# Uncertainty quantification & propagation

FISPACT-II provides predictions of inventory, radiological quantities and their uncertainties using nuclear data covariance information. Central to the method is a novel fast pathways search algorithm using directed graphs. The pathways output provides (1) an aid to identifying important reactions, (2) fast estimates of uncertainties, (3) reduced models that retain important nuclides and reactions for use in the code’s Monte-Carlo sensitivity analysis module. Described are the methods that are being implemented for improving uncertainty predictions, quantification and propagation using the covariance data that the recent nuclear data libraries contain. In the TENDL library, above the upper energy of the resolved resonance range, a Monte Carlo method in which the covariance data come from uncertainties of the nuclear model calculations is used. The nuclear data files are read directly by FISPACT-II without any further intermediate processing. Variance and covariance data are processed and used by FISPACT-II to compute uncertainties in collapsed cross-sections, and these are in turn used to predict uncertainties in inventories and all derived radiological data. These are summarised in the NDS paper on FISPACT-II uncertainty methods.

Pathways-based uncertainty

Propagation of uncertainty in inventories and responses is challenging for cases where the nuclide inventory depends strongly upon the production paths, particularly where there may be a great many. Using the uncertainties in reaction rates determined using full variance-covariance data in the nuclear data files, a sampling method can be used to determine uncertainties in various quantities, but this can be computationally impractical. Particularly in cases where many reactions and decays are responsible for the production of some nuclides, this requires full sensitivity analyses followed by potentially a great many nuclear data samples.

To both make these calculations more practical and allow full uncertainty quantification on observables, rigorous pathway analysis and pruning algorithms were developed. These identify and weight the possible combinations of reactions and decays which result in the production of dominant nuclides by using the UNCERTAINTY keyword. The nuclide set can be extended to others of interest using other keywords such as ROUTES, LOOKAHEAD and PATHRESET. Once the pathways are calculated, the uncertainty along each is calculated using the uncertainties along each as an uncorrelated sum of squares of individual links within the path. The weighted sum of these gives the uncertainty in the production of the nuclide.

Sensitivity and Monte-Carlo covariance sampling

FISPACT-II uses a Monte-Carlo approach to sensitivity analysis. A series $S$ of inventory calculations is performed with the set of $I$ independent variables $\{X_i^s;~ i=1,\ldots,I;~ s=1,\ldots,S\}$ chosen from distributions with means $\langle X_i\rangle$ and standard deviations $\langle \Delta X_i\rangle$. These runs produce a set of $J$ dependent variables $\{Q_j^s;~ j=1,\ldots,J;~ s=1,\ldots,S\}$. For the EAF libraries, the cross sections are treated as the independent variables, but with the TENDL libraries it is possible to take into account covariance data to get a more complete assessment of the uncertainties of results.

The TENDL libraries contain covariance data between some reactions of given parents. Given a rank $D$ symmetric positive definite covariance matrix, we can find a $D \times D$ similarity transformation matrix $M$ such that $Y=M^TX$ and $cov(Y,Y) = M^T cov(X,X) M$ is diagonal with diagonal elements $var(Y)$. The transformed cross sections $Y$ are treated as the independent variables, and random samples with means $\langle Y_i\rangle$ and standard deviations $\langle \Delta Y_i\rangle$ are chosen, and the input sample cross sections are computed using $X=MY$.

The dependent variables are the numbers of atoms of nuclides $j$ or some related radiological quantity. The pathways summaries created by pathways analysis provide a good guide as to which cross sections and decays are likely to be important to include as independent variables in the sensitivity calculation.

Selecting a sensitivity calculation in FISPACT-II causes the series of $S$ runs with different independent variables to be undertaken to compute, process and output the set $\{Q^s_j\}$. The default independent variable distribution is taken to be log-normal, but other options are provided (normal, log-uniform and uniform). Any sequence of irradiation pulses, changes in cross section, etc. that are possible with FISPACT-II can be used in the sensitivity calculations. The code performs the base calculation with full output, then repeats $S$ times the sequence of steps with different sets $\{X_i^s\}$. The results of the base calculation are not included in the sensitivity calculation.

Sensitivity calculations provide both uncertainty and sensitivity output. Summary uncertainty output of means $\bar{X}_i$ and $\bar{Q}_j$ and standard deviations $\Delta X_i$ and $\Delta Q_j$ is produced. The code writes tables of means, standard deviations and Pearson correlation coefficients, and outputs the raw data $\{X_i^s,~ Q_j^s;~ i=1,\dots,I;~ j=1,\ldots,J;~s=1,\ldots,S\}$ to file for possible post-processing.

Depletion uncertainty

In comparison with the nuclear data uncertainty in the production of nuclides, which requires a complete understanding of all reactions and decays which can be combined to create the final product, depletion of a nuclide only requires the reactions which remove that nuclide and its decay, if unstable. The uncertainty in the rate of removal of any nuclide can be easily computed as an uncorrelated sum (potentially correlated with channel-to-channel covariances) of reaction rates that remove the nuclide. For a nuclide without any creation, such U235 in a conventional LWR, the uncertainty in the fuel can be given by the integrated uncertainty in the removal reaction rates over the irradiation period. FISPACT-II will calculate these specific rate uncertainties and integrate them for a depletion uncertainty of initial nuclides.

In addition, an approximation for the depletion uncertainty including depletion of created inventory is calculated, by assuming constant creation and depletion rates, i.e. $\frac{dN}{dt} = – \mathsf{D} N + \mathsf{C}$, where $\mathsf{D}$ is a specific rate of depletion and $\mathsf{C}$ is a constant rate of creation. During each time-step and taking the number of atoms removed as:

$$\mathcal{D}_{i}N = N_i + \mathsf{C}_i\Delta t_i – \frac{\mathsf{C}_i}{\mathsf{D}_i} – \left( N_i – \frac{\mathsf{C}_i}{\mathsf{D}_i}\right) \exp \left( – \mathsf{D}_i \Delta t_i\right),$$

and the calculated uncertainty in the specific rate $\mathsf{D}$ allows us to calculate a range of depletion values to determine an uncertainty.