6. Sensitivity and Uncertainty Analysis

Introduction by B. T. Rearden and J. D. McDonnell

SCALE provides a suite of computational tools for sensitivity and uncertainty analysis to (1) identify important processes in safety analysis and design, (2) provide a quantifiable basis for neutronics validation for criticality safety and reactor physics analysis based on similarity assessment, and (3) quantify the effects of uncertainties in nuclear data and physical parameters for safety analysis [SU-RWJ+11, SU-WIJ+13].

Sensitivity Analysis and Uncertainty Quantification

Sensitivity analysis provides a unique insight into system performance in that the predicted response of the system to a change in some input process is quantified. Important processes can be identified as those that cause the largest changes in the response per unit change in the input. In neutron transport numerical simulations, calculating important responses such as keff, reaction rates, and reactivity coefficients requires many input parameters, including material compositions, system geometry, temperatures, and neutron cross section data. Because of the complexity of nuclear data and its evaluation process, the response of neutron transport models to the cross section data can provide valuable information to analysts. The SCALE sensitivity and uncertainty (S/U) analysis sequences-known as the Tools for Sensitivity and Uncertainty Analysis Methodology Implementation (TSUNAMI)-quantify the predicted change in keff, reaction rates, or reactivity differences due to changes in the energy-dependent, nuclide-reaction–specific cross section data, whether continuous-energy or multigroup.

Uncertainty quantification is useful for identifying potential sources of computational biases and highlighting parameters important to code validation. When applying uncertainties in the neutron cross section data, the sensitivity of the system to the cross section data can be applied to propagate the uncertainties in the cross section data to an uncertainty in the system response. Additionally, SCALE provides the ability to stochastically sample uncertainties in nuclear data or any other model input parameter (e.g., dimensions, densities, temperatures) and propagate these input uncertainties to uncertainties not only as traditional TSUNAMI responses of keff, reaction rates, and reactivity, but also in any general output quantity such as burnup isotopics, dose rates, etc. Additionally, where the same input quantities are used in multiple models, the sampling techniques can be applied to quantify the correlation in uncertainties of multiple systems due to the use of the same uncertain parameters across these systems.

Validation of Codes and Data

Modern neutron transport codes such as the KENO Monte Carlo codes in the SCALE code system can predict keff with a high degree of precision. Still, computational biases of one percent or more are often found when using these codes to model critical benchmark experiments. The primary source of this computational bias is believed to be errors in the cross section data as bounded by their uncertainties. These errors can be tabulated in cross section covariance data. To predict or bound the computational bias for a design system of interest, the American National Standards for Nuclear Criticality Safety in Operations with Fissionable Material Outside Reactors (ANSI/ANS-8.1-1998) [SU-AmericanNSSCWGA8198] and the American National Standard for Validation of Neutron Transport Methods for Nuclear Criticality Safety Calculations (ANSI/ANS-8.24-2007) [SU-AmericanNSSCWGA82407] allow calculations to be used to determine subcritical limits for the design of fissionable material systems. The standards require validation of the analytical methods and data used in nuclear criticality safety calculations to quantify any computational bias and the uncertainty in the bias. The validation procedure must be conducted through comparison of computed results with experimental data, and the design system for which the subcritical limit is established must fall within the area of applicability of the experiments chosen for validation. The ANS-8.1 standard defines the area(s) of applicability as “the limiting ranges of material compositions, geometric arrangements, neutron-energy spectra, and other relevant parameters (e.g., heterogeneity, leakage, interaction, absorption, etc.) within which the bias of a computational method is established.”

TSUNAMI Techniques for Code Validation

The TSUNAMI software provides a unique means to determine the similarity of nuclear criticality experiments to safety applications [SU-BRH+04]. The TSUNAMI validation techniques are based on the assumption that computational biases are primarily caused by errors in cross section data, the potential for which are quantified in cross section covariance data. TSUNAMI provides two methods to establish the computational bias introduced through cross section data.

For the first method, instead of using one or more average physical parameters to characterize a system, TSUNAMI determines the uncertainties in the computed response that are shared between two systems due to cross section uncertainties. These shared uncertainties directly relate to the bias shared by the two systems. To accomplish this, the sensitivity to each group-wise nuclide-reaction–specific cross section is computed for all systems considered in the analysis. Correlation coefficients are developed by propagating the uncertainties in neutron cross section data to uncertainties in the computed response for experiments and safety applications through sensitivity coefficients. The bias in the experiments, as a function of correlated uncertainty with the intended application, is extrapolated to predict the bias and bias uncertainty in the target application. This correlation coefficient extrapolation method is useful where many experiments with uncertainties that are highly correlated to the target application are available.

For the second method, data adjustment or data assimilation techniques are applied to predict computational biases, and more general responses, including but not limited to keff, can be addressed simultaneously [SU-BRH+04]. This technique uses S/U data to identify a single set of adjustments to nuclear data and experimental responses, taking into account their correlated uncertainties, which would improve the agreement between the response values from the experimental results and computational simulations. The same data adjustments are then used to predict an unbiased response (e.g., keff) value for the application and an uncertainty on the adjusted response value. The difference between the originally calculated response value and the new post-adjustment response value represents the bias in the original calculation, and the uncertainty in the adjusted value represents the uncertainty in this bias. If experiments are available to validate the use of a particular nuclide in the application, the uncertainty of the bias for this nuclide may be reduced. If similar experiments are not available, the uncertainty in the bias for the given nuclide is high. Thus, with a complete set of experiments to validate important components in the application, a precise bias with a small uncertainty can be predicted. Where the experimental coverage is lacking, a bias can be predicted with an appropriately large uncertainty. The data assimilation method presents many advantages over other techniques in that biases can be projected from an agglomeration of benchmark experiments, each of which may represent only a small component of the bias of the target application. Also, contributors to the computational bias can be analyzed on an energy-dependent nuclide-reaction–specific basis. However, this technique requires additional data that are not generally available and must be quantified or approximated by the analyst, specifically the correlation coefficients that quantify the relative independence of experimental measurements that use the same equipment, whether nuclear fuel, reactivity devices, or measurement tools.

Sensitivity and Uncertainty Analysis Tools in SCALE

The TSUNAMI-1D and TSUNAMI-3D analysis sequences compute the sensitivity of keff and reaction rates to energy-dependent cross section data for each reaction of each nuclide in a system model. The one-dimensional (1D) transport calculations are performed with XSDRNPM, two-dimensional (2D) transport calculations are preformed using NEWT, and the three-dimensional (3D) calculations are performed with KENO V.a or KENO-VI. The Monte Carlo capabilities of TSUNAMI-3D provide for S/U analysis from either continuous-energy or multigroup neutron transport, where the deterministic capabilities of TSUNAMI-1D only operate in multigroup mode. SAMS (Sensitivity Analysis Module for SCALE) is applied within each analysis sequence to provide the requested S/U data. Whether performing a continuous-energy or multigroup calculation, energy-dependent sensitivity data are stored in multigroup-binned form in a sensitivity data file (SDF) for subsequent analysis. Additionally, these sequences use the energy-dependent cross section-covariance data to compute the uncertainty in the response value due to the cross section-covariance data.

TSAR (Tool for Sensitivity Analysis of Reactivity Responses) computes the sensitivity of the reactivity change between two keff calculations, using SDFs from TSUNAMI-1D, and/or TSUNAMI-3D. TSAR also computes the uncertainty in the reactivity difference due to the cross section covariance data.

TSUNAMI-IP (TSUNAMI Indices and Parameters) uses the SDFs generated from TSUNAMI-1D, TSUNAMI-3D, or TSAR for a series of systems to compute correlation coefficients that determine the amount of shared uncertainty between each target application and each benchmark experiment considered in the analysis. TSUNAMI-IP offers a wide range of options for more detailed assessment of system-to-system similarity. Additionally, TSUNAMI-IP can generate input for the USLSTATS (Upper Subcritical Limit Statistical Software) [SU-LBDH97] trending analysis and compute a penalty, or additional margin, needed for the gap analysis. USLSTATS is distributed as a graphical user interface with SCALE, but its use is documented in the TSUNAMI Primer [SU-RMB+09], not in this documentation chapter.

TSURFER (Tool for S/U Analysis of Response Functions Using Experimental Results) is a bias and bias uncertainty prediction tool that implements the generalized linear least-squares (GLLS) approach to data assimilation and cross section data adjustment that also uses the SDFs generated from TSUNAMI-1D, TSUNAMI-3D, or TSAR. The data adjustments produced by TSURFER are not used to produce adjusted cross section data libraries for subsequent use; rather, they are used only to predict biases in application systems.

The TSUNAMI Primer also documents the use of the graphical user interfaces for TSUNAMI, specifically ExSITE (Extensible SCALE Intelligent Text Editor) that facilitates analysis with TSUNAMI-IP, TSURFER, TSAR, and USLSTATS as well as VIBE (Validation, Interpretation and Bias Estimation) for examining SDF files, creating sets of benchmark experiments for subsequent analysis, and gathering additional information about each benchmark experiment.

Sampler is a “super-sequence” that performs general uncertainty analysis by stochastically sampling uncertain parameters that can be applied to any type of SCALE calculation, propagating uncertainties throughout a computational sequence. Sampler treats uncertainties from two sources: (1) nuclear data and (2) input parameters. Sampler generates the uncertainty in any result generated by any computational sequence through stochastic means by repeating numerous passes through the computational sequence, each with a randomly perturbed sample of the requested uncertain quantities. The mean value and uncertainty in each parameter is reported, along with the correlation in uncertain parameters where multiple systems are simultaneously sampled with correlated uncertainties.

Used in conjunction with nuclear data covariances available in SCALE, Sampler is a general technique to obtain uncertainties for many types of applications. SCALE includes covariances for multigroup neutron cross section data, as well as for fission product yields, and radioactive decay data, which allow uncertainty calculations to be performed for most MG computational sequences in SCALE. At the present time, nuclear data sampling cannot be applied to SCALE CE Monte Carlo calculations, although the fundamental approach is still valid.

Used in conjunction with uncertainties in input data, Sampler can determine the uncertainties and correlations in computed results due to uncertainties in dimensions, densities, distributions of material compositions, temperatures, or any quantities that are defined in the user input for any SCALE computational sequence. This methodology was developed to produce uncertainties and correlations in criticality safety benchmark experiments, [SU-MR15] but it has a wide range of applications in numerous scenarios in nuclear safety analysis and design. The input sampling capabilities of Sampler also include a parametric capability to determine the response of a system to a systematic variation of an input parameter.



B. L. Broadhead, B. T. Rearden, C. M. Hopper, J. J. Wagschal, and C. V. Parks. Sensitivity-and uncertainty-based criticality safety validation techniques. Nuclear science and engineering, 146(3):340–366, 2004.


J. J. Lichtenwalter, S. M. Bowman, M. D. DeHart, and C. M. Hopper. Criticality Benchmark Guide for Light-Water-Reactor Fuel in Transportation and Storage Packages. Technical Report, Nuclear Regulatory Commission, 1997.


W. J. Marshall and B. T. Rearden. Determination of Experimental Correlations Using the Sampler Sequence within SCALE 6.2. In ICNC 2015. Charlotte, NC (USA), 2015.


B. T. Rearden, D. E. Mueller, S. M. Bowman, R. D. Busch, and S. J. Emerson. TSUNAMI Primer: A Primer for Sensitivity/Uncertainty Calculations with SCALE. Technical Report ORNL/TM-2009/027, Oak Ridge National Laboratory, Oak Ridge, TN (USA), 2009.


B. T. Rearden, M. L. Williams, M. A. Jessee, D. E. Mueller, and D. A. Wiarda. Sensitivity and Uncertainty Analysis Capabilities and Data in SCALE. Nuclear Technology, 2(174):236–288, 2011.


M. L. Williams, G. Ilas, M. A. Jessee, B. T. Rearden, D. Wiarda, W. Zwermann, L. Gallner, M. Klein, B. Krzykacz-Hausmann, and A. Pautz. A Statistical Sampling Method for Uncertainty Analysis with SCALE and XSUSA. Nuclear Technology, pages 515–526, 2013.


American Nuclear Society Standards Committee Working Group ANS-8.1. American National Standard for Nuclear Criticality Safety in Operations with Fissionable Materials outside Reactors. Number ANSI/ANS-8.1-1998. American Nuclear Society, 1998.


American Nuclear Society Standards Committee Working Group ANS-8.24. American National Standard for Validation of Neutron Transport Methods for Nuclear Criticality Safety Calculations. Number ANSI/ANS-8.24-2007. American Nuclear Society, 2007.