Nuclear Data Section
International Atomic Energy Agency
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Memo CP-D/680 (Rev.)
Date:
To:
From:

17 May 2011
Distribution
N.Otsuka, S. Kopecky, P Schillebeeckx

Subject:
Reference:

LEXFOR entry for covariances in the AGS format
Memo CP-D/564(Rev.)

IRMM and NDS have recently completed the compilation of neutron transmission
data together with and the covariance information. The data resulted frommeasurements at the GELINA facility (EXFOR 23077) [1]. Two new headings areproposed
Dictionary 24 (Data Headings)
TOF-MIN
Lower boundary of time-of-flightTOF-MAX
Upper boundary of time-of-flight.
In order to fulfill the action A51 of the 2010 NRDC meeting, addition to the LEXFOR“Covariance” is proposed below. Headings ERR-1, ERR-2 etc. will be interpreted as“AGS vector” when CHLSK (Cholesky) is coded under the keyword COVARIANCE.

In the example adopted in the LEXFOR input, the uncertainty in normalization factor(0.5%) is not treated as coded information under the MONIT-ERR, because the coded
total uncertainty (ERR-T) does not include this component. If one receives the total
uncertainty including the normalization uncertainty, the normalization uncertainty
must be coded information under the heading MONIT-ERR.

Covariance
Definition
For a measured quantity at two points σ i and σ j (e.g., cross section at two incident
energies; i,j=1,2,…,m), covariance between them are defined as(1)
cov(σ i , σ j ) = (σ i − σ i ) σ j − σ j = σ iσ j − σ i σ j()

If the cross section depends on p parameters (source of uncertainties) {x k }(k=0,1,2,…,p),p∂σ(2)
σ i − σ i = ∑ ki xik − xik ,k = 0 ∂x i
Eq.(1) can be rewritten aspp
∂σ j
∂σ j
∂σ∂σ i=∆⋅∆⋅+cov(σ i , σ j ) = ∑ ki cov xik , x lj
cov xik , x lj. (3)
σσ
δ
∑
ij
0 i0 jlk∂x j∂x lj
k ,l = 0 ∂x i
k ,l =1 ∂x i()()()

where k,l = 0 gives the uncorrelated uncertainty Δ 0 σ i =(∂σ i /∂x0)Δx i 0.and δ ij is theKronecker’s delta.
…
Cholesky Decomposition of Covariance Matrix
Eq.(3) is expressed as(4)V = MM t + S α Vα S αtkkl(V=cov(σ i , σ j ), M={Δ 0 σ i }, S α ={∂σ i /∂x i }, V α =cov(x i , x j ))The p×p matrix V α =cov(x i k, x j l) is positive definite and symmetric, and therefore
there is a matrix L which satisfies V α =LLt (Cholesky decomposition):(5)
V = MM t + Sα LLt Sαt = MM t + Dα Dαt ,where D α =S α L is a m×p matrix. The ij-th component of the Eq.(5) ispcov(σ i , σ j ) = ∆ 0σ i ⋅ ∆ 0σ j ⋅ δ ij + ∑ ∆ k σ i ⋅ ∆ k σ j .

(6)

k =1

(D α ={Δ k σ i }, k=0, 1,2,…,p and i=1,2,…,m). Therefore the m×m covariance matrix V
can be expressed by p sets of the m-dimension vectors {Δ k σ i }. This vector (AGSvector [2,3]) gives a compact expression of the covariance matrix when m is veryhuge (e.g., high resolution time-of-flight spectra).
If the covariance matrix cov(x i k, x j l) is simplified to Δx i kΔx j l δ kl (e.g. cor(x i k, x j l)=δ kl
for any i and j), the summation of the last term of Eq.(3) is simplified to thesummation of (∂σ i /∂x i k)Δx i k (∂σ j /∂x j l) Δx j lδ kl , and Δ k σ i in Eq.(6) becomes the k-th
partial uncertainty of the quantity σ i by comparing Eq.(3) and Eq.(6).
Though AGS vector {Δ k σ i } cannot be interpreted as the partial uncertainty of thequantity σ i due to the parameter x k in general, the vectors are coded in the same way,and given under headings ERR-1, ERR-2,.... with the code CHLSK under the keyword
COVARIANCE.

Example:
REACTIONMISC-COL(48-CD-0(N,TOT),,TRN)(MISC1) Width of time-of-flight bin(MISC2) Uncorrelated uncertainty squared
CORRECTION Data are corrected for- dead time at sample-in detector(din);- background at sample-in detector (Bin);- dead time at sample-out detector (dout);- background at sample-out detector (Bout),
as follows:
T=N(din*Cin-Bin)/(dout*Cout-Bout), whereN: Normalization factor (N=1 +/- 0.5%)
T:Transmission;
Cin: Count at sample-in detector;
Cout: Count at sample-out detector.
Backgrounds were expressed by constant plus sum oftwo exponential: a0+a1*exp(-b1*tof)+a2*exp(-b2*tof)
ERR-ANALYS ERR-1 to ERR-4 gives "correlation vectors" in theAGS format.
(ERR-T) Total uncertainty (sigma)(ERR-S) Total uncorrelated uncertainty (sigma)(ERR-1) Correlated component due to dead timecorrection (sample-in)(ERR-2) Correlated component due to backgroundCorrection (sample-in)(ERR-3) Correlated component due to dead timecorrection (sample-out)(ERR-4) Correlated component due to background
Correction (sample-out)
COVARIANCE (CHLSK) Compiled in ERR-1 to ERR-4 in the AGS format
ENDBIBNOCOMMONDATA
EN
TOF-MINTOF-MAXMISC1DATA
ERR-T
ERR-S
MISC2
ERR-1
ERR-2
ERR-3
ERR-4
EV
NSEC
NSEC
NSEC
NO-DIM
NO-DIM
NO-DIM
NO-DIM
NO-DIM
NO-DIM
NO-DIM
NO-DIM
4.79930E+0873301.2
873429.2
128.0 1.14780E+0 6.65562E-2
6.65376E-2 4.42725E-3 1.33447E-5-9.12646E-4-1.37145E-5 1.28552E-3
4.79790E+0873429.2873557.2
128.0 9.70250E-1 5.63176E-2
5.63021E-2 3.16993E-3 1.09942E-5-8.47529E-4-1.16680E-5 1.00913E-3…

In this example, ERR-S gives Δ 0 σ i and ERR-1 to ERR-4 give Δ k σ i (k=1,2,3,4,i=1,2,…,m) in Eq.(6).
References[1] S. Kopecky, I. Ivanov, M. Moxon, P. Schillebeeckx, P. Siegler, I. Sirakov, “The
total cross section and resonance parameters for the 0.178 eV resonance of 113Cd”,
Nuclear Instruments and Methods B 267 (2009) 2345 - 2350
[2] C. Bastian, “General procedures and computational methods for generating
covariance matrices”, Proc. Int. Symposium on Nuclear Data Evaluation
Methodology, Brookhaven National Laboratory, USA, October 12-16, 1992,
p642.
[3] N. Otuka, A. Borella, S. Kopecky, C. Lampoudis, P. Schillebeeckx, “Database fortime-of-flight spectra including covarainces”, Proc. Int. Conf. on Nucl. Data for
Sci .and Technol., Jeju island, Korea, April 26-30, 2010.

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