos

(Ed.), International Conference: Neutrons in Research and

Industry, Crete, Greece, 1996, Proc. SPIE 2867 (1997),

627–630, ISBN 0-8194-2263-0.

[52] S. Neumann, R. Boettger, S. Guldbakke, M. Matzke, W.

Sosaat, Nucl. Instr. and Meth. A 476 (2002) 353, these

proceedings.

[53] M. Matzke, F. d’Errico, O. Hecker, M. Luszik-Bhadra,

Rad. Prot. Dosim. 85 (1999) 93–97.

[54] K. Knauf, M. Weyrauch, F. Simon, N. Kaldune, Report

PTB-6.42-98-1, Physikalisch-Technische Bundesanstalt,

Braunschweig, 1998.

[55] M. Luszik-Bhadra, F. d’Errico, O. Hecker, M. Matzke,

Nucl. Instr. and Meth. A 476 (2002) 457, these proceed-

ings.


propagation of uncertainties in unfolding procedures

Documents