# Assesment of covariance processing with GAIA for nuclear data uncertainty propagation

Pierre Sole, Vaibhav Jaiswal, Cédric Jouanne, Vivian Salino

► To cite this version:

Pierre Sole, Vaibhav Jaiswal, Cédric Jouanne, Vivian Salino. Assesment of covariance processing with GAIA for nuclear data uncertainty propagation. 6th International Workshop on Nuclear Data Evaluation for Reactor Applications, WONDER 2023, Jun 2023, Aix-en-provence, France. pp.05001, 10.1051/epjconf/202429405001. irsn-04890549

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# Assesment of covariance processing with GAIA for nuclear data uncertainty propagation

Pierre Sole<fnref n="1" />,<fnref n="2" />,* , Vaibhav Jaiswal<fnref n="1" />, Cédric Jouanne<fnref n="2" />, and Vivian Salino<fnref n="1" />

**Abstract.** Nuclear data uncertainties are provided as covariance matrices in standard nuclear data libraries and propagating them trough neutronics simulations helps quantify the associated uncertainties on the final result. However, processing these matrices often poses challenges. Currently, the IRSN nuclear data processing code GAIA processes cross sections via several modules like DOP (Reconstruction and Doppler), TOP (URR), and SAB (TSL), but lacks the capability to process covariances. This paper introduces a new module named COP (COvariance Processing). The COP module aims to process covariance matrices comprehensively, including cross section (File 33), angular distribution (File 34), and resonance parameters (File 32). The preliminary results obtained using the COP module of GAIA in comparison with the ERRORR module of NJOY and PUFF module of AMPX are presented.

## 1 Introduction

Neutronics simulations are crucial for designing and analyzing nuclear systems. These simulations rely on nuclear data libraries that encompass a wide range of physical parameters, including cross sections, decay constants, and fission yields. However both the measurement and theoretical calculation of nuclear data inherently introduce uncertainties that must be rigorously considered in these simulations. Covariance matrices are used to represent nuclear data uncertainties and their inter-dependencies.

The GAIA code is developed by IRSN in order to process nuclear data libraries for neutronics simulations. GAIA already embeds a first module that performs both cross section reconstruction from resonance parameters and Doppler broadening DOP [1]. A second module produces probability tables for the unresolved resonance range TOP [2]. In our ongoing pursuit of enhanced uncertainty analysis, the COP module (COvariance Processing) within GAIA has been developed to process covariance matrices.

This paper aims to convey key concepts of nuclear data uncertainty processing, offering insights into both the theoretical foundations and the practical implementation within the GAIA code. We begin with a detailed examination of the processing of cross-section covariances, as encapsulated in File 33 of the ENDF-6 format [3]. Subsequently, we delve into the complexities of angular distribution uncertainties for secondary particles, encapsulated in File 34 of the same format.

<footnote n="1">¹PSN-RES/SNC/LN, Institut de Radioprotection et de Sûreté Nucléaire</footnote>

<footnote n="2">²Université Paris-Saclay, CEA, Service d'Etudes des Réacteurs et de Mathématiques Appliquées,91191, Gif-sur-Yvette, France</footnote>

<footnote>*Corresponding author : pierre.sole@irsn.fr</footnote>
# 2 Processing of covariance matrices

## 2.1 Processing of cross sections covariances : MF33

In this work, we have extended the capabilities of the GAIA C++ parser to construct a comprehensive data structure from the uncertainty information found in ENDF evaluations, following the guidelines outlined in the ENDF-6 format manual [3].

ENDF-6 manual stipulates that tabulated cross sections are located within the so-called FILE3 (MF3), and the corresponding uncertainties are stored in FILE33 (MF33).

In MF33, evaluators enter covariances into a list of values conventionally named 'F'. The LB flag specifies how to interpret this list, particularly whether F represents a relative or absolute covariance matrix. Let $X_i$ be a nuclear data at energy $E_i$ and $Y_j$ another data type at energy $E_j$. The covariance between $X_i$ and $Y_j$ is calculated using the formula:

$$ \mathrm{Cov}(X_i, Y_j) = g^{\mathrm{LB}}(F, X_i, Y_j) \qquad (1) $$

The function we denoted "g" in Eq. 1, combines values stored in "F" with groupwise data in a manner determined by the LB flag, purposely to reconstruct the full covariance matrix. All possible LB flags as listed in the ENDF manual have been implemented in COP.

A significant proportion of evaluations include resonance parameters, which are stored in FILE2 (MF2) along with their associated uncertainties in FILE32 (MF32). These resonance parameters are primarily utilized for the reconstruction of specific cross sections, typically (n,n), (n,γ) and fission reactions, within a specified energy range. Methodologies exist to propagate uncertainties from MF32 into MF33 representation. Specific details and analysis of these propagation methodologies are beyond the scope of this paper.

## 2.2 Processing of angular distribution of secondary particles : MF34

For relatively low energies, differential elastic-scattering cross sections are commonly expressed as series of Legendre polynomial expansion coefficients in the center-of-mass system. The MF34 section characterizes the covariances between these energy-dependent expansion coefficients. Similar to MF33 and the utilization of groupwise cross sections (Eq. 2), MF34 introduces the calculation of groupwise Legendre coefficients using Eq. 3.

$$ x_g = \frac{\int_g \phi(E)x(E)dE}{\int_g \phi(E)dE} \qquad (2) $$

$$ a_l(g) = \frac{\int_g \phi(E)\sigma_{el}(E)a_l(E)dE}{\int_g \phi(E)\sigma_{el}(E)dE} \qquad (3) $$

One notable distinction between these two formulas is that, in addition to the flux, the grouped Legendre coefficients are weighted by the elastic cross section. This additional factor reflects the intricate dependence on elastic scattering.

Fig. 1 visually illustrates a typical processed absolute covariance matrix (P1-P1) for the first Legendre order of elastic scattering data for $^{56}$Fe taken from JEFF-3.3, showing the covariance between the first Legendre coefficient across different energy groups.

In Fig. 1, the upper curve represents the averaged data, while the lower part visualizes the degree of correlation between the data points. The uncolored entries correspond to null coefficients. Notably for this nucleus, within the energy range marking the end of the resolved resonance region (RRR) at 850 keV, the presence of null covariances indicate the use of a different nuclear model with no correlations to the model used in the RRR.
<figure><img src="image_2.png" /><figcaption>Figure 1: Absolute covariance matrix of $^{56}$Fe (JEFF-3.3) and groupwise coefficients for the First Legendre order of Elastic Scattering</figcaption></figure>

# 3 A comprehensive insight of the COP Module

## 3.1 Uncertainty processing

The initial covariance matrices provided by evaluators are usually defined on specific energy meshes, called *evaluator meshes*, and may vary for each nucleus and reaction. These evaluator meshes may not be optimized for the intended neutronics simulations. Processing main step is to calculate these matrices into a unique user-defined energy mesh, referred to as the **user mesh**.

The general workflow for processing energy-dependent uncertainties in the COP module can be summarized as follows:

Step 1: Create the *union energy mesh* as the combination of the **user mesh** and all the covariance meshes given in one File (MF33 or MF34 for instance).

Step 2: Calculate multigroup data on this *union mesh*.

Step 3: Compute the covariance matrices on the **user mesh** using equation (1).

To achieve Step 2, we have implemented the grouping data formulas as presented in Eq. 2 and 3. This implementation uses a 7-points Gauss quadrature scheme to calculate groupwise Legendre coefficients. These coefficients are then used to process MF34 covariances.

## 3.2 Ensuring Symmetric Positive Definiteness (SPD) of Covariance Matrices

Symmetric Positive Definiteness (SPD) is a fundamental property of covariance matrices, ensuring their physical meaningfulness and usability for uncertainty quantification. In our context, these matrices may be used for multivariate sampling. This sampling employ the Cholesky decomposition, a numerically efficient and stable method. For this reason, ensuring the SPD property is of utmost importance.
We address the positive definiteness issue by performing a singular value decomposition (using the Eigen library [4]) and adjust any negative eigenvalues to be slightly greater than zero, on the order of $10^{-15}$. This ensures that the resulting matrix is positive definite. This adjustment often results in minor changes. COP carefully ensures that the final matrix remains sufficiently close to the initial one by comparing the absolute difference between coefficients of the two matrices to a certain threshold. Should this threshold be exceeded, COP will return a warning message.

# 4 Comparative Analysis of existing Processing Modules

To assess the capabilities of COP, a comparative analysis was conducted with ERRORR and PUFF respective modules of NJOY [5] and AMPX [6]. Modules like ERRORR combine the uncertainty from MF32 and MF33 to derive the final cross section uncertainty. Hence comparisons were performed on the $^{56}$Fe that does not contain MF32 in studied libraries. This absence of MF32 allows us to assess the capabilities of the COP module in handling the specific MF33 type of covariance.

## 4.1 Workflow used to compare COP, PUFF and ERRORR

Group cross sections are necessary in order to compute covariance matrix, but the grouping step requires reconstructed cross sections, a step usually performed by specific modules: RECONR (NJOY), POLIDENT (AMPX) and DOP in (GAIA). In order to perform fair comparisons, the reconstruction step must be done by the same module. In other words we want the exact same linearized cross sections prior to the grouping step. The unique workflow that allows to reach this result is presented here.

The reconstruction step cannot start neither with RECONR nor DOP because these modules output a PENDF file and such a format is not convertible for PUFF. Therefore, starting from a nuclear data evaluation, the AMPX modules POLIDENT and BROADEN reconstruct and Doppler-broaden the cross sections, producing an output that can be used by PUFF. Finally the ERRORR and COP expected files format is assured by AMPX module MAKPEN (MAKe PENDF).

<figure><img src="image_3.png" /><figcaption>Figure 2: Workflow for generating covariance file (GENDF)</figcaption></figure>

By employing this standardized starting point, the focus of the comparison was solely on the capabilities of the covariance processing modules and any differences observed in the output of the various modules could be attributed primarily to the variance in their covariance processing methodologies.

## 4.2 Assessment of similarities and differences in processing techniques

The feature of making the matrix SPD can only be performed by COP, thus it has been deactivated to ensure fair comparisons. Processed covariances from MF33 exhibit remarkable
consistency among the three modules. Notably, for several nuclei tested across different libraries, the percentage difference between the matrices generated by ERRORR and PUFF, relative to the COP matrix, averages around 0.0001%. This minor disparity can be attributed to the limited output format of these modules and the precision variations between Fortran and C++ implementations.

However, uncertainties on angular distributions are quite a different matter. PUFF does not process MF34, ERRORR handles uncertainty of the first Legendre order and COP can fully process the MF34 section, meaning uncertainties given between different Legendre orders. When comparing the first-order matrices produced by ERRORR and COP, we observe variations in magnitudes dependent on the energy mesh. The differences observed in Fig. 3 can be traced back to the implementation of Eq. 3 in the two codes. Even if the formula used in ERRORR is similar in structure, it diverges from COP in two critical aspects: it does not apply a weighting by the elastic cross section, $\sigma_{el}(E)$, and it employs a more restricted quadrature scheme. These variations are key to understanding the discrepancies noted.

<figure><img src="image_4.png" /><figcaption>Figure 3: Comparison of grouped First order Legendre coefficients in COP and ERRORR</figcaption></figure>

# 5 Uncertainty propagation

## 5.1 Methodology

The SERPENT Monte Carlo code [7, 8], employs the General Perturbation Theory to calculate sensitivity coefficients to nuclear data and the well-established "sandwich" formula to propagate uncertainties.

The formula evaluates the uncertainty for a given response, $\epsilon$, by leveraging the relative sensitivity coefficients, denoted as $S_{rel}$, which are calculated by SERPENT. This is achieved through the operation: $\epsilon = S_{rel} * RCOV * S_{rel}^T$. The relative covariance matrix RCOV may include a series of block covariance matrices to encompass correlations among various nuclear data elements, thereby facilitating a comprehensive uncertainty propagation analysis. The data to which both RCOV and $S_{rel}$ are associated must be consistent.

One advantage of SERPENT lies in its flexibility to work with different formats of uncertainty files: it can read covariance matrices in a straightforward, human-readable format or directly utilize the covariance libraries provided by the SCALE code package. The code
SCALE 6.2.4 encompasses both 56- and 252-group structures and are endorsed as recommended resources for various applications [9]. These libraries contain adjusted matrices derived from the ENDF/B library.

In the following section, we will explore the congruence between the propagated uncertainty outcomes from COP covariance matrices, specifically using the SPD algorithm, and the corresponding results from SCALE. Given the consistent results obtained from processing File MF33 using COP, PUFF, and ERRORR methodologies, it is evident that comparative analysis among these methodologies would likely yield minor distinctions. Hence, our focus turns to the comparison with SCALE, where we anticipate more meaningful insights.

## 5.2 Results for cross sections uncertainty propagation

The ENDF/B-VII.1 library was utilized for running SERPENT and processing covariance matrices with the COP module. This choice was made because the $^{56}$Fe data in the SCALE covariance library, identified as "scale.rev08.56groupcov7.1," originates from the ENDF/B-VII.1 library. The ICSBEP database contains experiments with $^{56}$Fe. HMF-014 [10] was chosen simulating $10^4$ particles over $10^5$ batches. The multiplication factor obtained for this simulation was $k_{eff} = 0.99608 \pm 42$ pcm.

<table><caption>Table 1: Comparing $k_{eff}$ uncertainty coming from SCALE 56-groups library and generated by COP after processing $^{56}$Fe from ENDF/B-VII.1</caption><thead><tr><th rowspan="2">Reactions</th><th colspan="2">56groupcov7.1</th><th colspan="2">COP ENDF/B-VII.1</th></tr><tr><th>Δk (pcm)</th><th>Rel. Unc.</th><th>Δk (pcm)</th><th>Rel. Unc.</th></tr></thead><tbody><tr><td>TOTAL</td><td>127.2</td><td>1.4 × 10<sup>-3</sup></td><td>126.85</td><td>1.4 × 10<sup>-3</sup></td></tr><tr><td>MT2-2 (n,n)</td><td>118.14</td><td>1.5 × 10<sup>-3</sup></td><td>117.79</td><td>1.5 × 10<sup>-3</sup></td></tr><tr><td>MT4-4 (n,n')</td><td>35.13</td><td>2.7 × 10<sup>-3</sup></td><td>35.14</td><td>2.6 × 10<sup>-3</sup></td></tr><tr><td>MT16-16 (n,2n)</td><td>0.017</td><td>0.13</td><td>0.015</td><td>0.13</td></tr><tr><td>MT102-102 (n,γ)</td><td>0.836</td><td>0.028</td><td>0.864</td><td>0.027</td></tr><tr><td>MT103-103 (n,p)</td><td>0.0161</td><td>0.067</td><td>0.0132</td><td>0.075</td></tr></tbody></table>

The uncertainty quantification using COP, as compiled in Table 1, falls within the relative uncertainty margins of the quantification using covariances from the SCALE library.

## 5.3 Results for angular distribution uncertainty propagation

In this section, we explore the propagation of angular distribution uncertainties using the MF34 processing feature of COP along with SERPENT. Notably, we transitioned from the ENDF/B-VII.1 nuclear data library, which lacked MF34 data, to the JEFF-3.3 library. This shift was necessary to enable the examination of angular distribution uncertainties of the $^{56}$Fe, as JEFF-3.3 provides Legendre coefficients up to the 6th order for this isotope. Additionally, we opted for a different benchmark, PMF-028 [10]. With its spherical configuration, it proves to be far more sensitive to angular distributions than HMF-014, given that neutrons not reflected toward the center do not contribute to the effective multiplication factor.

The results in Table 2 present the propagation of uncertainties for various parameters associated with the angular distribution. Notably, the table displays the uncertainties in $k_{eff}$ arising from the first-order Legendre coefficient covariance (P1), the second order (P2), and the cross-covariances between orders (P1-P2), as explained in Sec. 2.2.
<table><caption>Table 2: Uncertainty on the $k_{eff}$ coming from the two first Legendre orders</caption><thead><tr><th>Order</th><th>Δk (pcm)</th><th>Rel. Unc.</th></tr></thead><tbody><tr><td>TOTAL</td><td>375.1</td><td>1.8 × 10<sup>-3</sup></td></tr><tr><td>P1-P1</td><td>356.8</td><td>2.0 × 10<sup>-3</sup></td></tr><tr><td>P1-P2</td><td>-23.71</td><td>5.8 × 10<sup>-3</sup></td></tr><tr><td>P2-P2</td><td>97.32</td><td>2.0 × 10<sup>-3</sup></td></tr></tbody></table>

# 6 Conclusion

The newly introduced GAIA covariance processing module has demonstrated its effectiveness through comprehensive comparisons with existing modules. Particularly noteworthy is the significance of appropriately incorporating the MF34 covariance type. The discrepancies observed between the ERRORR and COP methodologies emphasize the critical nature of the comparative phase, crucial for achieving a robust and accurate covariance processing outcome. As highlighted in the initial sections of this paper, a comprehensive treatment of uncertainty necessitates the consideration of resonance parameter uncertainties. It is worth noting that this enhancement is currently under development for COP, promising further refinement in uncertainty quantification. The advancements presented herein contribute not only to the field of nuclear data analysis but also pave the way for improved precision in various applications reliant on accurate covariance data.

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[10] Nuclear Energy Agency, *ICSBEP Handbook 2018*, 2019, doi:10.1787/ea7c647e-en