A-priori and A-posteriori Covariance Data in Nuclear Cross Section Adjustments:
Issues and Challenges
G. Palmiotti,
1, ∗
M. Salvatores,
2, 1
and G. Aliberti
3
1
Idaho National Laboratory, Idaho Falls, ID 83415, USA
2
Consultant
3
Argonne National Laboratory, Argonne, IL 60439, USA
(Received 22 June 2014; revised received 12 August 2014; accepted 14 August 2014)
In order to provide useful feedback to evaluators a set of criteria are established for assessing
the robustness and reliability of the cross section adjustments that make use of integral experiment
information. Criteria are also provided for accepting the “a posteriori” cross sections, both as new
“nominal” values and as “trends”. Some indications of the use of the “a posteriori” covariance
matrix are indicated, even though more investigation is needed to settle this complex subject.
I. INTRODUCTION
The role for cross section adjustment is more and more
perceived as that of providing useful feedback to eval-
uators and differential measurement experimentalists in
order to improve the knowledge of neutron cross sections
to be used in a wider range of applications. This new role
for cross section adjustment requires tackling and solving
a new series of issues: definition of criteria to assess the
reliability and robustness of an adjustment; requisites to
assure the quantitative validity of the covariance data;
criteria to alert for inconsistency between differential and
integral data; definition of consistent approaches to use
both adjusted data and “a posteriori” covariance data to
improve quantitatively nuclear data files. In order to sat-
isfy these requirements, several methodology issues will
be illustrated in the following with practical examples.
Among them:
• Assessment of adjustments;
• Definition of criteria to accept new central values
of cross sections after adjustments;
• Avoid compensation among different input data in
the adjustments;
• Validation of the “a priori” and use of the “a pos-
teriori” covariance matrix.
∗
Corresponding author: Giuseppe.Palmiotti@inl.gov
II. ADJUSTMENT ASSESSMENT
In this section we will introduce several criteria to as-
sess the robustness of an adjustment as well parameters
helpful to select experiments to be used in the adjust-
ment. However, before proceeding to the assessment we
need to define several variables that intervene in the ad-
justment formulation. Therefore, if we use in the adjust-
ment N
E
integral parameters and N
σ
neutron cross sec-
tions, we define the following quantities: E
i
(i =1,N
E
):
Experimental value of measured integral parameter i;
C
i
(i =1,N
E
): “a priori” calculated value of integral pa-
rameter i; C
i
′ (i =1,N
E
): “a posteriori” calculated value
of integral parameter i; σ
j
(j =1,N
σ
): “a priori” cross
sections; σ
j
′ : “a posteriori” cross sections; S
ij
: Sensitiv-
ity coefficients for integral parameter i and cross section
j ; M
EC
=(M
E
+ M
C
): integral parameter covariance
matrix; M
E
: integral parameter covariance matrix due
to measurement covariance; M
C
: integral parameter co-
variance matrix due to calculation covariance; M
σ
: “a
priori” cross section covariance matrix; M
σ
′ : “a poste-
riori” cross section covariance matrix; χ
2
: “a priori” chi
square; χ
2
: “a posteriori” chi square; I : unity matrix;
G =(M
EC
+ SM
σ
S
T
): total integral covariance matrix.
With this nomenclature we can define the main adjust-
ment formulas starting from the χ
2
before adjustment [1]
χ
2
=(σ
− σ)
T
M
−1
σ
(σ
− σ)+(E − C)
T
M
−1
EC
(E − C).
(1)
The cross sections modifications that minimize the χ
2
thanks to the adjustment are calculated as
σ
− σ = M
σ
S
T
G
−1
(E − C), (2)
and the associated “a posteriori” cross section covariance
Available online at www.sciencedirect.com
Nuclear Data Sheets 123 (2015) 41–50
0090-3752/© 2014 Elsevier Inc. All rights reserved.
www.elsevier.com/locate/nds
http://dx.doi.org/10.1016/j.nds.2014.12.008