Nuclear Data Section
International Atomic Energy Agency
P.O.Box 100, A-1400 Vienna, Austria
Memo CP-D/564
Date:
To:
From:

20 May 2009
Distribution
N. Otsuka

Subject:

Compilation of covariance matrix given in the AGS format

Compilation of experimental covariance matrix for transmission and reaction yield
will be proposed in the NRDC 2009 meeting by Drs. Stefan Kopecky and Peter
Schillebeeckx (EC-JRC-IRMM, Geel). By using their expression of covariance matrix- AGS (Analysis of Geel Spectra) - total size covariance matrix is reduced from n2 ton(k+1), where n is the total number of TOF (time of flight) channel (~104). A note forthis formalism in Appendix 3 might be helpful for discussion.
They propose compilation of matrices D and S defined in Eqs.10 and 11 in the note asa function of TOF. I prepared a coding sample (Appendix 2) according to theirtemplate for EXFOR compilation (Appendix 1). Some possible technical questions
are summarized below:- Can we add TOF (in second) as an independent variable instead of incident energy?
- Can we allow negative values under ERR-n? Sensitivity coefficient in Eq. 12 (∂σi /∂xl) can be negative.
- We do not need new headings for uncertainties in the coding sample because the sizeof covariance matrices of systematic effect sources (defined in matrices W in Eq. 7)
is 1 x 1 for all 4 sources, and the matrix S then gives usual partial uncertainty (SeeEq.12). Can we still regard the matrix S as partial uncertainty (ERR-n) if W is larger
than 2 x 2 ?

Appendix 1: EXFOR template from Geel
1. EXPERIMENT DESCRIPTION
Main Reference
FacilityNeutron production
Primary neutron production target
Time resolution primary beam (ns)Moderator material
Surface Dimensions(mm x mm or diameter in mm)
Thickness (mm)Experimental details
Measurement type
Flight path length (m)(moderator – target (detector): face to face distance)
Angle(with respect to cp beam)
Beam dimensions(mm x mm or diameter in mm)
Detector
TypeMaterial
Surface Dimensions(mm x mm or diameter in mm)
Thickness (mm)SampleType (metal, powder)
Chemical composition
Atomic abundance of main element
WeightWeight per unit area (g/cm2)
Geometry
Surface dimensions(mm x mm or diameter in mm)
Thickness (mm)Thickness of main element (at/b)
Containment description
Uncertainties (at 1 sigma level)
Normalization (relative )Other sources creating correlated uncertainty
components(see data format)

GELINA

12,34Uranium
1 ns
H2O
2 containers 100 x 100 mm
40 mm
Transmission
26.444 +/- 0.006 m
99 degDiameter 40 mm
Scintillator
Li-glass (NE 905)
120 mm diameter
12.7 mm
MetalCd
100 %2.0993 +/- 0.0001 g
0.04172 g/cm2Diameter 80.04 +/- 0.03 mm
0.050 mm (nominal)
2.2352 10-4 at/b
No container
Dead Time correction sample
in [1]
Background correction sample
in parameter one [2]Dead Time correction sample
out [3]Background correction sample
out parameter one [4]References[1] S. Kopecky et al., submitted to NIMB
[2] J.M. Salome and R. Cools, Nucl. Instrum. Methods Phys. Res. A 179 (1981), 13[3] D. Tronc, J.M. Salome, K.H. Böckhoff, Nucl. Instrum. Methods Phys. Res. A 228 (1985), 217[4] M. Flaska et al. Nucl. Instrum. Methods Phys. Res. A 531 (2004), 392

2. DATA FORMAT
According to AGS format
Time-of-Flight[ns]
LowHigh3.1188E+06
3.1444E+06
3.1700E+06
3.1956E+06
3.2212E+06…
6.5116E+06
6.6140E+06
6.7164E+06
6.8188E+06
6.9212E+06
7.0236E+06

3.1444E+06
3.1700E+06
3.1956E+06
3.2212E+06
3.2468E+06…
6.6140E+06
6.7164E+06
6.8188E+06
6.9212E+06
7.0236E+06
7.0246E+06

Observable
Transmission
9.1309E-01
9.0347E-01
9.0134E-01
8.8804E-01
8.8347E-01…
5.1049E-01
5.1780E-01
5.2519E-01
5.3394E-01
5.4117E-01
5.6613E-01

UncertaintyTotal
σ5.2664E-03
5.2812E-03
5.2989E-03
5.2316E-03
5.2576E-03…
1.7897E-03
1.8092E-03
1.8266E-03
1.8541E-03
1.8762E-03
1.9202E-02

Uncorrelated
σ22.7570E-05
2.7719E-05
2.7902E-05
2.7195E-05
2.7461E-05…
3.0786E-06
3.1479E-06
3.2107E-06
3.3090E-06
3.3897E-06
3.6855E-04[1]
6.5601E-06
6.3922E-06
6.3386E-06
6.2074E-06
6.1074E-06…
2.7113E-06
2.7654E-06
2.8231E-06
2.8821E-06
2.9320E-06
3.0999E-06

Correlated Components[2][3]-1.0072E-04-7.4295E-06-1.0226E-04-7.2767E-06-1.0281E-04-7.2228E-06-1.0196E-04-7.1256E-06-1.0321E-04-7.0280E-06…
…
-5.1153E-05-4.0677E-06-5.0203E-05-4.1206E-06-4.9156E-05-4.1785E-06-4.8495E-05-4.2320E-06-4.7730E-05-4.2778E-06-4.8282E-05-4.4270E-06

Appendix 2: Coding sample
ENTRY
SUBENT
BIB
TITLE

00001
20090507
00001001
20090507
1466
The total cross section and resonance parameters for
the 0.178 eV resonance of 113CdAUTHOR(S.Kopecky, I.Ivanov, M.Moxon, P.Schillebeeckx,P.Siegler, I.Sirakov)
INSTITUTE (2ZZZGEL)REFERENCE (J,NIM/B,,,2009) Submitted to NIM/B#doi: 10.1016/j.nimb.2009.04.010
REL-REF(N,,J,NIM/A,179,13,1981)
J.M.Salome+, Details of GELINA facility(N,,J,NIM/A,228,217,1985)
D.Tronc+, Details of GELINA facility(N,,J,NIM/A,531,392,2004)
M.Flaska+, Details of neutron production
FACILITY(LINAC,2ZZZGEL) GELINA
INC-SOURCE Diameter of neutron beam = 40 mm(PHOTO) (g,n) on uranium target(2 containers, 100 x 100 mm, 40 mm thick)(THCOL) Water moderator
METHOD(TRN,TOF) Flight path (moderator - target (detector)
face to face distance) = 26.444 +/- 0.006 m
DETECTOR(SCIN,GLASD) Li-glass scintillator (NE 905)
120 mm diameter, 12.7 mm thick
SAMPLE
Physical type:
Metal
Chemical composition:
Element (Cd)
Purity of main element:
100%
Weight per area:
0.04172 g/cm2Diameter:
80.04 +/- 0.03 mm
Thickness:
0.050 mm (nominal)
Thickness of main element: 2.2352E-04 atom/b
CORRECTION Data are corrected for- dead time at sample-in detector(din);- background at sample-in detecto(Bin);- dead time at sample-out detector (dout);- background at sample-out detector (Bout),
as follows:
T=(din*Cin-Bin)/(dout*Cout-Bout), whereT:
Transmission;
Cin: Count at sample-in detector;
Cout: Count at sample-out detector.
ERR-ANALYS (ERR-T) Total uncertainty(ERR-S) Sum of uncorrelated uncertainties
due to counting statistics

0000100000001
0000100100001
0000100100002
0000100100003
0000100100004
0000100100005
0000100100006
0000100100007
0000100100008
0000100100009
0000100100010
0000100100011
0000100100012
0000100100013
0000100100014
0000100100015
0000100100016
0000100100017
0000100100018
0000100100019
0000100100020
0000100100021
0000100100022
0000100100023
0000100100024
0000100100025
0000100100026
0000100100027
0000100100028
0000100100029
0000100100030
0000100100031
0000100100032
0000100100033
0000100100034
0000100100035
0000100100036
0000100100037
0000100100038
0000100100039
0000100100040
0000100100041
0000100100042
0000100100043
0000100100044
0000100100045[4]9.3673E-05
9.4143E-05
9.4387E-05
9.2229E-05
9.2872E-05…
2.6553E-05
2.6437E-05
2.6257E-05
2.6329E-05
2.6267E-05
2.7783E-05

(sample-in and sample-out)
0000100100046(ERR-1) Uncertainty due to dead-time correction
0000100100047(sample-in)
0000100100048(ERR-2) Uncertainty due to background correction
0000100100049(sample-in)
0000100100050(ERR-3) Uncertainty due to dead-time correction
0000100100051(sample-out)
0000100100052(ERR-4) Uncertainty due to background correction
0000100100053(sample-out)
0000100100054
COVARIANCE If i=j,0000100100055
cov(T_i,T_j) = (ERR-S)i**2 +(ERR-1)i**2+(ERR-2)i**2
0000100100056+(ERR-3)i**2+(ERR-4)i**2
0000100100057,else
0000100100058
cov(T_i,T_j) = (ERR-1)i*(ERR-1)j0000100100059+ (ERR-2)i*(ERR-2)j0000100100060+ (ERR-3)i*(ERR-3)j0000100100061+ (ERR-4)i*(ERR-4)j0000100100062.
0000100100063
HISTORY(20090403R) On. Received by e-mail from S.Kopecky
0000100100064(20090423C) On.
0000100100065(20090507A) On. ERR-ANALYSIS updated,
0000100100066
CORRECTION and COVARIANCE added.
0000100100067(20090520A) On. TITLE added, AUTHOR corrected
0000100100068
ENDBIB
6600000100100069
NOCOMMON000000100100070
ENDSUBENT
690
0000100199999
SUBENT
00001002
20090507
0000100200001
BIB
330000100200002
REACTION(48-CD-0(N,TOT),TRN)
0000100200003
STATUS(TABLE) Data on MS word file0000100200004
ENDBIB
300000100200006
COMMON
230000100200007
TOF-L
TOF-L-ERR
0000100200008
TOF-L(-ERR): flight path length (in meter)
M
M
0000100200009
TOF-MIN/MAX: time-of-flight (nano-second) 0000100200010
26.4440.006
ENDCOMMON
300000100200011
DATA
91010000100200012
TOF-MINTOF-MAX
DATA
ERR-T
ERR-S
ERR-1
0000100200013
ERR-2
ERR-3
ERR-4
0000100200014
NSNS
NO-DIM
NO-DIM
NO-DIM
NO-DIM
0000100200015
NO-DIM
NO-DIM
NO-DIM
0000100200016
3.1188E+06 3.1444E+06 9.1309E-01 5.2664E-03 5.2507E-03 6.5601E-060000100200017-1.0072E-04-7.4295E-06 9.3673E-05
0000100200018
3.1444E+06 3.1700E+06 9.0347E-01 5.2812E-03 5.2649E-03 6.3922E-060000100200019-1.0226E-04-7.2767E-06 9.4143E-05
0000100200020
3.1700E+06 3.1956E+06 9.0134E-01 5.2989E-03 5.2822E-03 6.3386E-060000100200021-1.0281E-04-7.2228E-06 9.4387E-05
0000100200022
3.1956E+06 3.2212E+06 8.8804E-01 5.2316E-03 5.2149E-03 6.2074E-060000100200023-1.0196E-04-7.1256E-06 9.2229E-05
0000100200024…
6.7164E+06 6.8188E+06 5.2519E-01 1.8266E-03 1.7918E-03 2.8231E-060000100200211-4.9156E-05-4.1785E-06 2.6257E-05
00001002002126.8188E+06 6.9212E+06 5.3394E-01 1.8541E-03 1.8191E-03 2.8821E-060000100200213-4.8495E-05-4.2320E-06 2.6329E-05
0000100200214
6.9212E+06 7.0236E+06 5.4117E-01 1.8762E-03 1.8411E-03 2.9320E-060000100200215-4.7730E-05-4.2778E-06 2.6267E-05
0000100200216
7.0236E+06 7.0246E+06 5.6613E-01 1.9202E-02 1.9198E-02 3.0999E-060000100200217-4.8282E-05-4.4270E-06 2.7783E-05
0000100200218
ENDDATA
206
00000100200219
ENDSUBENT
218
00000100299999
ENDENTRY
200000199999999
ENDTRANS
10Z999999999999

Appendix 3: Quantitites given in the AGS format

Note: A compact expression of experimental covariance
Naohiko Otuka
IAEA Nuclear Data Section
This is a note for compact expression of experimental covariance (AGS format)[1, 2].

Formalism
Let us consider a quantity at i-th incident energy bin σi (1 ≤ i ≤ n) as a function of M uncorrelatedquantities yim (1 ≤ m ≤ M ) and K × L correlated quantities xlr (1 ≤ l ≤ L, 1 ≤ r ≤ Kl , K =K1 + K2 + ... + KL ):σi ≡ σ({yim }, {xlr }).
(1)
The total uncertainty of σi is decomposed to uncorrelated and correlated terms(∆σi )2tot =MX
m=1=MX
m=1(∆σi )2m +ÃLX(∆σi )2l(2)l=1!2!µÃ¶KlL XX∂σi∂σi∂σimll∆y+∆xq∆xr clqr
ll∂yim i∂x∂xqr
l=1 q,r=1(3)where clqr is the correlation matrix of the quantity xl . We assume that there is no correlation between xkq andxlr if k 6= l.
The covariance matrix of σi is then
Vij=MX
m=1=(∆σi )m (∆σj )m δij +LXl(∆σi )l (∆σj )l Cij

l=1KLlX XMX∂σi ∂σj∂σi ∂σj l
Uijm +Wmm∂yi ∂yj∂xlq ∂xlr qr
m=1
l=1 q,r=1(V, U ∈ Rn×n , W ∈ RKl ×Kl )

(4)
(5)

, where two covariance matrices
Uijm ≡ (∆yim )2 δij ,(6)

lWqr

(7)

≡∆xlq ∆xlr clqr

are introduced (δij : Kronecker’s delta).
The matrix W l is positive definite and symmetric and, therefore, can be decomposed to an upper triangle
Tmatrix R (R ∈ RKl ×Kl ) and its transposed matrix RT as W l = Rl Rl (Cholesky decomposition). ThenVij=MX∂σi ∂σjmm Uij +∂yim ∂yj
m=1= Dij +KlL X
Xl lSipSjp

l=1 p=1

1KlLXX∂σi ∂σj l lR R
∂xlq ∂xlr qp pr
l=1 p,q,r=1(8)(9)

, whereDijlSip≡≡MX∂σi ∂σjMX∂yim ∂yj
m=1m=1mm Uij = δijKlX∂σi∂xlq
q=1lRqp(∆σi )2m(D ∈ Rn×n ,diagonal),(S l ∈ Rn×Kl )

(10)
(11)

. The covariance matrix V is now expressed by the n elements of the diagonal matrix D and n × K elementsof the matrices S l (1 ≤ l ≤ L). This reduces total number of elements from√ n × n to n × (1 + K)
lll
2llll
If Kl =1 for all l, Wqr → W = (∆x ) , cqr → c = 1, Rqp → R = W l = ∆xl , andl
Sip
→ Sil =∂σ i∆xl ≡ (∆σi )l ,∂xl

(12)

which sigature is positive or negative. Therefore the covariance matrix and its diagonal component areVij= δijMX(∆σi )2m +m=1MX(∆σ)2tot = Vii =LX(∆σi )l (∆σj )l(13)l=1(∆σi )2m +m=1LX(∆σi )2l(14)l=1

EXFORl (1 ≤ l ≤ L, 1 ≤ p ≤We need one column for Dii (1 ≤ i ≤ n) and K = K1 + ... + KL columns for Sip
Kl , 1 ≤ i ≤ n).
If Kl = 1 for all L and the incident energy Ei is known, we can use the current EXFOR format:

ENE1E2...
Ei...
EnERR-SDii(∆σ1 )0(∆σ2 )0...
(∆σi )0...
(∆σn )0ERR-1Si1(∆σ1 )1(∆σ2 )1...
(∆σi )1...
(∆σn )1...
...
...
...
...
...
...
...

ERR-LSiL(∆σ1 )L(∆σ2 )L...
(∆σi )L...
(∆σn )L, where we allow to use the heading ERR-S for total uncorrelated uncertainty(∆σi )20 =MX(∆σi )2m(15)m=1.

Example - Transmission
Transmission T (,TRN in EXFOR) is related to total cross section σT as follows:
T (E) = exp (−LρσT (E))
(L: target thickness, ρ: target density), and expressed in terms of the 6 observables• Ciin : Count at sample-in detector (uncorrelated)2(16)• Ciout : Count at sample-out detector (uncorrelated)• din : Dead time correction for sample-in detector (correlated)• dout : Dead time correction for sample-out detector (correlated)• B in : Background correction for sample-in detector (correlated)• B out : Background correction for sample-out detector (correlated)
as follows:
Ti = Ti (Ciin , Ciout , din , dout , B in , B out ) =din Ciin − B in
dout Ciout − B out(17)

This is the case of M = 2, L = 4 and Kl = 1 (1 ≤ l ≤ 4) in the formalism. Covariance matrices foruncorrelated quantities are³

UijCin = ∆Ciin´2³δij , UijCout = ∆Ciout´2δij ,(18)while covariance matrices for correlated quantities and these Cholesky decompositions are³

W din = ∆dini´2³, W dout = ∆douti´2³, W Bin = ∆Biin´2³, W Bout = ∆Biout´2(19)

andRdin = ∆din , Rdout = ∆dout , RBin = ∆B in , RBout = ∆B out .

(20)

Therefore the covariance matrix V is expressed by n elements of the matrix DDij=∂Ti ∂Tj Cin∂Ti ∂Tj Cout
Uij +Uinin∂Ciout ∂Cjout ij∂Ci ∂Cj= (∆Ti )2Cin δij + (∆Ti )2Cout δij

(21)
(22)

and n × 4 elements of S matrices:
Sidin =
Sidout =SiBin =SiBout =∂Ti din∂Ti
R = in ∆din ≡ (∆Ti )dinin∂d∂d∂Ti dout∂TiR= out ∆dout ≡ (∆Ti )doutout∂d∂d∂Ti∂Ti Bin
R=∆B in ≡ (∆Ti )Bin
∂B in
∂B in∂Ti Bout
∂TiR=∆B out ≡ (∆Ti )Boutout∂B∂B out

(23)
(24)
(25)
(26)

Covariance matrix can be reconstructed from these elements as follows:
Vij= Dij Sidin Sjdin + Sidout Sjdout + SiBin SjBin + SiBout SjBout(27)= (∆Ti )2Cin δij + (∆Ti )2Cout δij+ (∆Ti )din (∆Tj )din + (∆Ti )dout (∆Tj )dout + (∆Ti )Bin (∆Tj )Bin + (∆Ti )Bout (∆Tj )Bout (28). The diagonal element Vii gives the total uncertainty of T i(∆Ti )2tot = (∆Ti )2Cin + (∆Ti )2Cout + (∆Ti )2din + (∆Ti )2dout + (∆Ti )2Bin + (∆Ti )2Bout
This can can be compiled as follows
3(29)

EN
E1E2...
Ei...
En, where

ERR-S(∆T1 )C(∆T2 )C...
(∆Ti )C...
(∆Tn )CERR-1(∆T1 )din(∆T2 )din...
(∆Ti )din...
(∆Tn )din...
...
...
...
...
...
...

ERR-4
(∆T1 )Bout(∆T2 )Bout...
(∆Ti )Bout...
(∆Tn )Bout(∆Ti )2C = (∆Ti )2Cin + (∆Ti )2Cout(30)References[1] C.Bastian, Proceedings of the International Symposium on Nuclear Data Evaluation Methodology,
Upton, US, 1992, p.642 (1993).
[2] C.Bastian, Proceedings of the International Conference on Neutron Research and Industry, Crete,
Greece, 1996, p.611 (1997).

4

Distribution:
a.mengoni@iaea.org
blokhin@ippe.ru
chiba@earth.sgu.ac.jp
claes.nordborg@oecd.org
ganesan@barc.gov.in
gezg@ciae.ac.cn
hasegawa@nea.fr
henriksson@near.fr
hongwei@ciae.ac.cn
jhchang@kaeri.re.kr
kaltchenko@kinr.kiev.ua
katakura.junichi@jaea.go.jp
kato@nucl.sci.hokudai.ac.jp
kirarlyb@atomki.hu
l.vrapcenjak@iaea.org
manuel.bossant@oecd.org
manokhin@ippe.ru
mmarina@ippe.ru
mwherman@bnl.gov
cc:stefan.kopecky@ec.europa.eu
peter.schillebeeckx@ec.europa.eu

nicolas.soppera@oecd.org
nklimova@kinr.kiev.ua
n.otsuka@iaea.org
nrdc@jcprg.org
oblozinsky@bnl.gov
ogritzay@kinr.kiev.ua
otto.schwerer@aon.atsamaev@obninsk.ru
s.babykina@polyn.kiae.su
s.dunaeva@iaea.org
stakacs@atomki.hu
stanislav.hlavac@savba.sk
taova@exped.vniief.ru
tarkanyi@atomki.hu
varlamov@depni.sinp.msu.ru
vlasov@kinr.kiev.ua
vmclane@optonline.net
v.zerkin@iaea.org
yolee@kaeri.re.kr