Global Assessment of Nuclear Data, GANDR
D.W. Muir

Introduction

In the planning of data development activities, the most frequently asked question is "Are the existing data are accurate enough?" If they are not, the next question usually is "What measurements are needed to remedy the situation?"

Frequently neutron transport calculations must be performed in order to predict the values of practical quantities of interest, such as reactor control-rod worth, biological dose to accelerator operators, and activation of the components of a fusion reactor. In transport applications, questions like those above are difficult to answer, because there is a very complex relationship between the accuracy of the basic input data and the accuracy of the output of the calculations.

Because of this complex relationship, research groups around the world have long recognized [1] the potential contribution of sensitivity and uncertainty analysis, based on perturbation theory, to help address these questions. However, past progress has been slowed by the need to track what has seemed to be an impossibly large amount of correlation information. With the remarkable developments that have taken place in computer capacity in recent years, this approach has become a much more interesting possibility.

In view of this need and these encouraging technological developments, the IAEA has initiated a project to develop a system based on sensitivity and uncertainty analysis for the Global Assessment of Nuclear Data Requirements (GANDR).  In this summary, we present a report on the goals and status of the GANDR project, and we provide a technical description of a functioning hardware and software system now in operation in the Nuclear Data Section, developed in support of the project.

Although the GANDR system is being developed in with global data assessment in mind as the eventual application, the system has been shown to be adaptable to more specific near term needs as well. For example, the GANDR system was employed in a recent evaluation [2] of nuclear data and covariances of the four tungsten isotopes. For this application, the standard GANDR initialization of the data covariances with a non-informative prior was replaced with initial covariances reflecting nuclide-specific constraints imposed by nuclear physics considerations.

To provide a detailed description of the GANDR project and the associated software and hardware, we have produced a six-volume set of reports in Word format. These cover a wide range of topics ranging from theoretical and programming aspects to user input instructions and sample problems. We list below the topics that are covered in the reports.

Volume 1.  Project Overview

Volume 2.  The ZOTTVL Program

Volume 3.  Auxiliary Programs GAPREP, GAPOST and GABROW

Volume 4.  User Input Instructions and Five Sample Problems Based on EXFOR Data

Volume 5.  Preparation of the EXFOR Master Library for the GANDR Project

Volume 6.  Multigroup Sensitivities

These six reports, and much additional information, have been made available on the IAEA Nuclear Data Center web site. For details, see below.

GANDR Project

As discussed above, the eventual goal of the GANDR project is to develop a capability to quantify the expected benefit of any proposed nuclear data measurement in reducing the uncertainty of applied calculations. This information is expected provide useful guidance in choosing among proposals involving different target materials or different measurement techniques.

However, reaching this goal will require the availability of a complete and consistent evaluation of nuclear data covariances, based on the current state of data measurements. Unfortunately, the formatted covariance data contained in the current generation of internationally distributed nuclear reaction data files are really not adequate to serve as a uniform and consistent baseline for this kind of planning.

One problem is that the quality and completeness of the existing information is highly variable from material to material. A second major problem is that, by design, the covariances in the existing international data evaluations do not reflect the impact of available, high-quality integral data. The main reason for this is that the introduction of integral data would generate very strong cross-material correlations and thereby greatly expand the volume of these widely distributed data files.

Therefore, an important intermediate goal of the GANDR project is to perform a global evaluation of data covariances from first principles, in a unified evaluation framework. While consistency and wide coverage are crucial, it is possible (and indeed necessary) to impose a certain level of coarseness on the actual evaluation process, as detailed below. Compromise is possible here because the goal is a practical tool for comparing the benefits of different experiments, not a complete characterization of all aspects of data uncertainty.

As planned, the global uncertainty evaluation will incorporate significant quantities of both high-quality differential data and clean integral data. By "clean," we mean data from simple-geometry integral experiments, preferably data that has been internationally recommended for the testing of differential data. Inclusion of integral data is important, because judgments concerning the need for new measurements should be based on all currently available experimental information, not just a portion of it.

An additional benefit of incorporating integral data into the GANDR evaluation process is that it reduces the need for exhaustive attention to all compiled differential data in this process. If accurate integral data are taken into account, older differential data with low accuracy would have no significant impact on the final data uncertainties and can safely be neglected.

Scope of the Evaluation

In order to keep the evaluation task manageable, we make the simplifying assumption that the fine details of evaluated nuclear data, such as the shape of energy-dependent cross section in the neighborhood of a single resonance, fine structure of neutron scattering angular distributions, and most emitted-particle angle-energy distributions, are sufficiently well constrained by nuclear physics that they can be treated as known quantities.

In contrast, nuclear physics provides relatively little guidance on the absolute magnitude of integrated neutron cross sections. The accuracy of such data rests ultimately on the absolute accuracy of experiments and, in many cases, on a relatively small number of measurements. Unfortunately, data measurements are susceptible to various types of systematic error, such as problems in the determination of the neutron flux, the sample size, the background or the detector efficiency.

In characterizing this uncertainty in integrated cross sections, we assume that the true value of each cross section of interest differs from the best existing evaluation by a smoothly varying multiplicative function A(E). These functions are defined on a fixed, 74-point energy grid that spans the range from 0.00001 eV to 150 MeV. The GANDR evaluation of data uncertainties is limited to determining the uncertainty of the nodal values of the functions A(E) as a function of material and reaction type.

For further economy, nuclear reactions are organized into 25 standard reaction types, some of which are groups of ENDF-format reaction types. For example, inelastic scattering to nuclear levels with excitation energies ranging from 1-6 MeV are treated as a single reaction in the GANDR framework. Because of the existence of threshold reactions, we have found it sufficient to allocate 700 parameters to each target material.

Finally, "global" coverage is defined, for the present purposes, as including the data for 130 different target materials. We follow the lead of the large regional evaluations on the question of which materials to treat as elemental and which as isotopic. The total number of GANDR parameters is thus 700 x 130, or 91000. For further details regarding the definition of the GANDR parameters, see Tables 1-1 through 1-5 in Volume 1.

The covariance matrix describing the uncertainties of the parameters and their correlations thus contains 91000 x 91001 / 2 = 4.14 billion unique numbers. Storage of a single copy of the complete GANDR covariance matrix requires, in double precision, a storage capacity of 33.1 gigabytes. In the current GANDR hardware environment, two separate copies of this large covariance matrix are maintained, on separate hard disk drives, one copy containing the current covariances and the other containing the covariances prior to the most recent update.

ZOTTVL Least Squares Solver

Motivated by the considerations discussed above, an early focus of the GANDR project has been the development of a new generalized linear least squares solver. The earlier IAEA evaluation program ZOTT99, ref. [3], has been modified extensively in order to treat problems having a much larger scope. The new, Very Large, version of ZOTT is ZOTTVL.

Now complete, ZOTTVL makes extensive use of data-paging techniques in order to handle the updating of the 91000 parameters, taking explicit account of the 4.14 billion covariances of these parameters. The data paging methodology relies heavily on partitioning the large matrices into smaller submatrices that reflect the inherent 700-parameter x 130-material structure of the information. For additional programming details, see Volume 2.

Differential and integral data tend to group naturally into blocks of several hundred to several thousand in size, each block corresponding to one actual experimental setup, with strong data correlations within a block but with relatively weak correlations between blocks. Thus the array dimensions in ZOTTVL are set assuming that the largest block of generally correlated experimental information to be encountered is of dimension 3600. The GANDR system can accommodate the input of an arbitrary number of such blocks of data.

Also as discussed in Volume 2, timing tests have been run with the new code. In these tests, a series of test problems were run in which every one of the GANDR parameters and covariances are updated, in response to a varying number of artificial integral data as input. It was found that, to a good approximation, the total running time on a Dell Precision 530 Workstation varies with the number of integral data nb according to the following simple formula, holding for nb up to about 1200: CPU time = (nb + 60) minutes. The constant term relates to the CPU time consumed in performing the significant task of transferring of the 33 gigabytes of data from one disk drive into core and then back out to the other disk drive in each test problem.

While these running times are certainly not trivial, these tests establish that it is practical to utilize generalized least squares to update this enormous data set on a modern personal computer, retaining and updating the full 91000 x 91000 covariance matrix in the process.

The number of materials in the "GANDR Library" is an input parameter to ZOTTVL, so it is easy to switch from the full-library update mode, as we have just discussed, to a single-material mode simply by setting this input parameter equal to one. In the single-material mode, running times of ZOTTVL are measured, not in hours, but in minutes. All computer runs discussed below correspond to using ZOTTVL in the single-material mode.

Other GANDR Programs

In addition to ZOTTVL, we have written a pre-processor program GAPREP that reads GANDR relevant user input and prepares all of the input files required to update the parameters and covariances in response to a set of data measurements, using ZOTTVL as the least-squares solver. Another important function of the GAPREP program is to assist the user in supplementing the uncertainty estimates from EXFOR (which contains only the diagonal of the data covariance matrix), with estimates of correlated uncertainty components, both of within an experiment and between experiments.

In addition, the program GAPREP includes features that support an iterative approach to solving moderately non-linear least squares problems, as will be encountered later in the GANDR project. The method employed, Gauss-Newton iteration, takes into account the fact that sensitivities of integral data generally depend to some extent on the differential nuclear data being evaluated. This involves the management of a set of data files external to ZOTTVL, which is a purely linear solver.

Similarly, a post-processor GAPOST has been developed to display the results of each evaluation update, both as edited text listings and in publication-quality plots. See examples in Figs. 1 and 2 below. In order to enforce consistency between GAPREP and GAPOST, the codes share a library of low level subroutines. These shared routines are stored on the GANDR master source file as "comdecks" that can be "called" by more that one program.

An additional small program GABROW browses the large GANDR binary data files and displays selected portions on a computer terminal. For more information on the GAPREP, GAPOST and GABROW codes, see Volume 3.

The current GANDR system includes, in addition to the GANDR suite of computer programs, a Master EXFOR Library (MEL) of measured microscopic nuclear data. The MEL was created by extracting all cross section data in the international EXFOR database (as of May 2005) for the 130 GANDR materials. The MEL is stored in ASCII format for convenient access by the GANDR programs. See Volume 5 for details.

Also, the Master PENDF Library (MPL) has been updated, using NJOY99.259 to process the ENDF/B-VII evaluations for the GANDR materials into resonance reconstructed, point-ENDF form.

With the cooperation and support of the OECD Nuclear Energy Agency, the GANDR suite of programs has recently been expanded to include two new programs, SEPREP and SEPOST.  These programs compute the sensitivity of multigrouped nuclear cross sections to the pointwise parameters of the GANDR library, using the methods described in Volume 6.  This new feature makes it possible to introduce into a data assessment a data set containing a mixture of point-energy data and energy-averaged data, such as are contained in the new evaluation of the International Neutron Cross Section Standards.

With the GANDR system in its present form, the user can, relatively easily, select a material and reaction type and then combine data from ENDF and from EXFOR to obtain an estimate of the corresponding data covariances. Of course, the quality of this estimate will depend crucially on the quality of the EXFOR uncertainty values. To improve the quality may require a substantial effort by the user in reviewing and supplementing the EXFOR uncertainty information.

Online Documentation

A. The complete six-volume set of GANDR reports discussed above can be accessed from the right side pannel, under "GANDR user manuals".

B. The GANDR-4.0 installation package G4INSTALL (g4install.tar.gz), containing the GANDR Fortran codes and Unix scripts for 16 test problems, can be accessed from the right side pannel, under "GANDR previous versions".

C. The large (280 Mb) auxiliary data package G4FILES (g4files.tar.gz), which includes the current versions of the two Master Libraries MEL and MPL, can be accessed from the right side pannel, under "GANDR previous versions".

References

1. L.N. Usachev and Yu.G. Bobkov, "Perturbation Theory and Experiment Planning in Relation to Nuclear Data for Reactors" [in Russian], Atomizdat, Moscow (1980).

2. R. Capote et al., Evaluation of Cross-Sections of Tungsten Isotopes including Covariance Estimation, International Conference on Nuclear Data for Science and Technology, Nice, France, 22-27 April 2007.

3. D.W. Muir, "ZOTT99, Data Evaluation using Partitioned Least Squares," Code package IAEA1371/01, NEA Computer Program Service (1999).

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