3.1 Adjustment methods provide a means for combining the results of neutron transport calculations with neutron dosimetry measurements (see Test Method E1005 and NUREG/CR-5049) in order to obtain optimal estimates for neutron damage exposure parameters with assigned uncertainties. The inclusion of measurements reduces the uncertainties for these parameter values and provides a test for the consistency between measurements and calculations and between different measurements (see 3.3.3). This does not, however, imply that the standards for measurements and calculations of the input data can be lowered; the results of any adjustment procedure can be only as reliable as are the input data.
3.2 Input Data and Definitions :
3.2.1 The symbols introduced in this section will be used throughout the guide.
3.2.2 Dosimetry measurements are given as a set of reaction rates (or equivalent) denoted by the following symbols:
These data are, at present, obtained primarily from radiometric dosimeters, but other types of sensors may be included (see 4.1).
3.2.3 The neutron spectrum (see Terminology E170) at the dosimeter location, fluence or fluence rate Φ(E) as a function of neutron energy E , is obtained by appropriate neutronics calculations (neutron transport using the methods of discrete ordinates or Monte Carlo, see Guide E482). The results of the calculation are customarily given in the form of multigroup fluences or fluence rates.
where:
E_{j} and E_{j}_{+1 } are the lower and upper bounds for the j-th energy group, respectively, and k is the total number of groups.
3.2.4 The reaction cross sections of the dosimetry sensors are obtained from an evaluated cross section file. The cross section for the i-th reaction as a function of energy E will be denoted by the following:
Used in connection with the group fluences, Eq 2, are the calculated group-averaged cross sections σ_{ij}. These values are defined through the following equation:
3.2.5 Uncertainty information in the form of variances and covariances must be provided for all input data. Appropriate corrections must be made if the uncertainties are due to bias producing effects (for example, effects of photo reactions).
3.3 Summary of the Procedures:
3.3.1 An adjustment algorithm modifies the set of input data as defined in 3.2 in the following manner (adjusted quantities are indicated by a tilde, for example, ã_{i}):
or for group fluence rates
or for group-averaged cross sections
The adjusted quantities must satisfy the following conditions:
or in the form of group fluence rates
Since the number of equations in Eq 11 is much smaller than the number of adjustments, there exists no unique solution to the problem unless it is further restricted. The mathematical algorithm in current adjustment codes are intended to make the adjustments as small as possible relative to the uncertainties of the corresponding input data. Codes like STAY'SL, FERRET, LEPRICON, and LSL-M2 (see Table 1) are based explicitly on the statistical principles such as "Maximum Likelihood Principle" or "Bayes Theorem," which are generalizations of the well-known least squares principle. Using variances and correlations of the input fluence, dosimetry, and cross section data (see 4.1.1, 4.2.2, and 4.3.3), even the older codes, notably SAND-II and CRYSTAL BALL, can be interpreted as application of the least squares principle although the statistical assumptions are not spelled out explicitly (see Table 1). A detailed discussion of the mathematical derivations can be found in NUREG/CR-2222 and EPRI NP-2188.TABLE 1 Available Unfolding Codes
Program | Solution Method | Code Available
From | Refer-
ences | Comments |
SAND-II | semi-iterative | RSICC Prog. No. CCC-112, CCC-619, PSR-345 |
1^{A} | contains trial spectra library. No output uncertainties in the original code, but modified Monte Carlo code provides output uncertainties (
2, 3, 4) |
| | | | |
SPECTRA | statistical, linear estimation | RSICC Prog. No. CCC-108 |
5, 6 | minimizes deviation in magnitude, no output uncertainties. |
| | | | |
IUNFLD/
UNFOLD | statistical, linear estimation | |
7 | constrained weighted linear least squares code using B-spline basic functions. No output uncertainties. |
| | | | |
WINDOWS | statistical, linear estimation, linear programming | RSICC Prog. No. PSR-136, 161 |
8 | minimizes shape deviation, determines upper and lower bounds for integral parameter and contribution of foils to bounds and estimates. No statistical output uncertainty. |
| | | | |
RADAK,
SENSAK | statistical, linear estimation | RSICC Prog. No. PSR-122 |
9, 10,11,12 | RADAK is a general adjustment code not restricted to spectrum adjustment. |
| | | | |
STAY'SL | statistical linear estimation | RSICC Prog. No. PSR-113 |
13 | permits use of full or partial correlation uncertainty data for activation and cross section data. |
| | | | |
NEUPAC(J1) | statistical, linear estimation | RSICC Prog. No. PSR-177 |
14, 15 | permits use of full covariance data and includes routine of sensitivity analysis. |
| | | | |
FERRET | statistical, least squares with log normal a priori distributions | RSICC Prog. No. PSR-145 |
2, 3 | flexible input options allow the inclusion of both differential and integral measurements. Cross sections and multiple spectra may be simultaneously adjusted. FERRET is a general adjustment code not restricted to spectrum adjustments. |
| | | | |
LEPRICON | statistical, generalized linear least squares with normal a priori and a posteriori distributions | RSICC Prog. No. PSR-277 |
16, 17, 18 | simultaneous adjustment of absolute spectra at up to two dosimetry locations and one pressure vessel location. Combines integral and differential data with built-in uncertainties. Provides reduced adjusted pressure vessel group fluence covariances using built-in sensitivity database. |
| | | | |
LSL-M2 | statistical, least squares, with log normal a priori and a posteriori distributions | RSICC Prog. No.
PSR-233 |
19 | simultaneous adjustment of several spectra. Provides covariances for adjusted integral parameters. Dosimetry cross-section file included. |
| | | | |
UMG | Statistical, maximum entropy with output uncertatinties | RSICC Prog. No.
PSR-529 |
20, 21 | Two components. MAXED is a maximum entropy code. GRAVEL (
22) is an iterative code. |
| | | | |
NMF-90 | Statistical, least squares | IAEA NDS |
23, 24 | Several components, STAY'NL, X333, and MIEKE. Distributed by IAEA as part of the REAL-84 interlaboratory exercise on spectrum adjustment (
25). |
| | | | |
GMA | Statistical, general least squares | RSICC Prog. No.
PSR-367 |
26 | Simultaneous evaluation with differential and integral data, primarily used for cross-section evaluation but extensible to spectrum adjustments. |
^{A} The boldface numbers in parentheses refer to the list of references appended to this guide.
3.3.1.1 An important problem in reactor surveillance is the determination of neutron fluence inside the pressure vessel wall at locations which are not accessible to dosimetry. Estimates for exposure parameter values at these locations can be obtained from adjustment codes which adjust fluences simultaneously at more than one location when the cross correlations between fluences at different locations are given. LEPRICON has provisions for the estimation of cross correlations for fluences and simultaneous adjustment. LSL-M2 also allows simultaneous adjustment, but cross correlations must be given.
3.3.2 The adjusted data ã_{i}, etc., are, for any specific algorithm, unique functions of the input variables. Thus, uncertainties (variances and covariances) for the adjusted parameters can, in principle, be calculated by propagation the uncertainties for the input data. Linearization may be used before calculating the uncertainties of the output data if the adjusted data are nonlinear functions of the input data.
3.3.2.1 The algorithms of the adjustment codes tend to decrease the variances of the adjusted data compared to the corresponding input values. The linear least squares adjustment codes yield estimates for the output data with minimum variances, that is, the "best" unbiased estimates. This is the primary reason for using these adjustment procedures.
3.3.3 Properly designed adjustment methods provide means to detect inconsistencies in the input data which manifest themselves through adjustments that are larger than the corresponding uncertainties or through large values of chi-square, or both. (See NUREG/CR-3318 and NUREG/CR-3319.) Any detection of inconsistencies should be documented, and output data obtained from inconsistent input should not be used. All input data should be carefully reviewed whenever inconsistencies are found, and efforts should be made to resolve the inconsistencies as stated below.
3.3.3.1 Input data should be carefully investigated for evidence of gross errors or biases if large adjustments are required. Note that the erroneous data may not be the ones that required the largest adjustment; thus, it is necessary to review all input data. Data of dubious validity may be eliminated if proper corrections cannot be determined. Any elimination of data must be documented and reasons stated which are independent of the adjustment procedure. Inconsistent data may also be omitted if they contribute little to the output under investigation.
3.3.3.2 Inconsistencies may also be caused by input variances which are too small. The assignment of uncertainties to the input data should, therefore, be reviewed to determine whether the assumed precision and bias for the experimental and calculational data may be unrealistic. If so, variances may be increased, but reasons for doing so should be documented. Note that in statistically based adjustment methods, listed in Table 1 the output uncertainties are determined only by the input uncertainties and are not affected by inconsistencies in the input data (see NUREG/CR-2222). Note also that too large adjustments may yield unreliable data because the limits of the linearization are exceeded even if these adjustments are consistent with the input uncertainties.
3.3.4 Using the adjusted fluence spectrum, estimates of damage exposure parameter values can be calculated. These parameters are weighted integrals over the neutron fluence
or for group fluences
with given weight (response) functions w(E) or w_{ j}, respectively. The response function for dpa of iron is listed in Practice E693. Fluence greater than 1.0 MeV or fluence greater than 0.1 MeV is represented as w(E) = 1 for E above the limit and w(E) = 0 for E below.
3.3.4.1 Finding best estimates of damage exposure parameters and their uncertainties is the primary objective in the use of adjustment procedures for reactor surveillance. If calculated according to Eq 12 or Eq 13, unbiased minimum variance estimates for the parameter p result, provided the adjusted fluence Φ˜ is an unbiased minimum variance estimate. The variance of p can be calculated in a straightforward manner from the variances and covariances of the adjusted fluence spectrum. Uncertainties of the response functions, w_{j}, if any, should not be considered in the calculation of the output variances when a standard response function, such as the dpa for iron in Practice E693, is used. The calculation of damage exposure parameters and their variances should ideally be part of the adjustment code.
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