Statistical Uncertainty Quantification of Probability Tables for Unresolved Resonance Cross Sections

. The probability table method is an important method to treat the self-shielding effect in the unresolved resonance region. A probability table is generated by using many “ladders” that represent pseudo resonance structures. This study developed a quantification method of the statistical uncertainty of a probability table. The product of the probability table and average cross section in each probability bin was considered as the target of the statistical uncertainty of the probability table. The central limit theorem (CLT), bootstrap method, and jackknife method were adopted to calculate the statistical uncertainty. The statistical uncertainties calculated using these methods were compared with the reference results. The calculation results showed that the statistical uncertainties obtained by CLT were similar to those of the other methods. Because the CLT calculation time was faster than that of the other methods, CLT was deemed as the best method for calculating the statistical uncertainty of the probability table. The statistical uncertainty quantification of the probability table developed in this study was implemented in nuclear data processing code FRENDY version 2. FRENDY version 2 can generate a probability table using the tolerance of the statistical uncertainty of the probability table.


Introduction
The resonance parameters in an unresolved resonance region are given as average values, e.g., average level spacing and average neutron width [1]. The averaged cross section is only reconstructed in the unresolved resonance region. The averaged cross section is insufficient for self-shielding calculations. However, the self-shielding effect in the unresolved resonance region significantly affects the fast-and intermediate-spectrum reactors [2]. The probability table method is widely used in continuous-energy Monte Carlo calculation codes and multigroup calculation codes to deal with this problem [3,4].
The probability table method is calculated using "ladders" that represent pseudo resonance structures. A ladder is generated using the average level spacing, average neutron width, and corresponding random numbers. Many ladders are required to generate an accurate probability table. The conventional nuclear data processing codes require a number of ladders as an input parameter to generate the probability table. The probability table generation requires a long computational time. A reduction in the number of ladders contributes to the reduction in the nuclear data processing time. However, the optimum number of ladders has not yet been investigated. Users need to manually set the number of ladders to a sufficiently large value to obtain a sufficiently converged probability table. Our previous study investigated the optimum number of ladders [5].
In the present study, we focused on the statistical uncertainty quantification of a probability table. The statistical uncertainty of a probability table provides good information for the estimation of convergence. It also contributes to the reduction in the processing time because it can optimize the number of ladders. of  the  statistical  uncertainty of a probability table   A probability table consists of probability , and average cross section , , where is the incident neutron energy and is the probability bin. In the continuous-energy Monte Carlo calculation codes, the cross section in an unresolved resonance region is obtained using random numbers. If incident neutron energy is −1 ≤ < and random number is , −1 ≤ < , , average cross section , is used as the cross section at . The cross section changes whenever neutron energy is introduced into the unresolved resonance region.

Calculation
The probability table represents the resonance structure. , changes from the top of the resonance peak to the bottom of the resonance. In the probability table method, not only , but also , is important for neutronic calculation. If , is very large, the effect of the statistical uncertainty of the probability table on the neutronics calculation will be larger even if , is smaller than that in the other probability bins. In this study, the product of probability , and average cross section , is used as the target of the statistical uncertainty of the probability table.
Many methods for statistical uncertainty quantification have been developed. In the current study, we adopted three calculation methods, namely, the central limit theorem (CLT) [6], bootstrap method [7], and jackknife method [8], for statistical uncertainty quantification of the probability table. CLT is commonly used for the uncertainty quaintification. If the number of targets is sufficiently large, the distribution of the sample means tends toward a normal distribution even if the distribution of the target is not a normal distribution. CLT cannot accurately calculate the statistical uncertainty if the number of targets is relatively small. The bootstrap and jackknife methods can calculate the statistical uncertainty with a smaller number of samples compared with CLT. The bootstrap method enables practical estimation of the variance and confidence interval of a sample estimate using a large number of resamples obtained from the original data [9]. The jackknife method is another resampling method. This method estimates the statistical uncertainty using subsamples from the original data. If the size of the original sample is n, the size of the subsample in the jackknife method is n -1, which is obtained by omitting one observation from the original sample. The jackknife method is simpler than the bootstrap method.
From the calculation-time perspective, CLT is faster than the other methods because it does not resample the data. However, if the statistical uncertainty of the target does not follow a normal distribution, the bootstrap or jackknife method should be used to accurately estimate the statistical uncertainty.
The calculation flow of the statistical uncertainty of the probability table is described as follows: (1) Generation of the kth ladder (2) Calculation of probability , and average cross section , , where, is the ladder index (3) Calculation of the statistical uncertainty in each probability bin , using the first to the kth ladder data, i.e., , Continuing to the next ladder The abovementioned calculation steps were implemented in a nuclear data processing code FRENDY version 2 [10]. This code was developed by Japan Atomic Energy Agency (JAEA), and it was released in the JAEA website [11] in January 2022. This code can generate an ACE-formatted file [12], which is used in the continuous-energy Monte Carlo calculation codes, e.g., PHITS [13], MCNP [14], and Serpent [15], and GENDF-and MATXS-formatted files, which are used for multigroup calculation codes.
FRENDY calculates the statistical uncertainty at every 10 ladder generations to reduce the calculation time of the statistical uncertainty. FRENDY compares the statistical uncertainty in each probability bin. The maximum statistical uncertainty in all probability bins is considered as the statistical uncertainty at the kth ladder index.
The probability table method deals with the total, elastic scattering, fission, and radiative capture reactions. The statistical uncertainty of all reactions must be considered. The calculation time of the statistical uncertainty affects the total processing time when the statistical uncertainties of all reactions are calculated. FRENDY calculates only the statistical uncertainty of the total reaction, and calculation of the statistical uncertainty of the other reactions is skipped when the statistical uncertainty of the total reaction is larger than the statistical uncertainty tolerance. FRENDY calculates the statistical uncertainty of the other reactions when the statistical uncertainty of the total reaction is less than or equal to the tolerance value. The probability table generation is finished when the statistical uncertainty of all reactions is less than or equal to the tolerance value.

Comparison of the statistical uncertainty quantification methods
U-235, U-238, and Sr-90 of JENDL-4.0 [16] are used for comparison of the statistical uncertainty quantification methods. Consideration of the probability table of fissile nuclides significantly affects the neutronic calculation. A light nuclide does not have a resonance parameter in the unresolved resonance region. The lightest nuclide of JENDL-4.0, which contains a resonance parameter in the unresolved resonance region, is denoted as Mn. We find that the average level spacing of these light nuclides is larger than that of U-235. In the current study, Sr-90 is used to compare the statistical uncertainty of the probability table. The temperature and number of probability bins for calculating the probability table are 293.6 K and 20, respectively. This study does not consider the impact of the number of probability bins on the statistical uncertainty of the probability table. The difference in the number of probability bins will affect the convergence of the statistical uncertainty. The impact of the number of probability bins will investigate in the future.
The number of resampling in the bootstrap method is 10,000, and the number of subsamples in the jackknife method is equal to the number of ladders.
We generate 1,000 probability table sets using 1,000 different random seeds to calculate the resonance results. The relative standard deviation of the 1,000 different probability table sets is considered as the reference results. Figure 1 shows the comparison of the relative standard deviation of the probability table in each nuclide at the lowest and highest incident neutron energy in the unresolved resonance region. The relative standard deviation of the probability table is calculated in each probability bin. Figure 1 shows the maximum relative standard deviation over all probability table bins because the maximum statistical uncertainty is important for estimating the convergence of the probability table. The relative standard deviation exponentially decays, and the gradient of each nuclide is identical. We need to note that the other reactions, i.e., elastic scattering, fission, and radiative capture reactions, also exhibit a similar trend. These results indicate that the reference results are appropriately calculated using the 1,000 different probability table sets.
The statistical uncertainties of the probability table using CLT, the bootstrap method, and the jackknife method are also calculated in the 1,000 different probability table sets. These uncertainty quantification methods calculate the statistical uncertainty of the probability table in each probability table set. Therefore, 1,000 statistical uncertainty sets are calculated. The average value of the statistical uncertainties calculated by each method is compared with the reference result.  Tables 1-3 list the lowest and highest incident neutron energy in the unresolved resonance region, respectively. The statistical uncertainty of each calculation method exhibits good agreement with the reference results.
The statistical uncertainty calculated by CLT is similar to that of the other calculation methods even if the number of ladders is small. This result indicates that the distribution of the target, i.e., the product of the probability and average cross section, is close to the normal distribution. As presented in Section 2, the CLT calculation time is faster than that of the other methods. The statistical uncertainty of the probability table can be quickly calculated using CLT.   In our previous study, we estimated the effect of the difference in the number of ladders on the neutronics calculation [5]. The integral experiments that introduced a large self-shielding effect were used for comparison. Our previous study demonstrated that the difference was sufficiently small when the number of ladders was 100. Therefore, the default number of ladders in FRENDY is set to 100. Figure 2 shows the calculation time of the probability table and the statistical uncertainty of the probability table when the number of ladders is 100. The statistical uncertainty of the probability table is calculated using CLT. Figure 2 shows that the calculation time and statistical uncertainty vary significantly with the nuclide and incident neutron energy.
The statistical uncertainty of the probability table is less than 5% in all cases. The default tolerance of the statistical uncertainty of the probability table in FRENDY is set to 5%.
The statistical uncertainty becomes smaller if the incident neutron energy becomes higher. This result indicates that the optimum number of ladders depends on the incident neutron energy. The optimum number of ladders decreases when the incident neutron energy increases. The number of ladders at a higher energy level is reduced if the tolerance of the statistical uncertainty of the probability table is used as the input parameter.
In the U-235 case, the calculation time becomes longer when the incident neutron energy becomes  Table 4 lists the calculation time of the probability table when the number of ladders or tolerance of the statistical uncertainty is used as the input parameter. The number of ladders and tolerance of the statistical uncertainty are 100 and 5%, respectively. The calculation time of the probability table is reduced to less than one-half when tolerance is used as the input parameter. This result shows that optimization of the number of ladders using the statistical uncertainty of the probability table is an effective method of reducing the calculation time of the probability table.

Conclusions
A calculation method for the statistical uncertainty of the probability table was developed to estimate the optimum number of ladders. The product of the probability and average cross section was considered as the target of the statistical uncertainty of the probability table. CLT, bootstrap method, and jackknife method were adopted to calculate the statistical uncertainty of the probability table. The statistical uncertainties of the probability table calculated using these methods were compared with the reference results.
The statistical uncertainties calculated by CLT were similar to those calculated by the other methods even if the number of ladders was small. The statistical uncertainty of the probability table could be quickly calculated using CLT.
JAEA developed a nuclear data processing code, namely, FRENDY version 2, and released it in the JAEA website. We implemented the statistical uncertainty quantification function developed in this study in our processing code. FRENDY version 2 uses CLT for the statistical uncertainty quantification of the probability table. This code could generate a probability table using the tolerance of the statistical uncertainty of the probability table using this function. The default tolerance of the statistical uncertainty is 5%. Note that FRENDY version 2 does not use the number of ladders as the input parameter unless the user sets the maximum number of ladders in the input file. If users set both the tolerance of the statistical uncertainty and the maximum number of ladders in an input file, the latter is used as the input parameter for the probability table generation.
Users do not need to consider the optimum number of ladders if the tolerance of the statistical uncertainty of the probability table is used as the input parameter.