.. _sec-tsunami.tsurfer: TSURFER: An Adjustment Code to Determine Biases and Uncertainties in Nuclear System Responses by Consolidating Differential Data and Benchmark Integral Experiments =================================================================================================================================================================== *M. L. Williams, B. L. Broadhead, M. A. Jessee, J. J. Wagschal1, and R. A. Lefebvre* ABSTRACT The TSURFER code uses the generalized linear least-squares method to consolidate a prior set of measured integral responses (such as *k*\ :sub:`eff` ) and corresponding calculated values obtained using the SCALE nuclear analysis code system. The initial estimates for the computed and measured responses are improved by adjusting the experimental values and the nuclear data used in the transport calculations-taking into account their correlated uncertainties-so that the most self-consistent set of data is obtained. This procedure makes the refined estimates of the calculated and measured responses agree within first-order accuracy, while constraining the data variations to minimize a generalized chi-square parameter. Consolidation of the original integral experiment data and calculated results reduces the prior uncertainty in the response estimates because additional information has been incorporated. The method can also address one or more "\ *application responses*\ " for which no experimental measurements are available. TSURFER computes an updated estimate for the application responses and provides an estimate for the computational bias and application uncertainty. The methodology is useful in validation studies for criticality safety and reactor analysis. ACKNOWLEDGMENTS The authors would like to acknowledge R. L. Childs, formerly of ORNL, for his contributions. Development of the TSURFER code was funded by the U.S. Department of Energy Nuclear Criticality Safety Program. Introduction ------------ This report describes the TSURFER code (**T**\ ool for **S**\ ensitivity/\ **U**\ ncertainty analysis of **R**\ esponse **F**\ unctionals using **E**\ xperimental **R**\ esults-pronounced "\ *surfer,*\ " with silent "T" like TSUNAMI), which is a functional module in the SCALE sensitivity and uncertainty (S/U) analysis methodology (see the :ref:`TSUNAMI-1D chapter `). The main functions of the code are to (a) compute uncertainties in calculated integral responses such as *k*\ :sub:`eff`, due to uncertainties in the input nuclear data; (b) reduce discrepancies between the measured and calculated responses by adjusting the nuclear data and experimental values such that the overall consistency is maximized; (c) analyze measured responses from benchmark experiments to establish the bias and associated uncertainty in some application response that has been calculated. TSURFER utilizes the generalized linear least-squares (GLLS) methodology based on S/U techniques originally developed in the 1970's and 1980's for a variety of applications, including nuclear data evaluation, :cite:`TSURFER-pazy_role_1974` fast reactor design studies, :cite:`TSURFER-weisbin_application_1976,TSURFER-poco_1988` and reactor pressure vessel damage predictions :cite:`TSURFER-maerker_development_1981`. A recent GLLS application has been in the area of criticality safety analysis, in which critical benchmarks are used to validate the computational methodology for predicting subcritical quantities and configurations of fissile materials :cite:`TSURFER-broadhead_sensitivity-and_2004`. Similar validation procedures also could be performed for other integral responses of interest for nuclear reactor analysis. These include responses such as reactivity coefficients associated with coolant voiding or Doppler broadening, peak power values, or in-core instrumentation readings. Although not limited to this area, the application of TSURFER to criticality safety validation studies is emphasized here. Application to validation studies ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Historically the validity of a calculation performed for some application has been established by considering how well the same calculational methods and nuclear data perform for a set of representative benchmark experiments. While a simple comparison of the computed and experimental results is very informative, it does not fully take advantage of the valuable information provided by the measured integral responses. If the original sets of calculated and experimental responses are consolidated in a consistent manner (i.e., correctly accounting for uncertainties), then the "adjusted" results should be a better estimate for the true responses, since the revised response values are based upon more information than was available in either the original calculations or measurements alone. This is essentially a statement of Bayes Theorem from probability theory, which indicates how prior information (calculated responses) can be evolved by incorporating additional information (integral measurements) into more reliable posterior results (the adjusted responses). The equations used for the GLLS methodology are equivalent to those obtained from Bayes Theorem :cite:`TSURFER-hwang1988`. Cross-section libraries for neutron transport calculations are processed from fundamental evaluated nuclear data files such as ENDF/B. Because the "true" values of the nuclear data are not known precisely, it is reasonable to view the ENDF data as being selected from a probability distribution of allowable values. Nuclear data uncertainties are described by covariance matrices that contain variances in the group cross sections for a given nuclide and reaction type, as well as covariances arising from correlations between energy groups, and possibly between reactions and materials. Discrepancies in the ENDF nuclear data caused by uncertainties in the evaluation process propagate to errors in group cross sections, which in general cause computed responses to disagree with the corresponding measured values. The GLLS approach considers potential variations in data parameters and measured integral responses that minimize the differences in measured and calculated integral responses (such as *k*\ :sub:`eff`) for a set of benchmark experiments, *taking into account uncertainties and correlations in the ENDF data and in the integral measurements*. Since there are generally many more cross-section values than measured integral responses, the determination of the data modifications is an under-determined problem. If the data variations are not performed in a reasonable manner, non-physical changes may be obtained. Data adjustments in the GLLS methodology are constrained by the magnitude of the nuclear data uncertainties and their correlations. TSURFER also accounts for uncertainties and correlations in the integral response measurements, arising from experimental uncertainties in parameters such as enrichment, density, impurities, etc. As long as realistic data covariances are used, the resulting data adjustments may be considered the "best estimates"-within the limitations of the GLLS linearity approximation-for realistic data alterations that improve the computed integral responses. It can be shown that the GLLS equations provide the maximum likelihood estimator for the correct nuclear data if the evaluated nuclear data and system parameters obey a multivariate normal probability distribution :cite:`TSURFER-hwang1988`. Some previous studies have applied the GLLS methodology to produce an adjusted nuclear data library in order to improve calculations of nuclear reactors with similar characteristics as the experiments used in the adjustment :cite:`TSURFER-ttys_1988`. In criticality safety analysis this procedure runs the risk of applying the adjusted library to systems beyond the limits for which the data modifications are appropriate. The usual function of TSURFER is not to output an adjusted nuclear data library but rather to obtain an adjusted application response (*k*\ :sub:`eff`) and to provide a quantitative estimate for its accuracy. Hence, it is more appropriate to view TSURFER as a tool to establish biases and uncertainties in calculated responses. Nuclear data adjustments are a by-product of this procedure. Traditional validation of criticality safety calculations estimates the computational bias based on trends in the calculated *k*\ :sub:`eff` values versus system parameters such as hydrogen-to-fissile ratios (H/X) or energy of average lethargy causing fission (EALF). These trending parameters are frequently used as measures of "similarity" between critical systems, hence their use as bias-predictors. Recent studies have shown that data sensitivity coefficients, either alone or in combination with cross-section uncertainty information, are good indicators of system similarity. Thus S/U-based indices have also been used in trending analyses, analogously to the commonly used physical parameters.\ :sup:`5` The input data for S/U trending analysis (i.e., calculated and measured responses, sensitivity coefficients, cross-section and experimental uncertainties) are almost identical to those needed for GLLS analysis; therefore, it is not surprising that some aspects of the TSURFER calculations are also used for trending results. However, TSURFER provides an alternative to the trending approach to determine the bias and can address other validation issues. For example, TSURFER is useful when there are few or no existing experiments that are similar to a particular application area, since the GLLS technique can include individual experiments that separately validate portions of the application area, even though none can be considered entirely similar to the application :cite:`TSURFER-goluoglu_sensitivity_2004`. Types of responses ~~~~~~~~~~~~~~~~~~ A response corresponds to a particular integral response *type* (e.g., *k*\ :sub:`eff`, reaction rate ratio, material worth, radiation dose, etc.) in a particular *nuclear* *system* (e.g., a benchmark experiment or a proposed storage arrangement of reactor fuel assemblies or a reactor core). In the TSURFER input, responses may be classified either as "\ *experiments*\ " or "\ *applications*\ " or "\ *omitted*." An *experiment* response has both calculated and measured values input to TSURFER, and these play an active role in the GLLS procedure, which minimizes the difference between the two results. A value for the uncertainty in the measured response and any correlations with other experiments are also input for experimental responses. Examples of experimental integral responses are the multiplication factor for the GODIVA critical benchmark experiment, the measured :math:`\rho^{28}` (ratio of epithermal to thermal capture rate for :sup:`238`\ U) in the TRX-1 critical benchmark lattice, or the coolant voiding reactivity in a power reactor. *Applications* are responses for which a calculated value is known but no measured value is available. Applications often correspond to hypothetical systems being considered within the context of a design study or a criticality safety analysis for which the computational bias and uncertainty are desired. Examples of application responses are the multiplication factor (subcritical) for a proposed fuel assembly storage rack or for a shipping cask. An application response plays a passive role in the GLLS procedure. Since the application has no experimental results, it does not impact the active responses included in the consolidation procedure; conversely, the GLLS procedure may modify the calculated application value if it is "similar" to some of the experimental responses. In this case the application response shares similar data sensitivity characteristics with one or more of the active responses and hence will be indirectly affected by the same data adjustments that impact the similar experimental responses. This provides a systematic, well-defined method for utilizing experimental benchmark measurements to establish a bias and uncertainty in the calculation of application response. A response designated as *omitted* in the TSURFER input neither affects other responses nor is affected by them. These responses are completely isolated from the GLLS procedure. This capability is sometimes useful to easily "turn off" an active system to observe its impact on the application results or on the consistency (chi-square) of the set of remaining experimental responses. Sources of Response Uncertainty ------------------------------- Transport calculations of responses such as the neutron multiplication factor inherently have biases and uncertainties due to several factors that can be grouped into three classes: (A) numerical approximations in the transport code; (B) system modeling approximations; and (C) input data uncertainties. Class-A uncertainties (numerical) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Class-A uncertainties are sometimes referred to as "methods uncertainties." In Monte Carlo calculations these may be caused by imperfections in random number generation routines, approximations in techniques for scoring neutron multiplication (e.g., incomplete convergence of fission source distribution, neglect of correlations between generations, etc.), and biases from algorithms used to represent nuclear data and to sample probability distributions, as well as the basic statistical uncertainty that is fundamental to the Monte Carlo method. Deterministic methods have uncertainties from using finite space-energy-direction meshes, truncated (rather than infinite) expansions of functions, incomplete convergence of iterations, and especially self-shielding approximations for the multigroup cross sections. Computational benchmark studies often can establish a reasonable upper limit for these effects, which may be judged either as negligible or as requiring some conservative bias to be applied to the application calculations. Here it is assumed that class-A uncertainties in the calculated response can be made acceptably small (e.g., by running more histories or refining mesh sizes) or at least have been previously quantified and can be bounded by a margin applied to the computation. Hence class (A) uncertainties are considered as systematic "tolerance," and are not addressed in the present GLLS methodology used by TSURFER. .. _sec-tsunami.tsurfer.classb_unc: Class-B uncertainties (modeling/experimental) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Class-B uncertainties occur because the mathematical model used in the transport computations of an application or an experimental response does not correspond exactly to the "true" system. The response uncertainty caused by modeling effects may be associated with either (i) direct computational simplifications such as omitting or homogenizing some components in the calculation model or (ii) fundamental uncertainties in the material compositions, densities, and dimensions of the experiment. The former are systematic uncertainties similar in effect to Class-A numerical uncertainties and may be addressed in the same manner; that is, by bounding the magnitude of the uncertainty through the applied safety margins. However, the latter are true random uncertainties that in theory have probability distributions, and these may be addressed with the TSURFER code. Even "clean" critical benchmark experiments have uncertainties in the nominal system parameters-such as fuel enrichment, impurities, densities, critical dimensions, and numerous other components-that may lead to discrepancies in the measured and calculated responses for the system. In TSURFER the impact of these uncertainties is designated as the "experimental uncertainty" in the response, since this uncertainty will be present even if no simplifications or approximations are made in the model used for the transport computation. The terminology is sometimes a source of confusion. For example the measured *k*\ :sub:`eff` in a critical experiment usually is known to be unity with a very small uncertainty associated with the long, but finite, stable period. While there is little doubt about *k*\ :sub:`eff` for a critical experiment, there may be considerable uncertainty in the system parameter values describing the benchmark configuration. This contribution to the modeling uncertainty could be justifiably considered either "experimental" (because system parameters such as material compositions and dimensions are specified by the experimentalists) or "computational" (because uncertainties in the system parameters affect the calculation model), but in TSURFER they are designated as experimental uncertainties. In any case the uncertainty in each system parameter must be propagated to an uncertainty in the measured response. For a *k*\ :sub:`eff` response this may be done experimentally by physically varying the system parameter and measuring the reactivity effect or, more commonly, by performing auxiliary transport calculations to determine the eigenvalue variation. This is discussed in a somewhat more quantitative manner in :numref:`sec-tsunami.tsurfer.rep_exp_unc`. The response uncertainty components associated with the respective modeling uncertainties in system parameters determine the overall experimental uncertainty. Many benchmark experiment descriptions in the *International Handbook of Evaluated Criticality Safety Benchmark Experiments* :cite:`TSURFER-briggs_international_2006` include information about uncertainties in the system parameters and their estimated impact on the multiplication factor. The standard deviations in *k*\ :sub:`eff` due to uncertainties in various system parameters are assigned by the benchmark evaluators based on published or archived experiment descriptions, and sometimes on other considerations. A complication in specifying experimental uncertainties is how to treat correlations among the experiments. Response correlations in two benchmark experiments may be caused by factors such as use of the same fuel pins and container tank, and common instrumentation (same detectors, hydrometers, etc.). For example, if two different experiments use the same fuel material, then it is not reasonable to conclude that the enrichment in one is too high while the other is too low, even if both differences fall within the specified standard deviation. Reference :cite:`TSURFER-williams_eigenvalue_2001` has shown that these correlations may not be negligible when applying the GLLS technique to a set of benchmark experiments. Only a limited amount of experiment correlation data has been published, but more is expected in future revisions to the *International Handbook of Evaluated Criticality Safety Benchmark Experiments*. TSURFER allows experimental uncertainties caused by uncertainties in system modeling parameters to be input for individual components and correlation coefficients can be specified for the shared system parameters of each response. This approach provides the capability for users to describe the actual sources of benchmark experiment correlations without having to know the overall correlation between two different experiments. See :numref:`sec-tsunami.tsurfer.rep_exp_unc` and :numref:`sec-tsunami.tsurfer.glls`. Class-C uncertainties (nuclear data) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In many applications, the major source of uncertainty in the calculated response is due to uncertainties in evaluated nuclear data such as microscopic cross sections, fission spectra, neutron yield (nu-bar), and scattering distributions that are contained in ENDF/B. These arise from uncertainties in experimental nuclear data measurements, as well as uncertainties in the evaluation process itself, which in general combine differential experimental information with nuclear physics theory to generate the basic data in compilations like ENDF/B. Class-C uncertainties are governed by probability distributions. The actual probabilities are unknown, but the evaluated data values are assumed to represent the mean of the distribution, and the evaluated variance represents a measure of the distribution width. Correlations as well as uncertainties in nuclear data can have a significant impact on the overall uncertainty in the calculated response; thus, it is important to include covariances as well as variances in the TSURFER calculations. The uncertainties in fundamental nuclear data also impact resonance self-shielding of multigroup cross-section values, further contributing to the response uncertainty :cite:`TSURFER-williams_eigenvalue_2001`. In the SCALE S/U methodology the effects of implicit changes in self-shielded cross sections are included in the overall response *sensitivity coefficients*, rather than in the covariance data, so that the fundamental data uncertainties are isolated from problem-specific effects :cite:`TSURFER-little_low-fidelity_2008` Covariance information is currently quite limited in all evaluated nuclear data compilations such as ENDF/B. A more complete library of multigroup uncertainties has been created for SCALE using data from a variety of sources, including ENDF/B-VI and VII, JENDL3.1, and approximate covariances based on uncertainties in measured integral data and nuclear model calculations :cite:`TSURFER-williams_scale-6_2008,TSURFER-little_low-fidelity_2008`. A detailed description of the SCALE covariance libraries is found in the :ref:`COVLIB chapter `. The GLLS methodology in TSURFER is mainly concerned with treating Class-C uncertainties due to nuclear data, along with Class-B experimental uncertainties. Analysis Procedure ------------------ .. _sec-tsunami.tsurfer.other_scale: Functional relation to other SCALE modules ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TSURFER is a functional module within the overall SCALE S/U methodology. Other modules in SCALE and outside SCALE perform complementary calculations and provide data files used by TSURFER, as described below. - PUFF-IV: AMPX code that processes ENDF/B nuclear data covariances and generates multigroup covariance data; creates nuclear data uncertainty files for input to TSURFER. - TSUNAMI-:ref:`1D `/:ref:`2D/3D ` SCALE control sequence that computes sensitivity coefficients for *k*\ :sub:`eff` or other responses in a 1D/2D/3D model of the experiment or application system; creates sensitivity files used by TSURFER. - TSAR: SCALE functional module that computes sensitivity coefficients for eigenvalue-difference responses such as reactor reactivity parameters, using *k*\ :sub:`eff` sensitivities from a TSUNAMI sequence; creates sensitivity files used by TSURFER. - :ref:`TSUNAMI-IP ` SCALE functional module that computes similarity and completeness indices for a set of responses with sensitivity coefficients. Prior to running TSURFER, it may be advantageous to perform scoping studies with TSUNAMI-IP to determine if the selected set of benchmark experiments provides adequate "coverage" for data uncertainties that have a significant impact on the application response. .. _sec-tsunami.tsurfer.guidelines: Guidelines for TSURFER analysis ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Both active and passive responses may be included in the TSURFER calculation. For example, in criticality safety validation procedures, responses of interest typically correspond to the system multiplication factor, *k*\ :sub:`eff`. The desired sub-critical design configuration would be a passive application system, while the critical benchmarks used for validation are active experiment systems. TSURFER determines the application bias and uncertainty by propagating data variations obtained from GLLS analysis of the active systems to the calculated multiplication factor of the passive system. The bias represents the change of the application's original *k*\ :sub:`eff` as a result of the consolidation of all the active critical benchmark experiments and the adjusted nuclear data parameters. An increase in *k*\ :sub:`eff` computed for the applications system response indicates that the calculated value was initially too low (a negative bias), and a decrease that the application's multiplication factor was too high (a positive bias). TSURFER also computes uncertainties in the initial and adjusted estimates for the system responses (e.g., multiplication factors). These response uncertainties include effects of experimental uncertainties (class-B) and nuclear data uncertainties (class-C) but not the impact of simplifications made to the experiment specifications and numerical approximations (class-A)-which should be included in the safety margin. Several quantities can be examined to give confidence to the predicted results :cite:`TSURFER-broadhead_sensitivity_1999,TSURFER-broadhead_sensitivity_1999-1`. The first is the completeness parameter, R, given in the output of the TSUNAMI-IP code. It has been suggested that values of R greater than about 0.7-0.8 for a set of active experiments indicate adequate cross-section coverage for an application response in the GLLS procedure; however, this is a preliminary conclusion that may change as more experience is gained :cite:`TSURFER-goluoglu_sensitivity_2004`. The *chi-square* (:math:`\chi^2`) statistic indicates the overall consistency of the suite of benchmarks and is key to proper interpretation of the TSURFER results. The value of :math:`\chi^2` per degree of freedom represents the average discrepancy between the calculated and measured responses, expressed in units of the combined variance of the calculation plus experiment. Values of chi-square per degree of freedom ideally should be within about :math:`\pm`\ 20% of unity, indicating that the calculations and measurements on average agree within about one standard deviation. Results in which this test is not strictly met may still be valid, but in general these should be viewed with skepticism unless the reasons for the test failure are understood. An excessively large :math:`\chi^2` can lead to unreliable results in the GLLS adjustment. TSURFER provides the total :math:`\chi^2` value, as well as estimated contributions from each experiment (see :numref:`sec-tsunami.tsurfer.chi_filter`). Individual :math:`\chi`\ :sup:`2` values suggest which experiments may contain inconsistencies (i.e., the difference between the measured and calculated *k*\ :sub:`eff` is larger than their combined uncertainties). Several methods can be used to improve the initial value of chi-square. One approach is to reevaluate the measurement uncertainties and their correlations for identified discrepant experiments. If the experimental or data uncertainties are underestimated, the data adjustments will correspond to an excessive number of standard deviations, as reflected in high :math:`\chi`\ :sup:`2` values. Values of :math:`\chi`\ :sup:`2` that are too low suggest that the input uncertainty estimates may be too high, and again a reevaluation should be considered. Thus it is quite important to utilize *realistic* (not conservative) estimates for the uncertainties in nuclear data and experimental measurements. Even when best estimates are used for all input uncertainties, it is not uncommon to encounter a few active responses that are inconsistent with the others, especially when dealing with a large number of benchmark experiments. In this case the best alternative to improve :math:`\chi^2` is to remove the outliers, either by transforming those experiments into passive responses or by omitting them entirely from the GLLS adjustment. TSURFER provides a "chi-square filtering" procedure that automatically omits inconsistent experiments until a specified target value of chi-square is achieved. Several options are provided to select the experiments to be omitted, as discussed in :numref:`sec-tsunami.tsurfer.chi_filter`. Omitted experiments should be examined to ensure that simple errors in the problem description are not present. An internal consistency test such as described in :cite:`TSURFER-broadhead_sensitivity_1999` also may be useful. The consistency test is performed by changing one of the benchmark experiments that is similar (see :numref:`sec-tsunami.tsurfer.similarity`) to the application response into a passive, pseudo-application response. The predicted bias for this passive response should be close to that of the original application; furthermore, in this case the bias prediction can be checked because this passive response actually has a measured experimental value. Required data for TSURFER ~~~~~~~~~~~~~~~~~~~~~~~~~ Active and passive responses considered in the GLLS analysis should have sensitivity data provided for each nuclide-reaction pair that significantly impacts the response. The sensitivity coefficients are pre-calculated using other SCALE modules as described in :numref:`sec-tsunami.tsurfer.other_scale` and are stored in individual files for each response included in the TSURFER analysis. The sensitivity data files must be in one of the SCALE sensitivity data formats described in :ref:`tsunami.data_formats`. The locations of the sensitivity files are specified in the TSURFER input data. It is not required for all sensitivity files to have the same group structures; for example, the sensitivity coefficients for one response may have been computed using a 238-group cross-section library, while sensitivities for another response could have a 44-group structure. Whatever the group structure of the sensitivity data, it is mapped into the same group structure as the covariance file. At present the SCALE covariance files use the SCALE 56-group structure by default. A file of nuclear data covariances also must be input to the TSURFER calculation. The covariance data file must be in COVERX format described in the :ref:`sec-scale.coverx_format` section. SCALE includes a comprehensive applications-oriented covariance library that includes evaluated covariances taken from ENDF/B-VII, ENDF/B-VI, and JENDL nuclear data files :cite:`TSURFER-williams_scale-6_2008` described in the :ref:`COVLIB chapter `. Ideally, the covariance files should contain data for all nuclide-reaction pairs on the response sensitivity data files. However, cross-section covariance data are not available for all nuclide-reaction pairs. Nuclide-reaction pairs without available covariance data are omitted from the GLLS analysis, but it is assumed that either the cross-section covariance data values for these pairs are well known (i.e., small uncertainties) or that the sensitivity to these nuclide-reaction pairs is small. Where these assumptions hold, the cross sections for these nuclide-reaction pairs should not be adjusted and can be omitted from the GLLS analysis. For situations where these assumptions are judged to be invalid, the use of GLLS analysis is not appropriate. However, TSURFER provides several input options to define uncertainty values for nuclide-reaction pairs with missing covariance data to assess the impact of the additional covariance data on the GLLS analysis. These input options are discussed in more detail in :numref:`sec-tsunami.tsurfer.parameter` and :numref:`6-6-5-3`. TSURFER Computation Methodology ------------------------------- A recent detailed derivation of the GLLS formalism is given in :cite:`TSURFER-broadhead_sensitivity_1999`. The general formalism allows cross correlations between the initial integral experiment measurements and the original nuclear data, such as would be present if the calculations used a previously "adjusted" library of nuclear data. Since this is not normally done in SCALE, correlations between the benchmark experiment measurements and the cross-section data in the multigroup libraries are not considered in the TSURFER code; therefore, the GLLS equations presented here are somewhat simplified compared to the more general expressions in :cite:`TSURFER-williams_perturbation_1986`. At present, the SCALE cross-section-covariance data files characterize nuclear data uncertainties in terms of relative covariances. Therefore, response sensitivity data in TSURFER are defined in terms of relative changes in the nuclear data. An *absolute* response sensitivity is defined as an absolute change in response due to a relative change in the nuclear data, that is, .. math:: {{\tilde{S}}_{\alpha }}\text{=}\alpha \frac{\partial \text{R}}{\partial \alpha } In this equation, R represents the response, :math:`\alpha` represents the nuclear data, and the tilde will be used to represent absolute sensitivity and uncertainty data. Likewise, a *relative* response sensitivity is defined as a relative change in response due to a relative change in the nuclear data, that is, .. |cmm| replace:: **C**\ :math:`_{\bf{mm}}` .. |caa| replace:: :math:`\mathbf{C}_{{\mathbf{\alpha \alpha }}}` .. |ckk| replace:: **C**\ :math:`_{\mathbf{kk}}` .. |raa| replace:: :math:`\mathbf{R}_{{\mathbf{\alpha \alpha }}}` .. |rmm| replace:: **R**\ :math:`_{\bf{mm}}` .. math:: {{\text{S}}_{\alpha }}\text{=}\frac{\alpha }{\text{R}}\frac{\partial \text{R}}{\partial \alpha } The initial development that follows is for *relative*, rather than *absolute*, response sensitivity and uncertainty parameters. It is then shown how to express the quantities in absolute form for reactivity analysis and mixed relative-absolute form for combined *k*\ :sub:`eff` and reactivity analysis. A summary of the notation and definitions used in this section can be found in :ref:`6-6A`. The methodology consists of calculating values for a set of I integral responses (*k*\ :sub:`eff`, reaction rates, etc.), some of which have been measured in selected benchmark experiments. Responses with no measured values are the selected "applications," described previously. The set of measured response values {m\ :sub:`i` ; i=1,2,…., I} can be arranged into an I-dimension column vector designated as **m**. By convention the (unknown) experimental values corresponding to applications are represented by the corresponding calculated values. As discussed in :numref:`sec-tsunami.tsurfer.classb_unc`, the measured integral responses have uncertainties---possibly correlated---due to uncertainties in the system parameter specifications. The I :math:`\times` I covariance matrix describing the relative experimental uncertainties is defined to be **C**\ :math:`_{\bf{mm}}`. The calculated integral values for each experiment and application are obtained by neutron transport calculations, producing a set of calculated responses {k\ :sub:`i` ; i=1,2,…., I} arranged in a I-dimension vector **k**. The multigroup cross-section data for all nuclide-reaction pairs used in the transport calculations of all responses comprise a set {:math:`\alpha`\ :sub:`n` ; n=1,2,…., M}, where M is the number of unique nuclide-reaction pairs multiplied by the number of energy groups. It is convenient to arrange these data into a M-dimensional column vector :math:`\mathbf{\alpha}`, so that the dependence of the initial calculated responses upon the input nuclear data values can be indicated as **k** = **k**\ (:math:`\alpha`). The prior covariance matrix for the nuclear data is equal to the M :math:`\times` M matrix :math:`\mathbf{C}_{\mathbf{\alpha \alpha }}`, which contains relative variances along the diagonal and relative covariances in the off-diagonal positions. These data describe uncertainties in the *infinitely dilute* multigroup cross sections. Nuclear data uncertainties cause uncertainties in the calculated response values. In general, these uncertainties are correlated because the same nuclear data library is used for all the transport calculations. The covariance matrix describing uncertainties in the calculated responses due to class-C uncertainties is designated as **C**\ :math:`_{\mathbf{kk}}`. Using expressions for propagation of error (the so called sandwich rule), the following relationship is obtained for the relative uncertainty in the calculated responses: .. math:: :label: eq6-6-1 \mathbf{C}_{\mathbf{k k}}=\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}} \mathbf{S}_{\mathbf{k \boldsymbol{\alpha}}}^{\mathbf{T}} .. |ska| replace:: :math:`\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}` where :math:`\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}` is the relative sensitivity matrix, whose (i, n) element is equal to the relative sensitivity of the i\ :sub:`th` response with respect to the n\ :sub:`th` data value, that is, :math:`\frac{1}{\mathrm{R}_{\mathrm{i}}} \frac{\partial \mathrm{R}_{\mathrm{i}}}{\partial \alpha_{\mathrm{n}}} \alpha_{\mathrm{n}}`. Sensitivity coefficients appearing in the sensitivity matrix are computed using first-order perturbation theory. A description of the equations used to compute sensitivity coefficients using first order perturbation theory can be found in :cite:`TSURFER-williams_perturbation_1986` or in the :ref:`SAMS chapter `. In the SCALE methodology, the sensitivity coefficients consist of an "explicit" component that accounts for the direct impact of the data on the neutron transport calculation, as well as an "implicit" component that accounts for its impact on other self-shielded multigroup data :cite:`TSURFER-williams_eigenvalue_2001`. For example, a variation in the hydrogen multigroup elastic cross section has an explicit effect on *k*\ :sub:`eff` through its impact on neutron moderation and leakage in the transport solution and has an implicit effect on the self-shielded :sup:`238`\ U multigroup cross sections, which causes additional change in *k*\ :sub:`eff`. Because self-shielding effects are addressed through the sensitivity coefficients, the nuclear data uncertainties in the covariance matrix correspond to the infinitely dilute values. Each row i of the sensitivity matrix contains sensitivity coefficients for all nuclear data used in the transport calculation of response i. These data also can be arranged into an M-component sensitivity *vector* **S**\ :sub:`i` for a particular response "i", which may be either an experiment or application. For example, the sensitivity vector **S**\ :sub:`i` is an M dimensional vector whose n\ :sup:`th` element is equal to the sensitivity coefficient of response "i" to data element :math:`\alpha_n` as specified previously. It is often convenient to express covariance matrices in terms of standard deviations [represented as :math:`\sigma`\ :sub:`i` for variable *i*] and correlation coefficients [represented by :math:`\rho_{i,j}` for the variable pair (*i,j*)]. The correlation coefficient is related to the corresponding covariance value by the equation .. math:: \rho_{\mathrm{i}, \mathrm{j}}=\frac{\operatorname{Cov}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{i}} \sigma_{\mathrm{j}}} Correlation coefficients, which have values between -1 and 1, indicate the degree of correlation between the pair of variables, where a value of 1.0 indicates full correlation, 0.0 no correlation, and -1.0 full anti-correlation. Using matrix notation, relative standard deviations are arranged in a diagonal matrix :math:`\boldsymbol{\sigma}` and the correlation coefficients in a square matrix **R** (symmetrical, but generally non-diagonal). The previously defined covariance matrices can be expressed as follows: .. math:: :label: eq6-6-2 \mathbf{C}_{\mathbf{mm}}=\sigma_{\mathbf{m}} \mathbf{R}_{\mathbf{mm}} \sigma_{\mathbf{m}} .. math:: :label: eq6-6-3 \mathbf{C}_{\boldsymbol{\alpha \alpha}}=\sigma_{\boldsymbol{\alpha}} \mathbf{R}_{\boldsymbol{\alpha \alpha}} \sigma_{\boldsymbol{\alpha}} .. math:: :label: eq6-6-4 \mathbf{C}_{\mathbf{kk}}=\sigma_{\mathbf{k}} \mathbf{R}_{\mathbf{kk}} \sigma_{\mathbf{k}} :eq:`eq6-3-3` and :eq:`eq6-6-4` can be substituted into :eq:`eq6-6-1` and rearranged to relate the nuclear data correlations to the correlations in the computed responses: .. math:: :label: eq6-6-5 \mathbf{R}_{\mathbf{kk}}=\left[\boldsymbol{\sigma}_{\mathbf{k}}^{-1} \mathbf{~S}_{\mathbf{k} \boldsymbol{\alpha}} \boldsymbol{\sigma}_{\alpha}\right] \mathbf{R}_{\boldsymbol{\alpha \alpha}} \left[\boldsymbol{\sigma}_{\mathbf{k}}^{-1} \mathbf{~S}_{\mathbf{k} \boldsymbol{\alpha}} \sigma_{\alpha}\right]^{\mathbf{T}} . The bracketed term on the right side of the above equation is an I :math:`\times` M matrix whose elements equal the number of relative standard deviations (:math:`\boldsymbol{\sigma}_{\mathbf{k}}`) that the response changes, due to a one relative standard deviation change in the nuclear data (:math:`\boldsymbol{\sigma}_{\boldsymbol{\alpha}}`). Even if the nuclear data are not correlated---that is, :math:`\mathbf{R}_{\boldsymbol{\alpha}\boldsymbol{\alpha}}` is an identity matrix---\ :math:`\mathbf{R}_{\mathbf{kk}}` is generally not diagonal. The expressions thus far describe *relative* response sensitivities and covariances (i.e., uncertainties). Similar expressions can also be derived for absolute quantities. In this report, absolute response sensitivities and covariances are denoted by a tilde, such as :math:`{{\mathbf{\tilde{C}}}_{\mathbf{mm}}}`, :math:`\,{{\mathbf{\tilde{C}}}_{\mathbf{kk}}}`, and :math:`\mathbf{\tilde{S}}_{\mathbf{k} \boldsymbol{\alpha}}`, which are explicitly defined in Appendix A. TSURFER allows for a mixed relative and absolute response sensitivities and covariances to be used in the analysis. In the TSURFER input (described in :numref:`6-6-5`), each response is designated as an *absolute*-formatted response or a *relative*-formatted response using the input keywords *"absolute"* or *"relative"*. This flexibility allows for the simultaneous use of both relative-formatted *k*\ :sub:`eff` sensitivity data (generated by TSUNAMI modules) and absolute-formatted eigenvalue-difference sensitivity data (generated by TSAR) in the same analysis. In this report, mixed relative-absolute response sensitivities and covariances are denoted by a caret, such as :math:`{{\mathbf{\hat{C}}}_{\mathbf{mm}}}`, :math:`\,{{\mathbf{\hat{C}}}_{\mathbf{kk}}}`, and :math:`\mathbf{\hat{S}}_{\mathbf{k} \boldsymbol{\alpha}}`. .. _sec-tsunami.tsurfer.rep_exp_unc: Representation of experimental uncertainty components ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Experimental uncertainties (i.e., type-B uncertainties as described in :numref:`sec-tsunami.tsurfer.classb_unc`) may be entered directly in the TSURFER input, or alternatively, it may be specified by defining individual "uncertainty components." The latter approach is useful in defining experimental correlations between measured responses. In this case, an index " :math:`\ell` " is introduced to identify the response uncertainty components associated with a particular system parameter, p\ :math:`_{\ell}`. For example, the measured *k*\ :sub:`eff` uncertainty components for a particular critical experiment consisting of uranyl nitrate dissolved in water might correspond to the following eight p\ :math:`_{\ell}` contributors, as identified by the value of (:math:`\ell`): (1) isotopic composition; (2) fuel concentration in the solution; (3) solution density; (4) excess acid concentration in the solution; (5) fuel impurities; (6) dimension of the solution tank; (7) thickness of the solution tank; (8) composition of the tank :cite:`TSURFER-briggs_international_2006`. The relative standard deviation of a measured response m\ :sub:`i` due to an uncertainty in the system parameter p\ :math:`_{\ell}` is designated as the uncertainty component :math:`\sigma _{m,i}^{\left( \ell \right)}`. Assuming that uncertainties in system parameters are uncorrelated, the response uncertainty component is related to the uncertainty in system parameter p\ :math:`_{\ell}` by the expression .. math:: \sigma_{\mathrm{m}, \mathrm{i}}^{(\ell)}=\frac{1}{\mathrm{~m}_{\mathrm{i}}}\left(\frac{\partial \mathrm{k}_{\mathrm{i}}}{\partial \mathrm{p}} \mathrm{p}\right) \sigma_{\mathrm{p}}=\mathrm{S}_{\mathrm{m}_{\mathrm{i}} \mathrm{p}} \sigma_{\mathrm{p}} , where :math:`{{\sigma }_{{{p}_{\ell }}}}` is the relative standard deviation of system parameter p\ ::math:`_{\ell}` and :math:`\text{S}_{{{\text{m}}_{\text{i}}}{{\text{p}}_{\ell }}}^{{}}` is the relative sensitivity coefficient relating p\ :math:`_{\ell}` to the measured response m\ :sub:`i`. In principle, the system parameter values and uncertainties could be treated directly in the TSURFER calculation by providing the sensitivity coefficients :math:`\text{S}_{{{\text{m}}_{\text{i}}}{{\text{p}}_{\ell }}}^{{}}`, thus allowing the experiment parameters to be included in the GLLS adjustment. However, at the present time the response uncertainty components :math:`\boldsymbol{\sigma}_{m, i}^{(\ell)}` must be determined prior to the TSURFER calculation and are read into TSURFER. Values for the response uncertainty components sometimes can be found in the benchmark experiment specifications :cite:`TSURFER-goluoglu_sensitivity_2004`, or auxiliary sensitivity analysis may be necessary. The relative experimental standard deviation of the i\ :sub:`th` measured response is calculated from .. math:: :label: eq6-6-6 \sigma_{\mathrm{m}, \mathrm{i}}=\sqrt{\sum_{1}\left(\sigma_{\mathrm{m}, \mathrm{i}}^{(\mathrm{1})}\right)^{2}} The (i,i) diagonal element of the relative covariance matrix corresponds to the relative experimental variance in response i, which is equal the square of :math:`\sigma _{m,i}^{{}}` above. Note that similar expressions can be derived for uncertainty components using absolute sensitivities and uncertainties. For absolute-formatted responses, the uncertainty components on the TSURFER input must be entered in terms of absolute standard deviation. This is discussed in more detail in :numref:`sec-tsunami.tsurfer.response`. If a different response j is measured in a benchmark system that shares some or all of the same uncertainty components as response i, then the two experiment responses have correlated uncertainties. In such a case the (i, j) element of the relative covariance matrix :math:`{{\mathbf{C}}_{\mathbf{mm}}}` is equal to .. math:: :label: eq6-6-7 \mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\sum_{\ell}\left(\sigma_{\mathrm{m}, \mathrm{i}}^{(\ell)} \sigma_{\mathrm{i}, \mathrm{j}}^{(\ell)} \sigma_{\mathrm{m}, \mathrm{j}}^{(\ell)}\right) and the total correlation coefficient for response pair (i, j) is .. math:: :label: eq6-6-8 \mathrm{R}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})=\frac{\mathrm{C}_{\mathrm{mm}}(\mathrm{i}, \mathrm{j})}{\sigma_{\mathrm{m}, \mathrm{i}} \sigma_{\mathrm{m}, \mathrm{j}}}=\frac{\sum\left(\sigma_{\mathrm{m}, \mathrm{i}}^{(\ell)} \rho_{\mathrm{i}, \mathrm{j}}^{(\ell)} \sigma_{\mathrm{m}, \mathrm{j}}^{(\ell)}\right)}{\sigma_{\mathrm{m}, \mathrm{i}} \sigma_{\mathrm{m}, \mathrm{j}}} where :math:`\rho _{\text{i,j}}^{(\ell )}`. is the correlation coefficient for responses i and j due uncertainty component :math:`\ell`. TSURFER allows the user to input text-identifiers for the various experiment uncertainty components, along with the associated values for relative standard deviations, :math:`\sigma _{\text{m,i}}^{\text{(}\ell \text{)}}`. Response correlation coefficients :math:`\rho _{\text{i,j}}^{(\ell )}` can be input for each type of uncertainty component, by response pair (i, j). The previous discussion applies only to *experiment responses* for which measurements have been performed. In the case of an *application response* for which no experimental measurement is known, the uncertainty is set internally by TSURFER to the large value of 10\ :sup:`10`, to approximate the "infinite" uncertainty in the unknown measurement, and correlations to other responses are set to zero. The large uncertainty for an application response has the effect of letting the response "float" in a passive manner; that is, the application response has a negligible effect on the adjustment of active responses, but the GLLS consolidation of the active responses with finite uncertainties can impact the adjusted value for the application. .. _sec-tsunami.tsurfer.glls: Generalized linear least-squares equations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Discrepancies in the calculated and measured responses are defined by the I dimensional column vector .. math:: :label: eq6-6-9 \mathbf{d} = \left\{\mathrm{d}_{i}=\frac{\mathrm{k}_{\mathrm{i}}(\alpha)-\mathrm{m}_{i}}{\mathrm{k}_{\mathrm{i}}(\alpha)}, i=1, \dots, I\right\} In TSURFER the components of **d** corresponding to application responses are set to zero because applications have no measured values. Using the standard formula for propagation of error and assuming no correlations between k and m, the relative uncertainty matrix for the discrepancy vector **d** can be expressed as the I by I matrix: .. math:: :label: eq6-6-10 \mathbf{C}_{\mathrm{dd}}=\mathbf{C}_{\mathbf{k k}}+\mathbf{F}_{\mathbf{m} / \mathbf{k}} \mathbf{C}_{\mathbf{mm}} \mathbf{F}_{\mathbf{m} / \mathbf{k}} = \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}} \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}}+\mathbf{F}_{\mathbf{m} / \mathbf{k}} \mathbf{C}_{\mathbf{m} \mathbf{m}} \mathbf{F}_{\mathbf{m} / \mathbf{k}} where the expression in :eq:`eq6-6-1`  was substituted for |ckk|, and **F**\ :math:`_{\mathbf{m}/ \mathbf{k}}` is an I :math:`\times` I diagonal matrix containing m/k factors, that is, :math:`\frac{E}{C}` factors (ratio of experimental to calculated response values). The inverse of the matrix **C**\ :math:`_{\mathbf{dd}}` appears in several expressions presented later in this section. In TSURFER the inversion is performed using routines from the LINPACK software package. The goal of the GLLS method is to vary the nuclear data :math:`\left(\alpha \rightarrow \alpha^{\prime}\right)` and the measured integral responses :math:`\left(\mathrm{m} \rightarrow \mathrm{m}^{\prime}\right)`, such that they are most consistent with their respective uncertainty matrices, |caa| and |cmm|. This is done by minimizing chi-square, expressed as .. math:: :label: eq6-6-11 \begin{aligned} \chi^{2} &=\left[\frac{\boldsymbol{\alpha}^{\prime}-\boldsymbol{\alpha}}{\boldsymbol{\alpha}}\right]^{\mathrm{T}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}^{-1}\left[\frac{\boldsymbol{\alpha}^{\prime}-\boldsymbol{\alpha}}{\boldsymbol{\alpha}}\right]+\left[\frac{\mathbf{m}^{\prime}-\mathbf{m}}{\mathbf{m}}\right]^{\mathrm{T}} \mathbf{C}_{\mathbf{mm}}^{-1}\left[\frac{\mathbf{m}^{\prime}-\mathbf{m}}{\mathbf{m}}\right] \\ &=[\boldsymbol{\Delta} \boldsymbol{\alpha}]^{\mathrm{T}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}^{-1}[\boldsymbol{\Delta} \boldsymbol{\alpha}]+[\boldsymbol{\Delta} \mathbf{m}]^{\mathrm{T}} \mathbf{C}_{\mathbf{mm}}^{-1}[\boldsymbol{\Delta} \mathbf{m}] \end{aligned} where :math:`\Delta \alpha_{i}=\frac{\alpha_{i}^{\prime}-\alpha_{i}}{\alpha_{i}}` and :math:`\Delta \mathrm{m}_{i}=\frac{\mathrm{m}_{i}^{\prime}-\mathrm{m}_{i}}{\mathrm{~m}_{i}}`. :eq:`eq6-6-11` is rearranged to give .. math:: :label: eq6-6-12 \boldsymbol{\chi}^{2}=\left[\boldsymbol{\sigma}_{\boldsymbol{\alpha}}^{-1} \boldsymbol{\Delta} \boldsymbol{\alpha}\right]^{\mathbf{T}} \mathbf{R}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}^{-1}\left[\boldsymbol{\sigma}_{\boldsymbol{\alpha}}^{-1} \boldsymbol{\Delta} \boldsymbol{\alpha}\right]+\left[\boldsymbol{\sigma}_{\mathbf{m}}^{-1} \boldsymbol{\Delta} \mathbf{m}\right]^{\mathbf{T}} \mathbf{R}_{\mathbf{mm}}^{-1}\left[\boldsymbol{\sigma}_{\mathbf{m}}^{-1} \boldsymbol{\Delta} \mathbf{m}\right] :eq:`eq6-6-12` expresses the variations in the nuclear data and measured responses in units of their respective standard deviations; that is, :math:`\mathbf{\left[\boldsymbol{\sigma}_{\boldsymbol{\alpha}}^{-1} \Delta \boldsymbol{\alpha}\right]}` and :math:`\left[\boldsymbol{\sigma}_{\mathbf{m}}^{-1} \boldsymbol{\Delta} \mathbf{m}\right]` Chi-square is a quadratic form indicating the squared magnitude of the combined data variations with respect to their uncertainties. This is easily seen for the simple case in which |raa|\ :sup:`-1` and |rmm|\ :sup:`-1` in :eq:`eq6-6-12` are identity matrices, so that :eq:`eq6-6-12` reduces to just the diagonal contributions: .. math:: :label: eq6-6-13 \chi^{2} \rightarrow \sum_{\mathrm{n}=1}^{\mathrm{M}}\left(\frac{\alpha_{n}^{\prime}-\alpha_{n}}{\sigma_{\alpha_{n}}}\right)^{2}+\sum_{\mathrm{i}=1}^{\mathrm{I}}\left(\frac{\mathrm{m}_{\mathrm{i}}^{\prime}-\mathrm{m}_{\mathrm{i}}}{\sigma_{\mathrm{m}_{i}}}\right)^{2} The first term on the on the right side of :eq:`eq6-6-13` is equal to the sum of the squares of the individual nuclear data variations expressed in units of their standard deviations while the second term represents a similar quantity for the measured integral responses. In the general case where correlations exist, the inverse matrices in :eq:`eq6-6-12` are not diagonal, and the value of chi-square must be evaluated using the indicated matrix multiplication. Thus it can be seen that the GLLS method determines adjustments in the nuclear data and experimental measurements that (a) make the calculated and measured responses agree [i.e., :math:`k^{\prime}=k^{\prime} \left(\alpha^{\prime}\right) = m^{\prime}` within the limitations of first-order sensitivity theory], and (b) minimize :eq:`eq6-6-11` so that the adjustments are most consistent with the data uncertainties. Although many possible combinations of data variations may make :math:`k^{\prime}=m^{\prime}`, there is a unique set that also minimizes :math:`\chi^2`. .. note:: In TSURFER the term "chi-square" normally is meant to signify the minimum value of the quadratic form in :eq:`eq6-6-11`. The significance of this minimum value is discussed in :numref:`sec-tsunami.tsurfer.chi_filter`. The following variations minimize :eq:`eq6-6-11`, subject to the constraint :math:`k^{\prime} \left(\alpha^{\prime}\right) = m^{\prime}` :cite:`TSURFER-williams_perturbation_1986` and the linearity condition :math:`[\boldsymbol{\Delta} \mathbf{k}]=\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}[\mathbf{\Delta} \boldsymbol{\alpha}]` where :math:`\Delta \mathrm{k}_{i}=\frac{\mathrm{k}_{i}^{\prime}-\mathrm{k}_{i}}{\mathrm{k}_{i}}`: .. math:: :label: eq6-6-14 \boldsymbol{\Delta} \boldsymbol{\alpha}=-\left[\mathbf{C}_{\boldsymbol{\alpha \alpha}} \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}} \mathbf{C}_{\mathbf{dd}}^{-1}\right] \mathbf{d} .. math:: :label: eq6-6-15 \boldsymbol{\Delta} \mathbf{m}=\left[\mathbf{C}_{\mathbf{mm}} \mathbf{F}_{\mathbf{m} / \mathbf{k}} \mathbf{C}_{\mathbf{dd}}^{-1} \right] \mathbf{d} In the above equations the initial response discrepancy vector **d** is operated on by the transformation matrix in square brackets to obtain the desired variations in nuclear data and integral measurements; thus, it is the discrepancy components that drive the adjustments. If the linearity assumption is valid, then the changes in the calculated responses are found to be .. math:: :label: eq6-6-16 \boldsymbol{\Delta} \mathbf{k}=\mathbf{F}_{\mathbf{m} / \mathbf{k}} \boldsymbol{\Delta} \mathbf{m}-\mathbf{d}=\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}} \boldsymbol{\Delta} \boldsymbol{\alpha} . :eq:`eq6-6-16` relates the adjustments in calculated responses, measured responses, and nuclear data. As previously discussed, consolidation of the calculated and measured responses reduces the prior uncertainties for :math:`\alpha^{\prime}`, **m**, and **k** because additional knowledge has been incorporated. This is indicated by their modified covariance matrices :math:`\mathbf{C}_{\boldsymbol{\alpha}^{\prime} \boldsymbol{\alpha}^{\prime}}`, :math:`\mathbf{C}_{\mathbf{m}^{\prime} \mathbf{m}^{\prime}}`, :math:`\mathbf{C}_{\mathbf{k}^{\prime} \mathbf{k}^{\prime}}`, respectively, given by :cite:`TSURFER-williams_perturbation_1986` .. math:: :label: eq6-6-17 \mathbf{C}_{\boldsymbol{a}^{\prime} \boldsymbol{\alpha}^{\prime}}=\mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}-\left[\mathbf{C}_{\boldsymbol{\alpha} \mathbf{\alpha}} \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}} \mathbf{C}_{\mathbf{dd}}^{-1} \mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}\right] .. math:: :label: eq6-6-18 \mathbf{C}_{\mathbf{m}^{\prime} \mathbf{m}^{\prime}}=\mathbf{C}_{\mathbf{m m}}-\left[\mathbf{C}_{\mathbf{m m}} \mathbf{F}_{\mathbf{m} / \mathbf{k}} \mathbf{C}_{\mathbf{dd}}^{-1} \mathbf{F}_{\mathbf{m} / \mathbf{k}} \mathbf{C}_{\mathbf{mm}}\right] .. math:: :label: eq6-6-19 \mathbf{C}_{\mathbf{k}^{\prime} \mathbf{k}^{\prime}}=\mathbf{C}_{\mathbf{k k}}-\left[\mathbf{C}_{\mathbf{k k}} \mathbf{C}_{\mathbf{dd}}^{-1} \mathbf{C}_{\mathbf{k k}}\right] . If all the responses on the TSURFER input are relative formatted, then the adjusted data and response values edited by TSURFER are obtained from :eq:`eq6-6-14`-:eq:`eq6-6-16`, while the square roots of diagonal elements in :eq:`eq6-6-17`-:eq:`eq6-6-19` correspond to the edited values for adjusted uncertainties in the nuclear data and in the experiment responses, respectively. The adjustment formulas must be modified slightly to be consistent with the absolute-formatted responses. In the following expressions, absolute response covariance and response sensitivity data are denoted by a tilde [see Appendix A.]: .. math:: :label: eq6-6-20 \mathbf{\tilde{d}}=\mathbf{k}(\boldsymbol{\alpha})-\mathbf{m} .. math:: :label: eq6-6-21 \mathbf{\tilde{C}}_{\mathbf{dd}}=\mathbf{\tilde{C}}_{\mathbf{k k}}+\mathbf{\tilde{C}}_{\mathbf{mm}} =\mathbf{\tilde{S}}_{\mathbf{k} \boldsymbol{\alpha}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}} \mathbf{\tilde{S}}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}} +\mathbf{\tilde{C}}_{\mathbf{m m}} .. math:: :label: eq6-6-22 \Delta \boldsymbol{\tilde{\alpha}}=\boldsymbol{\alpha}^{\prime}-\boldsymbol{\alpha} =-\left[\mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}} \mathbf{\tilde{S}}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}} \mathbf{\tilde{C}}_{\mathbf{dd}}^{-\mathbf{1}}\right] \mathbf{\tilde{d}} .. math:: :label: eq6-6-23 \Delta \mathbf{\tilde{m}}=\mathbf{m}^{\prime}-\mathbf{m} =\left[\mathbf{\tilde{C}}_{\mathbf{mm}} \mathbf{\tilde{C}}_{\mathbf{dd}}^{-1}\right] \mathbf{\tilde{d}} .. math:: :label: eq6-6-24 \Delta \mathbf{\tilde{k}}=\mathbf{k}^{\prime}-\mathbf{k}=\left(\mathbf{m}^{\prime}-\mathbf{m}\right)-\mathbf{\tilde{d}} =\mathbf{S}_{\mathbf{k} \boldsymbol{\alpha}}\left(\boldsymbol{\alpha}^{\prime}-\boldsymbol{\alpha}\right) Relative covariances for the posterior values of the nuclear data and measured responses are given as .. math:: :label: eq6-6-25 \mathbf{C}_{\boldsymbol{\alpha}^{\prime} \boldsymbol{\alpha}^{\prime}}=\mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}} - \left[\mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}} \mathbf{\tilde{S}}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}}\right] \mathbf{\tilde{C}}_{\mathbf{d} \mathbf{d}}^{-1} \left[\mathbf{\tilde{S}}_{\mathbf{k} \boldsymbol{\alpha}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}\right] .. math:: :label: eq6-6-26 \mathbf{\tilde{C}}_{\mathbf{m}^{\prime} \mathbf{m}^{\prime}}=\mathbf{\tilde{C}}_{\mathbf{m m}} - \left[\mathbf{\tilde{C}}_{\mathbf{m m}} \mathbf{\tilde{C}}_{\mathbf{dd}}^{-1} \mathbf{\tilde{C}}_{\mathbf{mm}}\right] If all the responses on the TSURFER input are absolute formatted, the adjusted data and response values edited by TSURFER are obtained from :eq:`eq6-6-22`-:eq:`eq6-6-24`, while the square roots of diagonal elements in :eq:`eq6-6-25`-:eq:`eq6-6-26` correspond to the edited values for adjusted uncertainties in the nuclear data and in the experiment responses, respectively. The adjustment formulas again must be modified slightly given a set of mixed relative/absolute-formatted responses. In the following expressions, mixed response covariance and response sensitivity data are denoted by a caret (see Appendix A.), and :math:`{{\mathbf{\hat{F}}}_{\mathbf{m/k}}}` is an I :math:`\times` I diagonal matrix containing m/k factors for relative-formatted responses or a value of one for absolute-formatted responses: .. math:: :label: eq6-6-27 \begin{aligned} \hat{\mathbf{d}} =\left\{\begin{array}{cc} \frac{\mathbf{k}(\boldsymbol{\alpha})_{i}-\mathbf{m}_{i}}{\mathbf{k}(\boldsymbol{\alpha})_{i}} & relative \\ \mathbf{k}(\boldsymbol{\alpha})_{i}-\mathbf{m}_{i} & absolute \end{array}\right. \end{aligned} .. math:: :label: eq6-6-28 \boldsymbol{\Delta} \hat{\mathbf{m}}_{\mathbf{i}}=\left\{\begin{array}{ll} \frac{\mathbf{m}_{\mathrm{i}}^{\prime}-\mathbf{m}_{\mathrm{i}}}{\mathbf{m}_{\mathrm{i}}} & relative \\ \mathbf{m}_{\mathrm{i}}^{\prime}-\mathbf{m}_{\mathrm{i}} & absolute \end{array}\right. .. math:: :label: eq6-6-29 \Delta \hat{\mathbf{k}}_{\mathrm{i}}=\left\{\begin{array}{cc} \frac{\mathbf{k}^{\prime}\left(\boldsymbol{\alpha}^{\prime}\right)_{i}-\mathbf{k}(\boldsymbol{\alpha})_{i}}{\mathbf{k}(\boldsymbol{\alpha})_{i}} & relative \\ \mathbf{k}(\boldsymbol{\alpha})_{i}-\mathbf{k}(\boldsymbol{\alpha})_{i} & absolute \end{array}\right. .. math:: :label: eq6-6-30 \hat{\mathbf{C}}_{\mathbf{dd}}^{-1}=\hat{\mathbf{C}}_{\mathbf{kk}}+\hat{\mathbf{F}}_{\mathbf{m} / \mathbf{k}} \hat{\mathbf{C}}_{\mathbf{mm}} \hat{\mathbf{F}}_{\mathbf{m} / \mathbf{k}}=\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}} \mathbf{C}_{\mathbf{dd}} \hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}}+\hat{\mathbf{F}}_{\mathbf{m} / \mathbf{k}} \hat{\mathbf{C}}_{\mathbf{mm}} \hat{\mathbf{F}}_{\mathbf{m} / \mathbf{k}} .. math:: :label: eq6-6-31 \boldsymbol{\Delta} \hat{\boldsymbol{\alpha}} \quad=-\left[\mathbf{C}_{\mathbf{dd}} \hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}} \hat{\mathbf{C}}_{\mathbf{dd}}^{-1}\right] \hat{\mathbf{d}} .. math:: :label: eq6-6-32 \boldsymbol{\Delta} \hat{\mathbf{m}}=\left[\hat{\mathbf{C}}_{\mathbf{m m}} \hat{\mathbf{F}}_{\mathbf{m} / \mathbf{k}} \hat{\mathbf{C}}_{\mathbf{dd}}^{-1}\right] \hat{\mathbf{d}} .. math:: :label: eq6-6-33 \boldsymbol{\Delta \mathbf { k }} \quad=\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}} \boldsymbol{\Delta} \hat{\boldsymbol{\alpha}} Covariances for the posterior values of the nuclear data and measured responses are given as .. math:: :label: eq6-6-34 \mathbf{C}_{\boldsymbol{\alpha}^{\prime} \boldsymbol{\alpha}^{\prime}}=\mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}-\left[\mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}} \hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}}^{\mathbf{T}}\right] \hat{\mathbf{C}}_{\mathbf{dd}}^{-\mathbf{1}}\left[\hat{\mathbf{S}}_{\mathbf{k} \boldsymbol{\alpha}} \mathbf{C}_{\boldsymbol{\alpha} \boldsymbol{\alpha}}\right] .. math:: :label: eq6-6-35 \hat{\mathbf{C}}_{\mathbf{m}^{\prime} \mathbf{m}^{\prime}}=\hat{\mathbf{C}}_{\mathbf{mm}}-\left[\hat{\mathbf{C}}_{\mathbf{mm}} \hat{\mathbf{F}}_{\mathbf{m} / \mathbf{k}} \hat{\mathbf{C}}_{\mathbf{dd}}^{-1} \hat{\mathbf{F}}_{\mathbf{m} / \mathbf{k}} \hat{\mathbf{C}}_{\mathbf{mm}}\right] If responses on the TSURFER input are both relative formatted and absolute formatted, the adjusted data and response values edited by TSURFER are obtained from Eqs. :eq:`eq6-6-31`-:eq:`eq6-6-33`, while the square roots of diagonal elements in Eqs. :eq:`eq6-6-34`-:eq:`eq6-6-35` correspond to the edited values for adjusted uncertainties in the nuclear data and in the experiment responses, respectively. .. _sec-tsunami.tsurfer.chi_filter: Consistency relations and chi-square filtering ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. |cdd| replace:: :math:`\mathbf{C}_{\mathbf{dd}}` The variations for :math:`\Delta m` and :math:`{\Delta}{\alpha}` defined by Eq. :eq:`eq6-6-14` and Eq. :eq:`eq6-6-15` are those that give the smallest value of the quadratic form :math:`\chi`\ :sup:`2`. This minimum :math:`\chi`\ :sup:`2` value is found by substituting these equations into Eq. :eq:`eq6-6-11`: .. math:: :label: eq6-6-36 \chi_{\min }^{2}=\mathbf{d}^{\mathbf{T}} \mathbf{C}_{\mathbf{dd}}^{-1} \mathbf{d}=\mathbf{d}^{\mathbf{T}}\left[\mathbf{C}_{\mathbf{k} \mathbf{k}}+\mathbf{F}_{\mathbf{m} / \mathbf{k}} \mathbf{C}_{\mathbf{m} \mathbf{m}} \mathbf{F}_{\mathbf{m} / \mathbf{k}}\right]^{-1} \mathbf{d} It is interesting to observe that Eq. :eq:`eq6-6-36` does not depend upon adjustments in nuclear data or integral experiments and physically expresses a measure of the initial discrepancies (**d**) in all responses, compared to their combined calculation and experiment uncertainties (:math:`\mathbf{C}_{\mathbf{k k}}+\mathbf{F}_{\mathbf{m} / \mathbf{k}} \mathbf{C}_{\mathbf{m} \mathbf{m}} \mathbf{F}_{\mathbf{m} / \mathbf{k}}`). In fact, the parameter is identical to the chi-square statistic discussed in :numref:`sec-tsunami.tsurfer.guidelines` that denotes consistency between the calculations and measurements. Equation :eq:`eq6-6-36` can be viewed as an inherent limit on the consistency of the GLLS adjustment procedure. If the initial calculated and measured responses are not consistent with their stated uncertainties, then adjustments in nuclear data and experiment values obtained by TSURFER cannot be consistent either. TSURFER provides an option for "chi-square filtering" to ensure that a given set of benchmark experiments is consistent; that is, that the input responses have an acceptable :math:`\chi _{\min }^{2}` defined by :eq:`eq6-6-36`. The code progressively removes individual experiments until the calculated :math:`\chi _{\min}^{2}` is less than the input target value *"target_chi"*. Each iteration removes one experiment estimated to have the greatest impact on chi-square per degree of freedom. The method used to assess individual contributions to :math:`\chi_{\min }^{2}` is specified by input parameter *"chi_sq_filter"*, which refers to the different approaches described below. .. centered:: Independent Chi-Square Option (*chi_sq_filter=independent*). The consistency of the i-th measured and calculated response values, disregarding any other integral response, is equal to the discrepancy in the measured and calculated value squared divided by the variance of the discrepancy of the i-th response: .. math:: :label: eq6-6-37 \boldsymbol{\chi}_{\textbf {ind }, \mathbf{i}}^{2}=\frac{\left(\mathbf{k}_{\mathbf{i}}-\mathbf{m}_{\mathbf{i}}\right)^{2}}{\boldsymbol{\sigma}_{\mathbf{ki}}^{2}+\boldsymbol{\sigma}_{\mathbf{mi}}^{2}} Equation :eq:`eq6-6-37` is strictly valid only when no correlations exist, but it may be a useful approximation to estimate the experiment having the greatest impact on chi-square per degree of freedom. Hence, this expression is called the "\ *independent chi-square*\ " approximation in TSURFER. This approximation executes fast since no matrix inversions are required. .. centered:: Diagonal Chi-Square Option (*chi_sq_filter=diagonal*) The "\ *diagonal chi-square*\ " approach uses diagonal values of the original inverse |cdd| matrix to estimate the experiment having the greatest impact on chi-square per degree of freedom: .. math:: :label: eq6-6-38 \boldsymbol{\chi}_{\textbf{dia}, \mathbf{i}}^{2} \equiv\left(\mathbf{k}_{\mathbf{i}}-\mathbf{m}_{\mathbf{i}}\right)^{2} \mathbf{C}_{\mathbf{dd}}^{-1}(\mathbf{i}, \mathbf{i}) In this method the correlations in all responses are taken into account to some extent. The original :math:`\mathbf{C}_{\mathbf{dd}}^{-1}`. is used in each iteration; therefore, the diagonal chi-square method requires only a single matrix inversion. .. centered:: Iterative-Diagonal Chi-Square Option (*chi_sq_filter=iter_diag*). This approach is identical to the diagonal chi-square method, except that an updated value of is computed each iteration to re-evaluate the total chi-square from Eq. :eq:`eq6-6-36`. Thus one matrix inversion is performed per iteration. .. centered:: Delta Chi-Square Option (*chi_sq_filter=delta_chi*). The most rigorous method to determine the impact of an individual response on the overall consistency is called the "\ *delta chi-square*\ " method in TSURFER. This method :cite:`TSURFER-yeivin_relative_1980` calculates the change in chi-square whenever a particular response is omitted for the analysis; that is, omitting the i\ *th* response results in .. math:: :label: eq6-6-39 \Delta \chi_{\mathrm{i}}^{2}=\left[\mathbf{d}^{\mathbf{T}} \mathbf{C}_{\mathbf{dd}}^{-1} \mathbf{d}\right]-\left[\mathbf{d}_{\neq \mathrm{i}}^{\mathrm{T}}\left(\mathbf{C}_{\mathbf{dd}}^{\neq \mathrm{i}}\right)^{-1} \mathbf{d}_{\neq \mathrm{i}}\right] where :math:`\mathbf{d}_{\neq i}` and :math:`\mathbf{C}_{\mathbf{d d}}^{\neq \mathrm{i}}` are, respectively, the discrepancy vector and discrepancy covariance with response i omitted. While Eq. :eq:`eq6-6-39` is the most rigorous method, it also requires the most computation effort. A matrix inversion must be performed for every omitted response, in each iteration. For SCALE 6.3, the runtime of this method was reduced by using the matrix inversion lemma to compute the inverse of each submatrix, according to the method of :cite:`TSURFER-juarez_ruiz_relationship_2016`. It has been observed that independent chi-square and diagonal chi-square options execute fast but often eliminate more experiments than necessary to obtain the target chi-square value. The diagonal chi-square option is somewhat faster than the iterative-diagonal chi-square option but also sometimes omits more than the minimum number of experiments. The delta chi-square option is currently default in TSURFER. Expressions for computational bias ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The computational "bias" is defined here as the observed difference between a calculated and measured response. In conventional validation studies the expected bias in an application response (for which there is no measurement, by definition) often is estimated as the sample mean of the biases for a set of benchmark experiments and the uncertainty in the application bias is estimated by the sample standard deviation of the experimental biases. The GLLS technique provides another method to compute the bias of an application response. The application response bias :math:`\beta_a` is defined as the expected deviation of the original calculated response k\ :sub:`a` from the best estimate of the measured response, which is unknown but has some probability distribution. Note that if the application response actually *did* have a prior measured value m\ :sub:`a`, then the best estimate for the experiment value would be the final adjusted value m\ :sub:`a`\ ' obtained from the GLLS procedure. For this reason the notation m\ :sub:`a`\ ' is used here to represent the (unknown) best estimate for the application's projected measured response, so that .. math:: :label: eq6-6-40 \beta_{a}=E\left[k_{a}-m_{a}^{\prime}\right] where E is the expectation operator. The application's projected experiment value can be expressed as :math:`m_a^{\prime} = k_a \left( \alpha^{\prime} \right) - \delta m_a`, where :math:`\delta m_a` represents the difference between the best computed response obtained with the adjusted data :math:`\alpha^{\prime}` and the expected value of the experimental measurement. Therefore Eq. :eq:`eq6-6-40` can be expressed .. math:: :label: eq6-6-41 \beta_{a}=E\left[k_{a}-k_{a}\left(\alpha^{\prime}\right)+\delta m_{a}\right]=k_{a}-k_{a}\left(\alpha^{\prime}\right)+E\left[\delta m_{a}\right] Recall that all *experiment* responses are sure to have :math:`\delta m_i = 0`, because the GLLS procedure forces k'=m' within the approximation of first-order theory. However, :math:`\delta m_a \left(= k_a^{\prime} - m_a^{\prime} \right)` for the application is not guaranteed to be zero, since there is no known measured value. Nevertheless the application response calculated using the best cross sections :math:`\alpha^{\prime}` should approach the desired (unknown) experiment value if a "sufficient" number of experiments similar to the application of interest are considered :cite:`TSURFER-broadhead_sensitivity_1999` so that under these conditions :math:`E\left[ \delta m_a \right] \rightarrow 0` for the application as well. More details concerning the suitable degree of similarity and the sufficient number of experiments necessary for convergence of the GLLS methodology are discussed in other publications :cite:`TSURFER-poco_1988,TSURFER-maerker_development_1981,TSURFER-broadhead_sensitivity_1999,TSURFER-broadhead_sensitivity_1999-1`. TSURFER also provides an automated procedure to examine the convergence of the bias, which is described in :numref:`sec-tsunami.tsurfer.converge`. Assuming an adequate benchmark data base such that :math:`E\left[ \delta m_a \right] \rightarrow 0`, Eq. :eq:`eq6-6-41` simplifies to .. math:: :label: eq6-6-42 \beta_{a}=k_{a}-k_{a}^{\prime}\left(\alpha^{\prime}\right) \sim-\left(k_{a}\right) \mathbf{S}_{a}^{\mathbf{T}} \Delta \boldsymbol{\alpha} or, stated in absolute terms, .. math:: :label: eq6-6-43 \beta_{a} \sim-\mathbf{S}_{\mathbf{a}}^{\mathbf{T}} \Delta \boldsymbol{\alpha} . In the above equations :math:`\mathbf{S_a}` is the column vector of relative sensitivities for the application response. A negative bias indicates that the original computed value was too low; therefore, the adjusted application result will be higher than the original (:math:`k_a^{\prime} > k_a`). Similarly, a positive bias means that the original response was calculated too high, and therefore :math:`k_a^{\prime} < k_a`. .. _sec-tsunami.tsurfer.similarity: Expressions for response similarity parameters ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TSURFER estimates the similarity in a pair of responses using one of three internally computed similarity coefficients-respectively designated as E, G, and C-specified by the input parameter *"sim_type"*. These are essentially equivalent to the corresponding similarity coefficients described in :cite:`TSURFER-broadhead_sensitivity-and_2004`, although there are slight differences in the definitions of E and G. Similarity coefficients are defined so that a value of zero indicates no similarity between the systems, and unity is maximum similarity. It is also theoretically possible, but unusual, to have a negative similarity in the range [-1,0], indicating systems that are "anti-correlated" in some sense---in which case they are treated as completely dissimilar. Input parameter *"sim_min"* specifies the minimum similarity coefficient (compared to a specified reference application) of systems to be included in the GLLS procedure. TSURFER also optionally edits the I by I similarity matrix whose elements are the similarity coefficients for every response-pair combination (including both experimental and application responses). The three types of similarity coefficients used in TSURFER are described below. In these expressions, :math:`\mathbf{S}_{\mathbf{i}}` is defined as the sensitivity *vector* (not matrix) for a particular response "i" which may be an experiment or application. The magnitude of the sensitivity vector corresponds to the L2 norm: :math:`\left|\mathbf{S}_{\mathbf{i}}\right|=\sqrt{\mathbf{S}_{\mathbf{i}}^{\mathbf{T}} \mathbf{S}_{\mathbf{i}}}`. The *E* similarity parameter (*sim_type=E*) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The E-similarity coefficient relating two responses i and j is defined analogously to the cosine of the angle between two direction vectors: .. math:: :label: eq6-6-44 \mathrm{E}_{\mathrm{i}, \mathrm{j}} \equiv \frac{\mathbf{S}_{\mathrm{i}}^{\mathrm{T}} \mathbf{S}_{\mathrm{j}}}{\left|\mathbf{S}_{\mathrm{i}}\right|\left|\mathbf{S}_{\mathrm{j}}\right|} A value of E=1.0 corresponds to the case when **S\ i** and **S\ j** are "parallel", such as would occur when the two sensitivity vectors are proportional. A value of E = 0.0 corresponds to the case when **S\ i** and **S\ j** are "perpendicular", such as occurs when the two sensitivity vectors have no common components (i.e., for every non-zero component in one, the corresponding component is zero in the other). Thus, E indicates the "relative direction" of the two sensitivity vectors in an N-dimensional vector space, with the assumption that the larger the parallel component, the greater the similarity. In theory, E could also take on the negative values in the interval [-1,0] if the two responses are anti-parallel. In addition, the E coefficient is the same for absolute-formatted sensitivities (i.e., :math:`\mathbf{\tilde{S}}_{\mathbf{i}}^{{}}` and :math:`\mathbf{\tilde{S}}_{j}^{{}}` ) or mixed relative-absolute sensitivities (e.g., :math:`\mathbf{\tilde{S}}_{\mathbf{i}}^{{}}` and :math:`\mathbf{S}_{j}^{{}}`). The G similarity parameter (sim_type=G) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The G-similarity coefficient for responses i and j is defined as, .. math:: :label: eq6-6-45 \mathrm{G}_{i j} \equiv 1-\frac{\left|\mathbf{S}_{\mathbf{i}}-\mathbf{S}_{\mathbf{j}}\right|^{2}}{\left|\mathbf{S}_{\mathbf{i}}\right|^{2}+\left|\mathbf{S}_{\mathbf{j}}\right|^{2}}=\frac{\mathbf{S}_{\mathbf{i}}^{\mathbf{T}} \mathbf{S}_{\mathbf{j}}}{|\mathbf{S}|^{2}} , where :math:`\overline{|\mathbf{S}|^{2}} \equiv \frac{\left|\mathbf{S}_{\mathbf{i}}\right|^{2}+\left|\mathbf{S}_{\mathbf{j}}\right|^{2}}{\mathbf{2}}`. As seen in the last term, the G parameter is similar to the E parameter, except the denominators are different. The effect of the different normalization is that G will be unity only if S\ :sub:`i` and S\ :sub:`j` are *identical*, while E indicates maximum similarity if they are *proportional*. It is important to note that the expression for the G parameter in the TSUNAMI-IP manual is different from Eq. :eq:`eq6-6-45`. In both the TSURFER and TSUNAMI-IP formulations of G, the calculated parameter depends on the sensitivity format. It is recommended that Eq. :eq:`eq6-6-45` be used with relative-formatted sensitivities to calculate G for *k*\ :sub:`eff` responses. The *C* similarity parameter (*sim_type=C*) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The C similarity coefficient represents the correlation in two calculated responses due to the shared uncertainty from common nuclear data. While E and G similarity coefficients only depend on the sensitivity vectors of two responses, the C parameter also involves cross-section covariance data. The C-similarity parameter for responses i and j is the value of the correlation coefficient (:math:`\rho_{ij}`) in position (i, j) of the R\ :sub:`kk` correlation matrix; thus, .. math:: :label: eq6-6-46 \mathrm{C}_{\mathrm{ij}} \equiv \rho_{\mathrm{ij}}=\mathrm{R}_{\mathrm{kk}}(\mathrm{i}, \mathrm{j})=\frac{\mathbf{S}_{\mathbf{i}}^{\mathbf{T}} \mathrm{C}_{\boldsymbol{\alpha \alpha}} \mathbf{S}_{\mathbf{j}}}{\boldsymbol{\sigma}_{\mathbf{i}} \boldsymbol{\sigma}_{\mathrm{j}}} . The C coefficient has the usual interpretation of a correlation coefficient: 1.0 implies that the two responses are completely correlated by their nuclear data; 0.0 means no correlation; and -1.0 means full anti-correlation. The C coefficient is the same for absolute-formatted sensitivities (i.e., :math:`\mathbf{\tilde{S}}_{\mathbf{i}}^{{}}` and :math:`\mathbf{\tilde{S}}_{j}^{{}}` ) or mixed relative-absolute sensitivities (e.g., :math:`\mathbf{\tilde{S}}_{\mathbf{i}}^{{}}` and :math:`\mathbf{S}_{j}^{{}}`). .. _sec-tsunami.tsurfer.converge: Convergence of reference application response ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It is sometimes useful to consider how the GLLS procedure "converges" the estimated bias in an application response, as the number and similarity of integral experiment responses included in the analysis is increased :cite:`TSURFER-williams_perturbation_1986`. In TSURFER, the bias-convergence can be edited for any one of the application responses, called the "reference application," which is defined by the value of "\ *ref_app*\ " in the TSURFER input. Inserting the keyword *"calc_cumul_effect"* on the TSURFER input activates the option to edit the cumulative impact of increasing the number of benchmark experiments in the GLLS calculation. In this case, the range of similarity coefficients [0.0\ :math:`\rightarrow`\1.0] is subdivided in bins of constant width set by the TSURFER input parameter *"bin_width,"* and the experiment responses are sorted into the bins according their similarity to the reference application response. Any experiments with negative similarity coefficients are included in the first bin. Each bin of benchmark experiments is successively added to the GLLS calculation, going from low to high response similarity, until the whole suite of benchmarks is included. Ideally, the calculated reference application bias (:math:`\beta`\ :sub:`a`) should converge and stabilize at some value as the number and similarity of the experiment responses increases. Under these conditions the value E[:math:`\delta m_a`] in Eq. :eq:`eq6-6-41` is approximately zero. .. _6-6-5: TSURFER Input Description ------------------------- The user input for TSURFER is described in this section. The input consists of an optional title on a single line followed by one required and four optional blocks of data which are identified in :numref:`tab6-6-1` and individually described in subsequent subsections. These data blocks must begin with **READ KEYNAME**\ and end with **END KEYNAME**, where **KEYNAME** is the name of an individual data block. The *PARAMETER* data block, if requested, should be entered first after the optional title. The *HTML, COVARIANCE,* and *RESPONSE* data blocks may follow in any order. If the *CORR* data block is necessary to specify experiment correlations, it should be the last block of data on the input. All keyword inputs are internally translated to lowercase with the exception of sensitivity data filenames and their file paths. .. tabularcolumns:: |p{3cm}|p{7cm}|p{3cm}| .. table:: Table keynames and descriptions for TSURFER input data blocks. :class: longtable :name: tab6-6-1 +----------------------------+-----------------------+-----------------------+ | **Keyname** | **Description** | **Required/Optional** | +----------------------------+-----------------------+-----------------------+ | *PARAMETER* | Parameters that | Optional | | | specify the | | | | covariance data file, | | | | chi-square filtering | | | | options, similarity | | | | filtering options, | | | | output edit options, | | | | and approximate | | | | cross-section | | | | covariance data | | | | options can be | | | | entered in this | | | | section. | | +----------------------------+-----------------------+-----------------------+ | *RESPONSE* | File paths to | Required | | | sensitivity data | | | *EXPERIMENTS*\ :sup:`a` | files representing | | | | experiments or | | | *APPLICATIONS*\ :sup:`a` | applications are | | | | input in this | | | | section. Measured | | | | response values and | | | | measured response | | | | uncertainties are | | | | also input in this | | | | section. | | +----------------------------+-----------------------+-----------------------+ | *COVARIANCE* | User-input standard | Optional | | | deviation for | | | | nuclide-reaction | | | | pairs for which | | | | cross-section-covaria\| | | | nce | | | | data are not | | | | available can be | | | | entered in this | | | | section. | | +----------------------------+-----------------------+-----------------------+ | *HTML* | Parameters to | Optional | | | customize the | | | | HTML-formatted output | | | | can be entered in | | | | this section. | | +----------------------------+-----------------------+-----------------------+ | *CORR* | Correlations between | Optional | | | measured responses | | | | and measured response | | | | uncertainty | | | | components can be | | | | entered in this | | | | section. | | +----------------------------+-----------------------+-----------------------+ | :sup:`a` The TSUNAMI-IP block keynames *EXPERIMENTS* and *APPLICATIONS* | | are also allowed. By default, sensitivity data files listed in | | *RESPONSE* or *EXPERIMENTS* data blocks are designated experiment | | responses and files listed in the *APPLICATIONS* data block are | | designated application responses. The response designation can be easily | | changed by the *use* keyword in the response definition record described | | in :numref:`sec-tsunami.tsurfer.response`. | +----------------------------------------------------------------------------+ .. _sec-tsunami.tsurfer.parameter: Parameter block ~~~~~~~~~~~~~~~ The *PARAMETER* data block is used to specify various keyword options used to control the execution of the code. These options include the name of the cross-section covariance data file, output edits, default covariance data, and chi-square or similarity filtering options. The parameter block must begin with *READ PARAMETER* and end with *END PARAMETER*. The data input to the parameter data block consist of numerous keywords that are shown, along with their default values and descriptions, in  :numref:`tab6-6-2`. A keyword that ends with "\ *=*\ " must be followed by numeric data or a character string. Keywords that do not end with "\ *=*\ " are Boolean flags that are used to turn on certain features of the code, such as the computation of certain data or certain output edits. If the keyword is present for a Boolean entry, the value is set to true. Otherwise, the value is set to false. All *PARAMETER* block keywords are optional. .. tabularcolumns:: |p{2.5cm}|p{2cm}|p{7cm}| .. table:: Input data for the Parameter block of TSURFER. :align: center :class: longtable :name: tab6-6-2 +-----------------------+-----------------------+-----------------------+ | **Keyword** | **Default value** | **Description** | +-----------------------+-----------------------+-----------------------+ | *absolute* or *abs* | False | Use absolute | | | | sensitivities and | | | | uncertainties for all | | | | applications and | | | | experiments in the | | | | analysis unless | | | | specifically | | | | overridden by | | | | experiment or | | | | application input. | +-----------------------+-----------------------+-----------------------+ | *adj_cov_cut* | 0.000001 | Cutoff value for | | | | including an adjusted | | | | cross-section-covaria\| | | | nce | | | | matrix in the post | | | | adjustment analysis | | | | and data file. If a | | | | nuclide-reaction to | | | | nuclide-reaction | | | | covariance contains | | | | no values exceeding | | | | adj_cov_cut,the | | | | matrix is excluded | | | | from further | | | | analysis. Note that | | | | adj_cov_cut | | | | represents a | | | | variance, not a | | | | standard deviation. | +-----------------------+-----------------------+-----------------------+ | *adjcut* | 0.00001 | Cutoff value for the | | | | cross-section | | | | adjustment edit. If | | | | the maximum (absolute | | | | value) multigroup | | | | cross-section | | | | adjustment for a | | | | given | | | | nuclide-reaction pair | | | | is less than | | | | *adjcut*, then the | | | | nuclide-reaction pair | | | | is not included in | | | | the cross-section | | | | adjustment edit. | +-----------------------+-----------------------+-----------------------+ | *bin_width\ a* | 0.01 | Size of the | | | | similarity bins for | | | | the cumulative | | | | iteration edits. | +-----------------------+-----------------------+-----------------------+ | *cov_fix* | False | Replace zero and | | | | large (standard | | | | deviation >1000%) | | | | values on diagonal of | | | | cross-section-covaria\| | | | nce | | | | data with user-input | | | | values and default | | | | values. | +-----------------------+-----------------------+-----------------------+ | *coverx=* | 56groupcov7.1 | Name of cross-section | | | | covariance data file | | | | to use in analysis. | | | | See the COVLIB | | | | chapter of SCALE | | | | documentation for | | | | detailed description | | | | of the available | | | | covariance library. | +-----------------------+-----------------------+-----------------------+ | *calc_cumul_effect* | False | Perform cumulative | | :sup:`a` | | iteration edit. | +-----------------------+-----------------------+-----------------------+ | *chi_sq_filter* | delta_chi | Method used for | | :sup:`b` | | chi-square filtering. | | | | Allowed values are | | | | *independent*, | | | | *diagonal*, | | | | *iter_diag*, and | | | | *delta_chi*. | +-----------------------+-----------------------+-----------------------+ | *def_min=* | 0.001 | Minimum sensitivity | | | | criteria to adjust | | | | nuclear data. The | | | | minimum sensitivity | | | | criteria is only | | | | applied to | | | | nuclide-reaction | | | | pairs with | | | | **missing** | | | | covariance data and | | | | if *use_dcov* or | | | | *use_icov* is | | | | entered. | +-----------------------+-----------------------+-----------------------+ | *large_cov=* | 10.0 | Cutoff fractional | | | | standard deviation | | | | value for *cov_fix*. | | | | Cross-section-covaria\| | | | nce | | | | data with | | | | uncertainties larger | | | | than *large_cov* are | | | | replaced with | | | | user-input or default | | | | values. Default =10, | | | | which is 1000% | | | | uncertainty. | +-----------------------+-----------------------+-----------------------+ | *nohtml* | False | If *nohtml* is | | | | present, | | | | HTML-formatted output | | | | is not generated. | +-----------------------+-----------------------+-----------------------+ | *print=* | regular | Level of ouput edits | | | | discussed below. | | | | Options are *minimum* | | | | and *regular*. | +-----------------------+-----------------------+-----------------------+ | *print_adjustments* | False | Option to print | | | | cross-section | | | | adjustment edit. | +-----------------------+-----------------------+-----------------------+ | *print_adj_corr* | False | Option to print the | | | | adjusted response | | | | correlation matrix. | +-----------------------+-----------------------+-----------------------+ | *print_init_corr* | False | Option to print the | | | | initial response | | | | correlation matrix. | +-----------------------+-----------------------+-----------------------+ | *print_sim_matrix* | False | Option to print the | | | | initial response | | | | similarity matrix. | +-----------------------+-----------------------+-----------------------+ | *ref_app=* | First Application on | If application | | :sup:`a` | Input | systems are included, | | | | *ref_app* is the | | | | index to the | | | | reference application | | | | response. Additional | | | | output edits are | | | | given for the | | | | reference application | | | | described in | | | | :ref:`6-6-6`. | +-----------------------+-----------------------+-----------------------+ | *relative* or *rel* | True | Use relative | | | | sensitivities and | | | | uncertainties for all | | | | applications and | | | | experiments in the | | | | analysis. This is the | | | | default and keyword | | | | *relative* is not | | | | required. | +-----------------------+-----------------------+-----------------------+ | *return_adj_cov* | False | Option to create a | | | | COVERX-formatted | | | | covariance data file | | | | of the adjusted | | | | cross-section-covaria\| | | | nce | | | | matrix. If | | | | *return_adj_cov* is | | | | present, the adjusted | | | | covariance data file | | | | is returned to the | | | | working directory | | | | with the file name | | | | *job_name.adj.cov* | | | | where *job_name* is | | | | the name of the input | | | | file. | +-----------------------+-----------------------+-----------------------+ | *return_work_cov* | False | If *return_work_cov* | | | | is present, the | | | | working covariance | | | | library is copied to | | | | the return directory | | | | with the file name | | | | *job_name.wrk.cov* | | | | where *job_name* is | | | | the name of the input | | | | file. If | | | | *return_work_cov* is | | | | not present, the | | | | working covariance | | | | library remains in | | | | the temporary working | | | | directory with the | | | | file name | | | | *job_name.wrk*. | +-----------------------+-----------------------+-----------------------+ | *sim_min=* | -1 | Minimum similarity | | :sup:`c` | | coefficient of | | | | experimental | | | | responses with the | | | | reference application | | | | response to be | | | | included in the | | | | adjustment. | +-----------------------+-----------------------+-----------------------+ | *sim_type=* | None | Criteria to calculate | | :sup:`c` | | initial response | | | | similarity matrix. | | | | Allowed values are | | | | *none, E, C,* and | | | | *G.* | +-----------------------+-----------------------+-----------------------+ | *target_chi=* | 1.2 | Target chi-square per | | :sup:`b` | | degree of freedom for | | | | consistency | | | | acceptance. If | | | | *target_chi=0.0*, | | | | chi-square filtering | | | | is not performed. | +-----------------------+-----------------------+-----------------------+ | *udcov=* | 0.05 | User-defined default | | | | value of standard | | (optional) | | deviation in | | | | cross-section data to | | | | use for all groups | | | | for nuclide-reaction | | | | pairs for which | | | | cross-section-covaria\| | | | nce | | | | data are not | | | | available on the | | | | input covariance | | | | library. | +-----------------------+-----------------------+-----------------------+ | *udcov_corr=* | 1.0 | User-defined default | | | | correlation value to | | (optional) | | use for | | | | nuclide-reaction | | | | pairs for which | | | | cross-section-covaria\| | | | nce | | | | data are not | | | | available on the | | | | input covariance | | | | library. | +-----------------------+-----------------------+-----------------------+ | *udcov_corr_type=* | zone | User-defined default | | | | correlation in | | (optional) | | cross-section data to | | | | use for | | | | nuclide-reaction | | | | pairs for which | | | | cross-section-covaria\| | | | nce | | | | data are not | | | | available on the | | | | input SCALE | | | | covariance library. | | | | Allowed values are | | | | *long*, *zone*, and | | | | *short.* | +-----------------------+-----------------------+-----------------------+ | *udcov_therm=* | 0.0 | User-defined default | | | | value of standard | | (optional) | | deviation in | | | | cross-section data to | | | | use for thermal data | | | | for nuclide-reaction | | | | pairs for which | | | | cross-section-covaria\| | | | nce | | | | data are not | | | | available on the | | | | input covariance | | | | library. | +-----------------------+-----------------------+-----------------------+ | *udcov_inter=* | 0.0 | User-defined default | | | | value of standard | | (optional) | | deviation in | | | | cross-section data to | | | | use for intermediate | | | | data for | | | | nuclide-reaction | | | | pairs for which | | | | cross-section-covaria | | | | nce | | | | data are not | | | | available on the | | | | input covariance | | | | library. | +-----------------------+-----------------------+-----------------------+ | *udcov_fast=* | 0.0 | User-defined default | | | | value of standard | | (optional) | | deviation in | | | | cross-section data to | | | | use for fast data for | | | | nuclide-reaction | | | | pairs for which | | | | cross-section-covaria\| | | | nce | | | | data are not | | | | available on the | | | | input covariance | | | | library. | +-----------------------+-----------------------+-----------------------+ | *uncert_long* | False | Prints extended table | | | | of uncertainty in | | | | response due to | | | | covariance data. | +-----------------------+-----------------------+-----------------------+ | *use_dcov* | False | Use default | | | | cross-section-covaria\| | | | nce | | | | data for | | | | nuclide-reaction | | | | pairs not included on | | | | the input covariance | | | | data file. | +-----------------------+-----------------------+-----------------------+ | *use_diff_groups=* | true | Permit sensitivity | | | | data files to have | | | | different energy | | | | group structures. | | | | This parameter is now | | | | always true and does | | | | not need to be set. | +-----------------------+-----------------------+-----------------------+ | *usename* | False | Use the name of the | | | | sensitivity data file | | | | as the default | | | | response identifier | | | | in the TSURFER | | | | output. | +-----------------------+-----------------------+-----------------------+ | *use_icov* | False | Use user-defined | | | | cross-section-covaria\| | | | nce | | | | data input in the | | | | *COVARIANCE* input | | | | data block in place | | | | of the default values | | | | for user-input | | | | nuclide-reaction | | | | pairs that are not on | | | | the input covariance | | | | data file. | +-----------------------+-----------------------+-----------------------+ | :sup:`a` See :numref:`sec-tsunami.tsurfer.converge` for | | additional information on bias convergence analysis. | | | | :sup:`b` See :numref:`sec-tsunami.tsurfer.chi_filter` for | | additional inforomation on chi-square filtering methods. | | | | :sup:`c` See :numref:`sec-tsunami.tsurfer.similarity` for | | additional information on similarity coefficients. | +-----------------------------------------------------------------------+ The PARAMETER block keyword *print* controls the general level of the TSURFER output print. The minimum print level *"print=minimum"* summarizes the input values for experimental responses and uncertainties, edits chi-square values, and prints GLLS results for the application responses. The regular print option *"print=regular"* additionally shows GLLS results computed for all adjusted experimental responses. The *PARAMETER* block keywords-\ *use_dcov, udcov, udcov_therm, udcov_inter, udcov_fast, udcov_corr,* and *udcov_corr_type*-are used to specify the default covariance data for nuclide-reaction pairs that do not have covariance data available on the SCALE covariance data file. The Boolean flag keyword *use_dcov* activates the use of default covariance data for nuclide-reaction pairs with missing covariance data. The *udcov* keyword specifies a default relative standard deviation for all energy groups. The keywords *udcov_therm, udcov_inter,* and *udcov_fast* can be used to specify the default relative standard deviation for the thermal energy groups, intermediate energy groups, and fast energy groups, respectively. If either *udcov_therm, udcov_inter,* or *udcov_fast* are omitted from the input, the default uncertainty applied for their respective energy groups is the *udcov* value. The keyword *udcov_corr* specifies the correlation coefficient for the default covariance data, and *udcov_corr_type* specifies the correlation type. The correlation type options are (a) *long* - apply correlation coefficient in all energy groups, (b) *short* - apply correlation coefficient in adjacent groups, and (c) *zone* - apply correlation within fast, intermediate, and thermal groups, but no correlation is applied between different group ranges. For additional user control over the approximate cross-section covariance data, the *COVARIANCE* data block can be used to input uncertainty values for particular nuclide-reaction pairs. To utilize the covariance data generated by user-input in the *COVARIANCE* data block, the keyword *use_icov* must be entered in the *PARAMETER* data block. Approximate covariance data specified in the *COVARIANCE* data block are referred to as user-input data. The input for the COVARIANCE data block is described in more detail in :numref:`6-6-5-3`. When *use_dcov* and/or *use_icov* and *cov_fix* are specified in the *PARAMETER* data block, and a reaction has zero or large (standard deviation > 1000%) values on the diagonal of the covariance matrix, these values are replaced with the square of the user-input or default standard deviation value, and the corresponding off-diagonal terms are substituted according to the user-input or default correlation values. Warning messages are printed to identify which values were replaced and which standard deviation value was used in the replacement. The maximum relative standard deviation in which to apply the covariance correction can be specified by the user with the *large_cov* keyword. The *def_min* keyword is used to determine if the default or user-input covariance data is applied for nuclide-reaction pairs with missing covariance data. For each nuclide-reaction pair with missing covariance data, TSURFER calculates the maximum, absolute-value, groupwise response sensitivity over all active (i.e., experiment) and passive (i.e., application) responses on the input. If the maximum sensitivity value for a given nuclide-reaction pair is greater than *def_min*, the default or user-input covariance data is applied and the cross-section data for the nuclide-reaction pair is adjusted in the analysis. If the *relative* keyword is entered in the *PARAMETER* data block, the value of *def_min* is interpreted as a relative-formatted sensitivity. Likewise, if the *absolute* keyword is entered in the *PARAMETER* data block, the value of *def_min* is interpreted as an absolute-formatted sensitivity. If both *relative* and *absolute* are entered, the last keyword in the *PARAMETER* data block sets the format for both *def_min* and the response sensitivity data files. If both *relative* and *absolute* are omitted, the default format for *def_min* is relative. The minimum sensitivity criterion is slightly different if both relative-formatted *k*\ :sub:`eff` responses and absolute-formatted eigenvalue-difference (reactivity) responses are included in the analysis. In this case, the minimum sensitivity criteria can be entered for each response in the *RESPONSE* block described in the next section. During the GLLS analysis, TSURFER computes a new covariance data file that contains cross-section-covariance data only for the nuclide-reaction pairs that are listed in the response sensitivity data files. The new covariance data file, referred to as the working covariance data file, is written in COVERX format like the input SCALE covariance data file. The working covariance data file contains any default or user-input cross-section-covariance data for nuclide-reaction pairs that were not in the input SCALE covariance data file as well as any corrected cross-section-covariance data if the *cov_fix* keyword is entered on the input. The working covariance data file can be read by the data plotting tool in Fulcrum to visualize the cross section covariance data used in the analysis. .. _sec-tsunami.tsurfer.response: RESPONSE block ~~~~~~~~~~~~~~ In the *RESPONSE* data block, sensitivity data files are designated as either application responses or experiment responses. The *RESPONSE* data block is also used to specify experimental response values, experimental response uncertainties, and uncertainties of experimental response components. The TSUNAMI-IP block keynames *EXPERIMENTS* and *APPLICATIONS* are also allowed. Each data block must begin with *READ KEYNAME* and end with *END KEYNAME* where *KEYNAME* can be *APPLICATIONS, EXPERIMENTS,* or *RESPONSE*. By default, sensitivity data files listed in *RESPONSE* or *EXPERIMENTS* data blocks are designated as experiment responses, while files listed in the *APPLICATIONS* data block are designated as application responses. Multiple *RESPONSE*, *EXPERIMENTS*, and *APPLICATIONS* data blocks are allowed, and they can be entered in any order. However, the order of the sensitivity data files in the TSURFER input is important when defining experiment correlations. Two recommended input methods are (a) define all experiment and application responses in a single *RESPONSE* data block using the *use* keyword and specify the role of each response in the analysis or (b) define all experiment responses in a single *EXPERIMENTS* block, and define all application responses in a single *APPLICATIONS* block. Inside each data block, sensitivity data files are listed using response definition records. A response definition record is a single line of input that contains the sensitivity data filename, two required keywords and eight optional keywords shown in parentheses. The sensitivity data filename and keywords can be entered in any order, with the following format: *filename (use=R) (name=N) (type=T) ev=E uv=U (cv=C) (nu=P) (omit) (abs) (rel) (msen=M)* where *filename =* sensitivity data filename. The filename can include the file path. *R =* adjustment role. Allowed values are *appl*, *expt*, and *omit*. The default value is *expt* in the RESPONSE or EXPERIMENT block and *appl* in the APPLICATION block. N = 20-character maximum alphanumeric response identifier in TSURFER output. T = 8-character maximum alphanumeric identifier for the response type (e.g., "keff", "gpt", or "rho"). The response-type identifier is used in various output edits along with the response name identifier. E = experimental value of the response. U = uncertainty value of the response. C = calculated value of the response. P = number of uncertainty components to characterize the experiment uncertainty for this response. *omit* - Optional keyword used to omit the response from the analysis. This can also be done by the *use=omit* keyword specification. *abs* - Optional keyword that specifies absolute sensitivities, absolute experiment uncertainties, and absolute components of uncertainty that are used for this response. The keyword *absolute* is also valid. *rel* - Optional keyword that specifies relative sensitivities, relative experiment uncertainties, and relative components of uncertainty that are used for this response. The keyword *relative* is also valid. *M* = minimum sensitivity criteria for this response. This value will replace the def_min value in the PARAMETER block to determine if nuclide-reaction pairs with missing covariance data are included in the adjustment. Case-sensitive filenames and file paths are allowed for sensitivity data filename. However, **spaces are not allowed in the filenames or file paths.** The sensitivity data filename is limited to 80 characters, and the total length of the response definition must not exceed 255 characters. The *use* keyword specifies the role of the response in the GLLS analysis. "\ *use=expt"* designates the corresponding sensitivity data file as an experiment response. Likewise, "\ *use=appl"* designates the corresponding data file as an application response. In addition, the user can omit the sensitivity data file from the analysis by entering either '\ *use=omit"* or simply *omit* on the response definition record. If the *use* keyword is not included, the role of the response is determined by the data block name; that is, "\ *use=appl"* is implied for the *APPLICATIONS* block and "\ *use=expt"* is implied for the *EXPERIMENTS* and *RESPONSE* blocks. By default, TSURFER identifies responses in the output according to the title on the sensitivity data files. For files that have the same titles, or have long or non-descriptive titles, the *usename* keyword in the *PARAMETER* data block can be used to identify the response by their sensitivity data filename. Although filenames are unique, they can also be non-descriptive. For this reason, the *name* keyword on the response definition record can be used to create a new identifier for the response in the TSURFER output. Similarly, the *type* keyword can be used to identify the response type in the output. The default response type is *"keff"* for *k*\ :sub:`eff` responses and *"rho"* for eigenvalue-difference responses. It may be useful to include a sequence number in the response name, in order to more easily associate the response number to the input response data. For example, the response names for the first three responses entered could be *name=1_GODIVA, name=2_JEZEBEL,* and *name=3_ZPR4*. In the *CORR* data block and in the printed output, responses are identified by their sequence number (i.e., the order read in), so it is convenient to show this number in the response Name, especially when dealing with a large number of responses. The measured value of the response (*ev=*) and the measured uncertainty *(uv=*) are required for experiment responses. For application or omitted responses, the *ev* and *uv* keywords are permitted but are not required. The calculated response value is read from the sensitivity data file, but can be overridden by the *cv* keyword. The *nu=* keyword defines the number of uncertainty components that characterize the experiment response uncertainty. If the experiment response uncertainty is given in terms of uncertainty components, the *uv=* keyword specification is optional. An uncertainty component definition record follows the response definition record if the *nu=* keyword specification is given. The uncertainty component definition record has the following format: *ucmp1 val1 ucmp2 val2 ........ ucmpP valP*, where *uncmp1 =* 4-character alphanumeric identifier for the 1\ :sup:`st` uncertainty component, *val1* = experiment uncertainty for component *uncmp1,* *Uncmp2 =* 4-character alphanumeric identifier for the 2\ :sup:`nd` uncertainty component, *val2* = experiment uncertainty for component *uncmp2,* *uncmpP =* 4-character alphanumeric identifier for the P\ :sup:`th` uncertainty component, and *valP* = experiment uncertainty for component *uncmpP.* The uncertainty component definition record contains *nu=P* pairs of alphanumeric identifiers and numeric values. The measured uncertainty value is determined by Eq. :eq:`eq6-6-6`. The keywords *abs* and *rel* are used to determine the format of sensitivity and uncertainty data on the response definition record and the uncertainty component definition record. For a *k*\ :sub:`eff` response, the following four input definitions are equivalent: .. code-block:: none 1) name=exp_001 ev=1.001 uv=0.005000 rel C:\sensitivity\k_critical_a.sdf 2) name=exp_001 ev=1.001 uv=0.005005 abs C:\sensitivity\k_critical_a.sdf 3) name=exp_001 ev=1.001 nu=2 rel C:\sensitivity\k_critical_a.sdf enri 0.003000 sden 0.004000 4) name=exp_001 ev=1.001 nu=2 abs C:\sensitivity\k_critical_a.sdf enri 0.003003 sden 0.004004 In the example above, the measured *k*\ :sub:`eff` is 1.001 :math:`\pm` 0.5% or 1.001 :math:`\pm` 0.005005. (Although most critical experiments have measured *k*\ :sub:`eff`\ = 1, this contrived example reveals the difference between the *absolute* format and the *response* format.) The sensitivity data filename is given as *C:\sensitivity\k_critical_a.sdf*, and the experiment response is referred to as *exp_001* in the TSURFER output. In 1), the *relative* format is used to specify the relative standard deviation of the measured response as 0.005. In 2), the *absolute* format is used to specify the absolute standard deviation of the measured response as 0.005005. Because the TSUNAMI-generated sensitivity data file is in relative format, TSURFER internally renormalizes the relative sensitivities to absolute sensitivities. In 3), the *relative* format is used to specify the relative standard deviation of *k*\ :sub:`eff` due to two components (*enri* and *sden*) as 0.003 and 0.004, respectively. Using Eq. :eq:`eq6-6-6`, the relative standard deviation of *k*\ :sub:`eff` is computed to be 0.005. In 4), the *absolute* format is used to specify the absolute standard deviation of *k*\ :sub:`eff` due to two components (*enri* and *sden*) as 0.003003 and 0.004004, respectively. The absolute standard deviation of *k*\ :sub:`eff` is computed to be 0.005005. Like 2), the sensitivity data file is internally converted to contain absolute sensitivities. For a second example, the following input definitions are equivalent for an eigenvalue-difference, or reactivity, response: .. code-block:: none 1) C:\sensitivity\reactivity.sdf ev=15.0000 uv=3.0 abs 2) C:\sensitivity\reactivity.sdf ev=0.00015 uv=0.2 rel In this example, the measured reactivity is 15 pcm (percent-mille) :math:`\pm` 3 pcm or 0.00015 :math:`\pm` 20%. TSAR creates reactivity sensitivity files in either (a) *absolute* format where the calculated reactivity response and sensitivities are in pcm units or (b) *relative* format with relative sensitivities and the calculated reactivity response are not in pcm units. The TSURFER response definition records are designed to be consistent with the TSAR formats. In 1), the *absolute* format is used to specify the absolute standard deviation of the measured response as 3 pcm. In 2), the *relative* format is used to specify the measured reactivity response as 0.00015 with a relative standard deviation of 0.20 (or 20%). Because TSAR-generated reactivity sensitivity data files may be in absolute format or relative format, TSURFER internally renormalizes the reactivity sensitivity data file to the user-requested format. On occasion, it is desired to adjust a set of nuclear data with zero-valued reactivity responses (i.e., *ev=0.0*). For this case, the *absolute* format should be used because the relative standard deviation of the measured response approaches infinity. If the *abs* or *rel* keywords are not included on the response definition record, the default format is determined by the *abs* or *rel* keywords in the PARAMETER data block. If more than one formatting keyword is entered in either the PARAMETER data block or response definition record, the last keyword entered sets the format. As an example, the following inputs are equivalent: 1) .. highlight:: none :: read parameter relative end parameter read response C:\sensitivity\k_critical_a.sdf ev=1.0 uv=0.005 rel C:\sensitivity\reactivity.sdf ev=0.0 uv=3.0 abs end response 2) :: read parameter relative end parameter read response C:\sensitivity\k_critical_a.sdf ev=1.0 uv=0.005 C:\sensitivity\reactivity.sdf ev=0.0 uv=3.0 abs end response 3) :: read parameter absolute end parameter read response C:\sensitivity\k_critical_a.sdf ev=1.0 uv=0.005 rel C:\sensitivity\reactivity.sdf ev=0.0 uv=3.0 end response 4) :: read response C:\sensitivity\k_critical_a.sdf ev=1.0 uv=0.005 C:\sensitivity\reactivity.sdf ev=0.0 uv=3.0 abs end response In this example, two experiment responses are given. The first response is a relative-formatted *k*\ :sub:`eff` response. The second response is an absolute-formatted reactivity response. In 1), the format is determined by the formatting keyword on the response definition record. In 2), the relative-format is set as the default format by the *PARAMETER* block and the absolute format for the reactivity response is specified on its response definition record. In 3), the absolute-format is set as the default format by the *PARAMETER* block and the relative format for the *k*\ :sub:`eff` is set by its response definition record. Case 4) is the same as case 2) where the default relative format is applied if no *PARAMETER* block is included. The final optional keyword for the response definition record is *msen=M*. This record sets the minimum sensitivity criteria for nuclide-reaction pairs with missing covariance data. The keyword is useful when dealing with mixed formatted responses. For example, the following input contains three relative formatted *k*\ :sub:`eff` responses and two absolute-formatted reactivity responses. .. code-block:: none 1) read parameter use_dcov udcov=0.05 def_min=0.00001 relative end parameter read response C:\sensitivity\godiva.sdf ev=1.0 uv=0.005 C:\sensitivity\zpr.sdf ev=1.0 uv=0.005 C:\sensitivity\jezebel.sdf ev=1.0 uv=0.005 C:\sensitivity\void_react_1.sdf ev=0.0 uv=3.0 abs msen=0.1 C:\sensitivity\void_react_2.sdf ev=0.0 uv=3.0 abs msen=0.1 end response In this example, the *PARAMETER* block keywords initialize all the responses that follow as relative-formatted responses and the minimum sensitivity criteria for applying default covariance data is 0.00001 or 0.001%. This criteria is used for the three *k*\ :sub:`eff` responses. In the response definition records for the two reactivity responses, the absolute-format is specified and the minimum sensitivity criteria is 0.1 pcm. Therefore, default covariance data is used for a nuclide-reaction pair with missing cross-section covariance data if at least one *k*\ :sub:`eff` sensitivity for one of the three *k*\ :sub:`eff` responses is greater than 0.001% or at least one reactivity sensitivity for one of the two reactivity responses is greater than 0.1 pcm. Similar to the example above, this example has the following equivalent input: :: 2) read parameter use_dcov udcov=0.05 def_min=0.1 absolute end parameter read response C:\sensitivity\godiva.sdf ev=1.0 uv=0.005 rel msen=0.00001 C:\sensitivity\zpr.sdf ev=1.0 uv=0.005 rel msen=0.00001 C:\sensitivity\jezebel.sdf ev=1.0 uv=0.005 rel msen=0.00001 C:\sensitivity\void_react_1.sdf ev=0.0 uv=3.0 C:\sensitivity\void_react_2.sdf ev=0.0 uv=3.0 end response .. _6-6-5-3: COVARIANCE block ~~~~~~~~~~~~~~~~ The *COVARIANCE* data block allows the user to specify a covariance matrix for specific nuclide-reaction pairs for which covariance data are not present on the input SCALE covariance library or that have zero or large values on the diagonal. The *COVARIANCE* data block must begin with *READ COVARIANCE* and end with *END COVARIANCE*. The available *COVARIANCE* data block keywords and their default values are given in :numref:`tab6-6-3`. .. tabularcolumns:: |p{2cm}|p{2cm}|p{2cm}|p{3cm}|p{5cm}| .. table:: Input data for the Covariance block of TSURFER. :class: longtable :name: tab6-6-3 +-------------+-------------+-------------+-------------+-------------+ | **Input | **Requireme\| **Default | **Allowed | **Descripti\| | parameter** | nt** | value** | values** | on** | +=============+=============+=============+=============+=============+ | Nuclide | Required | none | Nuclide | Nuclide for | | | | | name or ZA | which | | | | | number | covariance | | | | | | data are to | | | | | | be entered | +-------------+-------------+-------------+-------------+-------------+ | Reaction | Required | none | Reaction | Reaction | | | | | name or ZA | for which | | | | | number | covariance | | | | | | data are to | | | | | | be entered | +-------------+-------------+-------------+-------------+-------------+ | *all=* | Optional | 0.0 | any number | Fractional | | | | | | standard | | | | | | deviation | | | | | | value to be | | | | | | applied to | | | | | | all groups | +-------------+-------------+-------------+-------------+-------------+ | *fast=* | Optional | 0.0 | any number | Fractional | | | | | | standard | | | | | | deviation | | | | | | value to be | | | | | | applied to | | | | | | fast groups | +-------------+-------------+-------------+-------------+-------------+ | *therm=* | Optional | 0.0 | any number | Fractional | | | | | | standard | | | | | | deviation | | | | | | value to be | | | | | | applied to | | | | | | thermal | | | | | | groups | +-------------+-------------+-------------+-------------+-------------+ | *inter=* | Optional | 0.0 | any number | Fractional | | | | | | standard | | | | | | deviation | | | | | | value to be | | | | | | applied to | | | | | | intermediat\| | | | | | e | | | | | | groups | +-------------+-------------+-------------+-------------+-------------+ | *corr=* | Optional | 1.0 | any number | Correlation | | | | | from -1.0 | between | | | | | to 1.0 | groups | +-------------+-------------+-------------+-------------+-------------+ | corr_type\ | Optional | *zone* | *long, | Type of | | *=* | | | short, | correlation | | | | | zone* | applied | | | | | | from | | | | | | group-to-gr\| | | | | | oup | | | | | | covariance | | | | | | values | | | | | | | | | | | | *long* - | | | | | | correlation | | | | | | is applied | | | | | | between all | | | | | | groups | | | | | | | | | | | | *short* - | | | | | | correlation | | | | | | is applied | | | | | | only | | | | | | between | | | | | | adjacent | | | | | | groups | | | | | | | | | | | | *zone* - | | | | | | correlation | | | | | | is applied | | | | | | within | | | | | | fast, | | | | | | intermediat\| | | | | | e, | | | | | | and thermal | | | | | | groups, but | | | | | | no | | | | | | correlation | | | | | | is applied | | | | | | between | | | | | | zones | +-------------+-------------+-------------+-------------+-------------+ | *end* | Required | | | Denotes end | | | | | | of input | | | | | | for current | | | | | | nuclide/rea\| | | | | | ction | | | | | | (must not | | | | | | start in | | | | | | column 1) | +-------------+-------------+-------------+-------------+-------------+ Any MT number or reaction name will be treated as a valid input, but only those present on the response sensitivity data files will produce useful information. The available reaction sensitivity types are shown in table *Reaction Sensitivity Types Computed by SAMS* in the TSUNAMI-IP manual. An energy-covariance matrix is created for the specified nuclide-reaction pair with the square of the entered standard deviation for the diagonal terms for all groups using the *all=* value. Groups in the fast, intermediate, and thermal energies are then set to the square of the standard deviation value entered for *fast=*, *inter=*, and *therm=*, respectively. The off-diagonal terms of the energy matrix are generated according to the input for *corr=*, and *corr_type=*, with default settings of *1.0* and *zone.* Data entered in this block do not override data present on the covariance data file. The SCALE 5.1 input format is supported where data are entered in triplets with the nuclide name or ZA identifier (e.g., u-235 or 92235), then the reaction MT name or number (e.g., 18 or fission), and then a standard deviation value. In this case, the *end* keyword must not be entered. These data are only used if *use_icov* is specified in the *PARAMETER* data block. When both *use_icov* and *cov_fix* are specified in the *PARAMETER* data block, and a reaction has zero or large (standard deviation > 1000%) values on the diagonal of the covariance matrix, these values are replaced with the square of the user input standard deviation value, and the corresponding off-diagonal terms are substituted according to the values of *corr* and *corr_type*. HTML block ~~~~~~~~~~ The optional *HTML* data block is used to customize HTML-formatted output. The *HTML* data block must begin with *READ HTML* and end with *END HTML*. The data input in the *HTML* data block consist of several keywords that are shown, along with their default values and descriptions, in :numref:`tab6-6-4`. A keyword that ends with "\ *=*\ " must be followed by text data. For color entries, any valid html color name can be entered or the hexadecimal representation can be used if preceded by a # sign. For example, to change the background color of the html output to white, *bg_clr=white* and *bg_clr=#ffffff* have the same effect, because *ffffff* is the hexadecimal representation of white. An extensive list of available colors for customized output is shown in :ref:`6-5b`. Please note that not all features are supported by all browsers. .. table:: Input data for HTML block of TSURFER :align: center :class: longtable :name: tab6-6-4 +-----------------------+-----------------------+-----------------------+ | **Keyword** | **Default value** | **Description** | +-----------------------+-----------------------+-----------------------+ | *bg_clr=* | *papayawhip* | Background color. | +-----------------------+-----------------------+-----------------------+ | *h1_clr=* | *maroon* | Color used for major | | | | headings. | +-----------------------+-----------------------+-----------------------+ | *h2_clr=* | *navy* | Color used for | | | | sub-headings. | +-----------------------+-----------------------+-----------------------+ | *txt_clr=* | *black* | Color for plain text. | +-----------------------+-----------------------+-----------------------+ | *lnk_clr=* | *navy* | Color for hyperlinks. | +-----------------------+-----------------------+-----------------------+ | *lnk_dec=* | *none* | Decoration for | | | | hyperlinks. (none, | | | | underline, overline, | | | | line-through, blink). | +-----------------------+-----------------------+-----------------------+ | *vlnk_clr* | *navy* | Color for visited | | | | hyperlinks. | +-----------------------+-----------------------+-----------------------+ | *ud_clr=* | *blue* | Color for values in | | | | tables that use | | | | default covariance | | | | data. | +-----------------------+-----------------------+-----------------------+ | *ui_clr=* | *red* | Color for values in | | | | tables that use | | | | user-input covariance | | | | data. | +-----------------------+-----------------------+-----------------------+ | *udfix_clr=* | *royalblue* | Color for values in | | | | tables that use | | | | default corrected | | | | covariance data. | +-----------------------+-----------------------+-----------------------+ | *uifix_clr=* | *green* | Color for values in | | | | tables that use | | | | user-input corrected | | | | covariance data. | +-----------------------+-----------------------+-----------------------+ CORR block ~~~~~~~~~~ The CORR block specifies correlation coefficients between different experiment responses. When present, this block must be the last data block in the input file. The correlation block must begin with *READ CORR* and end with *END PARAMETER*. Correlation coefficients for experimental response uncertainties may be entered either as the total correlation coefficient for a pair of responses; or for a particular uncertainty-component shared by two responses. Values for correlation coefficients are input in the form: corr_typ (i,j)=\ :math:`\rho` ....... {repeat for I=1,N } end where: ``corr_typ =`` 4-character alphanumeric identifier for the response uncertainty component previously defined in the READ RESPONSES block (i.e., *ucmp1*, *umcp2*, etc.). ``corr_typ`` may also equal *totl* indicating that the total correlation is entered. The ``corr_typ`` identifier may be omitted and the total correlation is assumed. *N* = number of responses in TSURFER input. :math:`\rho=` the correlation coefficient at the specified position in the correlation matrix for each uncertainty component. *i,j* = the row and column index to the correlation matrix for each uncertainty component. Correlation coefficients can be entered the following five ways: 1) Element-by-element - (i,j)\ :math:`=\rho` 2) By row - (i,j1:j2)\ :math:`=\rho` 3) By column - (i1:j2,j)\ :math:`=\rho` 4) By block - (i1:i2,j1:j2)\ :math:`=\rho`. This option can be used to set a large block of the correlation matrix to one number. All diagonal elements in the block are reset to 1.0. 5) By shorthand block - (i1:i2)\ :math:`=\rho`. This is identical to (i1:i2,i1:i2)\ :math:`=\rho`. All diagonal elements in the block are reset to 1.0. TSURFER initializes each correlation matrix as an N by N identity matrix. Therefore, all uncorrelated elements (i.e., values equal to 0) do not have to be entered. The correlation matrix can be specified using multiple lines of input with each line having a maximum of 255 characters. As each correlation coefficient is processed, the symmetric element of the correlation matrix is assigned to the same value. Therefore only the upper or lower triangular portion of each correlation matrix must be specified. For example, given the following *RESPONSE* block: :: read response name=1_k C:\sensitivity\k_critical_a.sdf ev=1.0 uv=0.005 name=2_k C:\sensitivity\k_critical_b.sdf ev=1.0 uv=0.005 name=3_k C:\sensitivity\k_critical_c.sdf ev=1.0 uv=0.005 name=4_k C:\sensitivity\k_critical_d.sdf ev=1.0 uv=0.005 end response then the following forms of the CORR block are equivalent in specifying the 4 x 4 total correlation matrix as: .. math:: \left[ \begin{matrix} 1 & .2 & .3 & .2 \\ .2 & 1 & .2 & .2 \\ .3 & .2 & 1 & .1 \\ .2 & .2 & .1 & 1 \\ \end{matrix} \right] 1) Specify the upper triangular portion of the matrix element by element: :: read corr totl (1,2)=.2 (1,3)=.3 (1,4)=.2 (2,3)=.2 (2,4)=.2 (3,4)=.1 end end corr 2) Specify the lower triangular portion of the matrix element by element: :: read corr totl (2,1)=.2 (3,1)=.3 (4,1)=.2 (3,2)=.2 (4,2)=.2 (4,3)=.1 end end corr 3) Use the colon character to specify multiple elements at one time in the upper triangular portion of the matrix: :: read corr totl (1,2)=.2 (1,3)=.3 (1,4)=.2 (2,3:4)=.2 (3,4)=.1 end end corr Values entered for the total correlation matrix will override any component correlations entered. Correlation coefficients may be specified in the *CORR* data block for responses that share one or more of the same uncertainty components. The value of *corr_typ* must correspond to one of the 4-character alphanumeric identifiers given to an uncertainty component. Only those uncertainty components that appear in more than one response description should be entered, since these are the only ones with correlations. An END keyword is required to terminate the data of an individual uncertainty component, and the input is repeated for each type of correlated uncertainty component. .. note:: The experiment covariance matrix should be positive definite to ensure a physical result for all possible sensitivities. If the input correlation values do not satisfy this constraint, a warning message is printed. Use of several fully correlated uncertainties can lead to an over-constrained system, which may result in a non-positive-definite covariance matrix. In order to help avoid this problem, correlation values usually should be limited to a maximum of 0.95, suggesting that a small random component is always present. As an example of correlation matrices for uncertainty components, consider the following input: :: read response name=1_k C:\sensitivity\k_critical_a.sdf ev=1.0 nu=2 enri=0.003 soln=0.004 name=2_k C:\sensitivity\k_critical_b.sdf ev=1.0 nu=2 enri=0.0005 soln=0.0012 name=3_k C:\sensitivity\k_application.sdf use=app end response read corr enri (1,2)=0.5 end soln (1,2)=0.8 end end corr In this example, two uncertainty components are used, identified as *enri* and *soln*. Using the propagation of error formula Eq. :eq:`eq6-6-6` , the relative standard deviation of the experiment responses *1_k* and *2_k* are determined as :math:`\sqrt{\left(0.003^{2}+0.004^{2}\right)}=0.005` and :math:`\sqrt{\left(0.0005^{2}+0.0012^{2}\right)}=0.0013`, respectively. The correlation matrix *enri* and *soln* are given as .. math:: \left[ \begin{matrix} 1 & .5 \\ .5 & 1 \\ \end{matrix} \right] and .. math:: \left[ \begin{matrix} 1 & .8 \\ .8 & 1 \\ \end{matrix} \right] Using Eq. :eq:`eq6-6-7`, the relative covariance between response *1_k* and *2_k* is calculated as: :math:`0.003 * 0.5 * 0.0005` (*enri* component) + :math:`0.0005 * 0.8^{*} 0.0012` (*soln* component) = 1.23E-6. .. _6-6-6: Sample Problem Input and Output Description ------------------------------------------- Input and text output ~~~~~~~~~~~~~~~~~~~~~ An example TSURFER input is given in :numref:`list6-6-1` and the text output is shown in :numref:`list6-6-2`-:numref:`list6-6-14`. In this sample problem, 40 *k*\ :sub:`eff` responses are specified in the *RESPONSE* block, 37 experiments and 3 applications. All calculation options are turned on including the use of default and user-input covariance data, similarity filtering to the reference application, chi-square filtering for consistency, and bias convergence analysis. Each section of the text output is described in order below. Some of the figures of the text output have been truncated from their original length. The examples of output are for illustrative purposes and only demonstrate the format of the TSURFER results. 1. Echo of Input (:numref:`list6-6-2`) - The TSURFER input data are printed for the *PARAMETER, HTML*, and *COVARIANCE* data blocks. Both user-specified and default values for the various keywords are edited. 2. Covariance Warnings (:numref:`list6-6-3`) - If the PARAMETER block keywords *use_dcov* and/or *use_icov* and/or *cov_fix* are entered, covariance warnings are listed that specify the nuclide-reaction pairs for which approximate covariance data is applied. 3. Listing of Input Responses (:numref:`list6-6-4`) - Various information is listed for each response. This includes the response index, name, title, adjustment role (e.g., *expt*), type, calculated response value, measured response value, and similarity coefficient to the reference application. The similarity coefficient column is only edited if a reference application is listed on the input. 4. Experiment Uncertainties (:numref:`list6-6-5`) - The experiment standard deviations, as well as any input uncertainty components, are edited for each measured response. When uncertainty components are given, the total standard deviation is computed from Eq. . 5. Chi-square summary (:numref:`list6-6-6`) - Different chi-squared values are edited based on the GLLS analysis. This includes the initial value of chi-squared, the target value of chi-squared based on the *target_chi* keyword, and the final value of chi-squared. The independent and diagonal chi-squares are also edited. 6. Correlation Matrices (:numref:`list6-6-7`-:numref:`list6-6-9`) - Correlation matrices are printed after the chi-squared edit in the following order: (1) response similarity matrix if *print_sim_matrix* is entered in the input, (2) the prior calculated response correlation matrix and prior measured response correlation matrix if *print_init\_*\ corr is entered in the input, and (3) the adjusted response correlation matrix if *print_adj_corr* is entered in the input. The value of the keyword **sim_type** designates the type of similarity coefficient appearing in the similarity matrix. See Sect. 6.6.4.3 for description of the types of similarity coefficients. The response correlation matrices are defined in Appendix A. 7. Cumulative Convergence Edit (:numref:`list6-6-10`) - The cumulative convergence edit follows the printout of the requested correlation matrices. TSURFER only performs the cumulative convergence calculation if the keyword *calc_cumul_effect* is entered in the input. Four columns of data are printed that specify the cumulative range number, the maximum similarity coefficient allowed for each adjustment, the number of experiments with similarity coefficients within the specified range, and the computed application bias for each range, shown as o/v(A-C)/C, where A represents the adjusted *k*\ :sub:`eff` value and C represents the original calculated *k*\ :sub:`eff` value. 8. Summary of Adjustments (:numref:`list6-6-11`) - The adjustment summary table is an 11-column table that summarizes the prior and posterior values of each response. The 11 columns include the response adjustment role (i.e., *expt, appl,* or *omit*), the name and type identifiers, the prior and posterior uncertainties of each response, the independent and diagonal chi-squared values, and the change in the response between the prior and posterior values. 9. Summary of Adjusted Responses (:numref:`list6-6-12`) - Following the adjustment summary table, the adjusted values of each response are listed in tabular format. The adjusted uncertainty values of the response are also included. 10. Application and Bias Summary (:numref:`list6-6-13`) - The application and bias summary table follows the adjusted response table. This edit is only printed if applications are specified on the TSURFER input. For each application, the following values are tabulated: the name and type of the response, the prior and posterior values of the application response, the prior and posterior values of the application uncertainty, and the application bias as determined by Eq. . If the application is a relative-formatted response, the fractional bias is also included in the table. Following this table, a second table is printed that lists the contribution to the reference application bias for each nuclide-reaction pair used in the analysis. The nuclide-reaction pairs are listed in descending order based on the fraction of bias L1-norm, defined as .. math:: f_{x}=\frac{\sum_{g}\left|S_{x, g} \Delta \alpha_{x, g}\right|}{\sum_{x^{\prime}} \sum_{g}\left|S_{x^{\prime}, g} \Delta \alpha_{x^{\prime}, g}\right|} 11. Multigroup Cross-Section Adjustment Table (:numref:`list6-6-14`) - The Multigroup cross-section adjustment tables are printed if the *print_adjustments* keyword is included in the TSURFER input. For each nuclide-reaction pair, a table is printed that includes the relative adjustment of each multigroup cross-section, and the prior and posterior values of the cross-section uncertainty. If an application is included in the TSURFER input, the bias contribution and fraction of bias L1-norm are also edited. The order of the nuclide-reaction pairs is determined by the fraction of bias L1-norm. .. code-block:: scale :name: list6-6-1 :caption: TSURFER sample input. =tsurfer TSURFER sample problem read parameter ' chi_sq_filter=delta_chi target_chi=3.0 ' calc_cumul_effect bin_width=0.01 ' ref=40 sim_type=c sim_min=0.3 ' use_dcov use_icov cov_fix coverx=44groupcov udcov=0.05 def_min=0.0 ' print=regular print_adjustments print_adjustments print_adj_corr print_init_corr return_work_cov return_adj_cov ' uncert_long=false ' end parameter read covariance u-235 elastic all=0.07 end u-238 elastic all=0.06 end end covariance read response nam=1_hst001-1 hst001-001.sdf ev=1.0 nu=4 pyra 0.0042 ucna 0.0021 B10a 0.0030 H/Ua 0.0042 nam=2_hst001-1 hst001-02.sdf ev=1.0 nu=4 pyra 0.0032 ucna 0.0025 B10a 0.0032 H/Ua 0.0040 nam=3_hst001-1 use=appl hst001-03.sdf nam=4_hst001-1 hst001-04.sdf ev=1.0 nu=3 pyra 0.0039 ucna 0.0015 H/Ua 0.0025 nam=5_hst001-1 hst001-05.sdf ev=1.0 nu=4 pyra 0.0040 ucna 0.0021 B10a 0.0025 H/Ua 0.0032 nam=6_hst001-1 hst001-06.sdf ev=1.0 nu=3 ucna 0.0021 B10a 0.0030 H/Ua 0.0042 nam=7_hst001-1 hst001-07.sdf ev=1.0 nu=4 pyra 0.0022 ucna 0.0022 B10a 0.0033 H/Ua 0.0042 nam=8_hst001-1 hst001-08.sdf ev=1.0 nu=4 pyra 0.0022 ucna 0.0025 B10a 0.0036 H/Ua 0.0045 nam=9_hst001-1 hst001-09.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=10_hst001-1 hst001-10.sdf ev=1.0 nu=2 ucnb 0.0028 H/Ub 0.0050 nam=11_hst001-1 hst002-01.sdf ev=1.0 nu=2 ucnb 0.0031 H/Ub 0.0032 nam=12_hst001-1 hst002-03.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 Figure 6.6.1. TSURFER sample input. nam=13_hst001-1 hst002-09.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=14_hst001-1 hst003-03.sdf ev=1.0 nu=2 ucnb 0.0011 H/Ub 0.0022 nam=15_hst001-1 hst003-08.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=16_hst001-1 hst003-18.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=17_hst001-1 hst004-003.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=18_hst001-1 hst021-030.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=19_hst001-1 hst025-02.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=20_hst001-1 hst025-04.sdf ev=1.0 nu=2 ucnb 0.0021 H/Ub 0.0042 nam=21_hst001-1 hst025-05.sdf ev=1.0 nu=4 pyrc 0.0042 ucnc 0.0021 B10c 0.0030 H/Uc 0.0042 nam=22_hst001-1 hst027-01.sdf ev=1.0 nu=4 pyrc 0.0045 ucnc 0.0025 B10c 0.0033 H/Uc 0.0040 nam=23_hst001-1 hst29i-01.sdf ev=1.0 nu=4 pyrc 0.0039 ucnc 0.0021 B10c 0.0030 H/Uc 0.0045 nam=24_hst001-1 hst29i-02.sdf ev=1.0 nu=4 pyrc 0.0042 ucnc 0.0030 B10c 0.0020 H/Uc 0.0022 nam=25_hst001-1 hst29i-03.sdf ev=1.0 nu=4 pyrc 0.0042 ucnc 0.0021 B10c 0.0030 H/Uc 0.0042 nam=26_hst001-1 hst29i-04.sdf ev=1.0 nu=4 pyrc 0.0022 ucnc 0.0031 B10c 0.0040 H/Uc 0.0036 nam=27_hst001-1 hst29i-05.sdf ev=1.0 nu=4 pyrc 0.0042 ucnc 0.0021 B10c 0.0030 H/Uc 0.0042 nam=28_hst001-1 hst29i-06.sdf ev=1.0 nu=4 pyrc 0.0032 ucnc 0.0021 B10c 0.0045 H/Uc 0.0032 nam=29_hst001-1 hst29i-07.sdf ev=1.0 nu=4 pyrc 0.0042 ucnc 0.0025 B10c 0.0030 H/Uc 0.0036 nam=30_hst001-1 hst30i-01.sdf ev=1.0 nu=3 ucnc 0.0021 B10c 0.0030 H/Uc 0.0042 nam=31_hst001-1 hst30i-02.sdf ev=1.0 nu=3 ucnc 0.0031 B10c 0.0030 H/Uc 0.0042 nam=32_hst001-1 use=appl hst30i-03.sdf ev=1.0 uv=0.003 nam=33_hst001-1 hst30i-04.sdf ev=1.0 nu=3 ucnc 0.0031 B10c 0.0040 H/Uc 0.0038 nam=34_hst001-1 hst30i-05.sdf ev=1.0 nu=3 ucnc 0.0021 B10c 0.0030 H/Uc 0.0042 nam=35_hst001-1 hst30i-06.sdf ev=1.0 nu=3 ucnc 0.0025 B10c 0.0040 H/Uc 0.00436 Figure 6.6.1. TSURFER sample input (continued). nam=36_hst001-1 hst30i-07.sdf ev=1.0 nu=3 ucnc 0.0031 B10c 0.0040 H/Uc 0.0038 nam=37_hst001-1 hst31i-01.sdf ev=1.0 nu=3 ucnc 0.0021 B10c 0.0030 H/Uc 0.0042 nam=38_hst001-1 hst31i-02.sdf ev=1.0 nu=3 ucnc 0.0041 B10c 0.0020 H/Uc 0.0032 nam=39_hst001-1 hst31i-03.sdf ev=1.0 nu=3 ucnc 0.0021 B10c 0.0030 H/Uc 0.0042 nam=40_hst001-1 use=appl hst31i-04.sdf end response read corr pyra (1,2)=0.2 (1,4)=0.6 (1,5)=0.5 (1,7)=0.2 (1,8)=0.6 (2,4)=0.6 (2,5)=0.5 (2,7)=0.2 (2,8)=0.6 (4,5)=0.6 (4,7)=0.5 (4,8)=0.5 (5,7)=0.2 (5,8)=0.6 (7,8)=0.6 end ucna (1,2)=0.2 (1,4)=0.2 (1,5)=0.2 (1,6)=0.2 (1,7)=0.2 (1,8)=0.2 (2,4)=0.2 (2,5)=0.2 (2,6)=0.2 (2,7)=0.2 (2,8)=0.2 (4,5)=0.6 (4,6)=0.6 (4,7)=0.5 (4,8)=0.5 (5,6)=0.8 (5,7)=0.2 (5,8)=0.6 (6,7)=0.8 (6,8)=0.6 (7,8)=0.6 end H/Ua (1,2)=0.2 (1,4)=0.2 (1,5)=0.2 (1,6)=0.2 (1,7)=0.2 (1,8)=0.2 (2,4)=0.2 (2,5)=0.2 (2,6)=0.2 (2,7)=0.2 (2,8)=0.2 (4,5)=0.6 (4,6)=0.6 (4,7)=0.5 (4,8)=0.5 (5,6)=0.8 (5,7)=0.2 (5,8)=0.6 (6,7)=0.8 (6,8)=0.6 (7,8)=0.6 end ucnb (9,10)=0.2 (9,11)=0.2 (9,12)=0.2 (9,13)=0.2 (9,14)=0.2 (9,15)=0.2 (9,16)=0.2 (9,17)=0.2 (9,18)=0.2 (9,19)=0.2 (9,20)=0.2 (10,11)=0.2 (10,12)=0.2 (10,13)=0.2 (10,14)=0.2 (10,15)=0.2 (10,16)=0.2 (10,17)=0.2 (10,18)=0.2 (10,19)=0.2 (10,20)=0.2 (11,12)=0.2 (11,13)=0.2 (11,14)=0.2 (11,15)=0.2 (11,16)=0.2 (11,17)=0.2 (11,18)=0.2 (11,19)=0.2 (11,20)=0.2 (12,13)=0.2 (12,14)=0.2 (12,15)=0.2 (12,16)=0.2 (12,17)=0.2 (12,18)=0.2 (12,19)=0.2 (12,20)=0.2 (13,14)=0.2 (13,15)=0.2 (13,16)=0.2 (13,17)=0.2 (13,18)=0.2 (13,19)=0.2 (13,20)=0.2 (14,15)=0.2 (14,16)=0.2 (14,17)=0.2 (14,18)=0.2 (14,19)=0.2 (14,20)=0.2 (15,16)=0.2 (15,17)=0.2 (15,18)=0.2 (15,19)=0.2 (15,20)=0.2 (16,17)=0.2 (16,18)=0.2 (16,19)=0.2 (16,20)=0.2 (17,18)=0.2 (17,19)=0.2 (17,20)=0.2 (18,19)=0.2 (19,20)=0.2 end H/Ub (9,10)=0.2 (9,11)=0.2 (9,12)=0.2 (9,13)=0.2 (9,14)=0.2 (9,15)=0.2 (9,16)=0.2 (9,17)=0.2 (9,18)=0.2 (9,19)=0.2 (9,20)=0.2 (10,11)=0.2 (10,12)=0.2 (10,13)=0.2 (10,14)=0.2 (10,15)=0.2 (10,16)=0.2 (10,17)=0.2 (10,18)=0.2 (10,19)=0.2 (10,20)=0.2 (11,12)=0.2 (11,13)=0.2 (11,14)=0.2 (11,15)=0.2 (11,16)=0.2 (11,17)=0.2 (11,18)=0.2 (11,19)=0.2 (11,20)=0.2 (12,13)=0.2 (12,14)=0.2 (12,15)=0.2 (12,16)=0.2 (12,17)=0.2 (12,18)=0.2 (12,19)=0.2 (12,20)=0.2 (13,14)=0.2 (13,15)=0.2 (13,16)=0.2 (13,17)=0.2 (13,18)=0.2 (13,19)=0.2 (13,20)=0.2 (14,15)=0.2 (14,16)=0.2 (14,17)=0.2 (14,18)=0.2 (14,19)=0.2 (14,20)=0.2 (15,16)=0.2 (15,17)=0.2 (15,18)=0.2 (15,19)=0.2 (15,20)=0.2 (16,17)=0.2 (16,18)=0.2 (16,19)=0.2 (16,20)=0.2 (17,18)=0.2 (17,19)=0.2 (17,20)=0.2 (18,19)=0.2 (19,20)=0.2 end ucnc (21:31,21:31)=0.2 (21:31,33:39)=0.2 (33:39,33:39)=0.2 end end corr end .. code-block:: none :caption: Echo of TSURFER input parameters. :name: list6-6-2 :class: long ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + + + T S U R F E R + + + + TITLE: tsurfer sample problem + + + ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ I N P U T D A T A PARAMETER VALUE DESCRIPTION absolute false Print uncertainty values and penalty assessments in absolute format. This is the default format. Relative format can be specified using the "rel" keyword in the APPLICATIONS, EXPERIMENTS, or RESPONSE input blocks. adjcut 1.0000E-05 Cutoff value for the cross-section adjustment edits. If the maximum group-wise relative adjustment for a given cross-section is less than adjcut, then it is omitted from the cross-section adjustment table. adj_cov_cut 1.0000E-06 Cutoff value for including an adjusted cross-section-covariance matrix in the post adjustment analysis and data file. If a nuclide-reaction to nuclide-reaction covariance contains no values exceeding adj_cov_cut,the matrix is excluded from further analysis. Note that adj_cov_cut represents a variance, not a standard deviation. bin_width= 1.000E-02 Size of similarity bins for cumulative iteration edits. cov_fix true Replace zero and large values on diagonal of cross-section covariance data with user input values and dcov value. cov_unit= 33 Logical unit for cross-section covariance data. calc_cumul_effect true Perform cumulative iteration edit. chi_sq_filter= delta_chi Method used for chi^2 filter analysis. Possible values are: independent - use independent chi^2 filtering method. diagonal - use diagonal chi^2 filtering method. iter_diag - use iterative diagonal chi^2 filtering method. delta_chi - use iterative delta-chi chi^2 filtering method. large_cov= 10.0000 Cutoff fractional standard deviation value for cov_fix. nohtml false Flag to cause HTML output to not be produced. print= regular Level of output edits for this analysis (minimum or regular). print_sim_matrix false Option to print similarity matrix. print_adjustments true Option to print 1-D cross-section adjustments. print_init_corr true Option to print initial response correlation matrix. print_adj_corr true Option to adjusted response correlation matrix. ref_app= 40 Index to reference application response. relative true Print uncertainty values and penalty assessments in relative format. This is the default format. Absolute format can be specified using the "abs" keyword in the APPLICATIONS, EXPERIMENTS, or RESPONSE input blocks. return_work_cov true Option to copy the working covariance data file back to the return directory. return_adj_cov true Option to copy the adjusted covariance data file back to the return directory. If return_adj_cov is false, the adjusted covariance data file is not created. sen_unit= 41 Logical unit for sensitivity data files. sim_type= C Criteria used to calculate sim matrix. Possible values are: E - Calculate the similarity matrix using Esum correlation coefficients. G - Calculate the similarity matrix using Gm correlation coefficients. C - Calculate the similarity matrix using Ck correlation coefficients. target_chi= 3.000E+00 Target chi-square per degree of freedom for consistency acceptance. udcov= 0.0500 User-defined default value of standard deviation in cross-section data to use for nuclide-reaction pairs for which covariance data are not available on the selected data file. udcov_corr= 1.0000 User-defined default correlation value to use for nuclide-reaction pairs for which covariance data are not available on the selected data file. udcov_corr_type= zone User-defined default correlation in cross-section data to use for nuclide-reaction pairs for which covariance data are not available on the selected data file. (long, zone, short) udcov_fast= 0.0000 User-defined default value of standard deviation in cross-section data to use for fast data for nuclide-reaction pairs for which covariance data are not available on the selected data file. udcov_inter= 0.0000 User-defined default value of standard deviation in cross-section data to use for intermediate data for nuclide-reaction pairs for which covariance data are not available on the selected data file. udcov_therm= 0.0000 User-defined default value of standard deviation in cross-section data to use for thermal data for nuclide-reaction pairs for which covariance data are not available on the selected data file. uncert_long false Prints extended table of uncertainty in response due to covariance data. use_dcov true Use user-defined default covariance data, udcov, for nuclide reaction pairs not included on the covariance data file. use_diff_groups true Allow sensitivity data files to have different energy group structures. usename false Use the name of the sensitivity data file in place of its title in all output. use_icov true Use user-defined data input in COVARIANCE input data block in place of udcov value for user input nuclide-reaction pairs that are not on the covariance data file. USER COVARIANCE DATA ZA NUCLIDE REACTION MT ALL THERMAL INTER FAST CORREL TYPE ------ ------- -------- ---- ---------- ---------- ---------- ---------- -------- ------ 92235 u-235 elastic 2 7.00E-02 0.00E+00 0.00E+00 0.00E+00 1.00 zone 92238 u-238 elastic 2 6.00E-02 0.00E+00 0.00E+00 0.00E+00 1.00 zone HTML Format Options PARAMETER VALUE DESCRIPTION --------- ----- ----------- bg_clr= papayawhip Background color h1_clr= maroon Color used for major headings h2_clr= navy Color used for sub-headings txt_clr= black Color for plain text lnk_clr= navy Color for hyperlinks lnk_dec= none Decoration for hyperlinks (none, underline, overline, line-through, blink) vlnk_clr= navy Color for visited hyperlinks <<<< GENERALIZED LEAST-SQUARE ANALYSIS >>>> Name of cross section COV file : 44groupcov Name of working cross section COV file : tsurfer.wrk.cov Name of adjusted cross section COV file : tsurfer.adj.cov Name of adjusted cross section PLT file : tsurfer.xs-adjust.plt Number of groups on COV file : 44 =>All sensitivity coefficients will be converted into COV group structure .. code-block:: none :name: list6-6-3 :caption: TSURFER covariance warnings edit. Generating working covariance matrix ... ----------------------------------------------------------------------------------- Covariance Warnings in creating working COVERX library ----------------------------------------------------------------------------------- WARNING: cov_fix applied for b-10 n,p Default standard deviation data value 0.0500 will replace 0.0000 for group 15 Default standard deviation data value 0.0500 will replace 0.0000 for group 16 Default standard deviation data value 0.0500 will replace 0.0000 for group 17 Default standard deviation data value 0.0500 will replace 0.0000 for group 18 Default standard deviation data value 0.0500 will replace 0.0000 for group 19 Default standard deviation data value 0.0500 will replace 0.0000 for group 20 Default standard deviation data value 0.0500 will replace 0.0000 for group 21 Default standard deviation data value 0.0500 will replace 0.0000 for group 22 Default standard deviation data value 0.0500 will replace 0.0000 for group 23 Default standard deviation data value 0.0500 will replace 0.0000 for group 24 Default standard deviation data value 0.0500 will replace 0.0000 for group 25 Default standard deviation data value 0.0500 will replace 0.0000 for group 26 Default standard deviation data value 0.0500 will replace 0.0000 for group 27 Default standard deviation data value 0.0500 will replace 0.0000 for group 28 Default standard deviation data value 0.0500 will replace 0.0000 for group 29 Default standard deviation data value 0.0500 will replace 0.0000 for group 30 Default standard deviation data value 0.0500 will replace 0.0000 for group 31 Default standard deviation data value 0.0500 will replace 0.0000 for group 32 Default standard deviation data value 0.0500 will replace 0.0000 for group 33 Default standard deviation data value 0.0500 will replace 0.0000 for group 34 Default standard deviation data value 0.0500 will replace 0.0000 for group 35 Default standard deviation data value 0.0500 will replace 0.0000 for group 36 Default standard deviation data value 0.0500 will replace 0.0000 for group 37 Default standard deviation data value 0.0500 will replace 0.0000 for group 38 Default standard deviation data value 0.0500 will replace 0.0000 for group 39 Default standard deviation data value 0.0500 will replace 0.0000 for group 40 Default standard deviation data value 0.0500 will replace 0.0000 for group 41 Default standard deviation data value 0.0500 will replace 0.0000 for group 42 Default standard deviation data value 0.0500 will replace 0.0000 for group 43 Default standard deviation data value 0.0500 will replace 0.0000 for group 44 ... Working covariance matrix created for future processing. .. code-block:: none :name: list6-6-4 :caption: TSURFER response list edit. :class: long Number of Input Sensitivity Files = 40 =>Number of Applications (passive) Included in GLLSM: 3 =>Number of Benchmarks (active) Included in GLLSM : 37 =>Number of Responses (active+passive) used in GLLSM: 40 =>Number of Input Systems Omitted from GLLSM(*) : 0 ** Description of Prior Responses ** RESP.# EXPERIMENT NAME SENS. TITLE USE TYPE CALC EXP Ck W/ REFERENCE APPLICATION 1 1_hst001-1 r1 expt keff 1.0015E+00 1.0000E+00 9.597E-01 2 2_hst001-1 r2 expt keff 9.9852E-01 1.0000E+00 9.569E-01 3 3_hst001-1 r3 appl keff 1.0024E+00 <( NA )> 9.589E-01 4 4_hst001-1 r4 expt keff 1.0008E+00 1.0000E+00 9.558E-01 5 5_hst001-1 r5 expt keff 1.0011E+00 1.0000E+00 9.640E-01 6 6_hst001-1 r6 expt keff 1.0046E+00 1.0000E+00 9.640E-01 7 7_hst001-1 r7 expt keff 9.9994E-01 1.0000E+00 9.593E-01 8 8_hst001-1 r8 expt keff 9.9975E-01 1.0000E+00 9.589E-01 9 9_hst001-1 r9 expt keff 9.9634E-01 1.0000E+00 9.561E-01 10 10_hst001-1 r10 expt keff 9.9521E-01 1.0000E+00 9.660E-01 11 11_hst001-1 rot2 tank in cen expt keff 1.0046E+00 1.0000E+00 9.723E-01 12 12_hst001-1 rot7 tank in cen expt keff 1.0010E+00 1.0000E+00 9.609E-01 13 13_hst001-1 rot38 tank in ce expt keff 1.0009E+00 1.0000E+00 9.661E-01 14 14_hst001-1 rot4 tank in cen expt keff 1.0019E+00 1.0000E+00 9.638E-01 15 15_hst001-1 rot14 tank in ce expt keff 1.0049E+00 1.0000E+00 9.621E-01 16 16_hst001-1 rot29 tank in ce expt keff 9.9888E-01 1.0000E+00 9.641E-01 17 17_hst001-1 ol3ne 15.5 in. s expt keff 1.0031E+00 1.0000E+00 3.882E-01 18 18_hst001-1 case 30 experime expt keff 9.9962E-01 1.0000E+00 9.858E-01 19 19_hst001-1 heu-sol-therm-02 expt keff 1.0023E+00 1.0000E+00 9.720E-01 20 20_hst001-1 heu-sol-therm-02 expt keff 1.0031E+00 1.0000E+00 9.744E-01 21 21_hst001-1 heu-sol-therm-02 expt keff 1.0060E+00 1.0000E+00 9.767E-01 22 22_hst001-1 heu-sol-therm-02 expt keff 9.9845E-01 1.0000E+00 9.660E-01 23 23_hst001-1 heu-sol-therm-02 expt keff 1.0050E+00 1.0000E+00 9.828E-01 24 24_hst001-1 heu-sol-therm-02 expt keff 1.0086E+00 1.0000E+00 9.937E-01 25 25_hst001-1 heu-sol-therm-02 expt keff 1.0014E+00 1.0000E+00 9.952E-01 26 26_hst001-1 heu-sol-therm-02 expt keff 9.9857E-01 1.0000E+00 9.988E-01 27 27_hst001-1 heu-sol-therm-02 expt keff 1.0045E+00 1.0000E+00 9.989E-01 28 28_hst001-1 heu-sol-therm-02 expt keff 1.0051E+00 1.0000E+00 9.971E-01 29 29_hst001-1 heu-sol-therm-02 expt keff 1.0057E+00 1.0000E+00 9.931E-01 30 30_hst001-1 heu-sol-therm-03 expt keff 9.9998E-01 1.0000E+00 9.798E-01 31 31_hst001-1 heu-sol-therm-03 expt keff 1.0014E+00 1.0000E+00 9.862E-01 32 32_hst001-1 heu-sol-therm-03 appl keff 9.9990E-01 <( NA )> 9.852E-01 33 33_hst001-1 heu-sol-therm-03 expt keff 1.0083E+00 1.0000E+00 9.820E-01 34 34_hst001-1 heu-sol-therm-03 expt keff 1.0036E+00 1.0000E+00 9.910E-01 35 35_hst001-1 heu-sol-therm-03 expt keff 1.0048E+00 1.0000E+00 9.936E-01 36 36_hst001-1 heu-sol-therm-03 expt keff 1.0029E+00 1.0000E+00 9.979E-01 37 37_hst001-1 heu-sol-therm-03 expt keff 1.0045E+00 1.0000E+00 9.944E-01 38 38_hst001-1 heu-sol-therm-03 expt keff 1.0051E+00 1.0000E+00 9.978E-01 39 39_hst001-1 heu-sol-therm-03 expt keff 1.0043E+00 1.0000E+00 9.967E-01 40 40_hst001-1 heu-sol-therm-03 appl keff 1.0017E+00 <( NA )> 1.000E+00 31 31_hst001-1 heu-sol-therm-03 expt keff 9.9987E-01 1.0000E+00 9.565E-01 32 32_hst001-1 heu-sol-therm-03 appl keff 9.9894E-01 <( NA )> 9.527E-01 33 33_hst001-1 heu-sol-therm-03 expt keff 1.0069E+00 1.0000E+00 9.782E-01 34 34_hst001-1 heu-sol-therm-03 expt keff 1.0033E+00 1.0000E+00 9.820E-01 35 35_hst001-1 heu-sol-therm-03 expt keff 1.0042E+00 1.0000E+00 9.831E-01 36 36_hst001-1 heu-sol-therm-03 expt keff 1.0026E+00 1.0000E+00 9.878E-01 37 37_hst001-1 heu-sol-therm-03 expt keff 1.0039E+00 1.0000E+00 9.842E-01 38 38_hst001-1 heu-sol-therm-03 expt keff 1.0040E+00 1.0000E+00 9.800E-01 39 39_hst001-1 heu-sol-therm-03 expt keff 1.0028E+00 1.0000E+00 9.868E-01 40 40_hst001-1 heu-sol-therm-03 appl keff 1.0007E+00 <( NA )> 1.000E+00 ave. calc - exp value |C-E| = 3.1229E-03 sample stand. dev. in |C-E| = 2.2236E-03 Chi-fission spectrum sensitivities are constrained. .. raw:: latex \begin{landscape} .. code-block:: none :name: list6-6-5 :caption: Uncertainty components edit. :class: long ** Input Experiment Response Uncertainties ** RESP.# TYPE UNC. UNITS TOTAL UNCERT. COMPONENTS (std. dev.): STD DEV pyra ucna b10a h/ua ucnb h/ub pyrc ucnc b10c h/uc 1 keff % dk/k 6.97782E-1 4.2000E-1 2.1000E-1 3.0000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2 keff % dk/k 6.53682E-1 3.2000E-1 2.5000E-1 3.2000E-1 4.0000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 3 keff % dk/k 0.00000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4 keff % dk/k 4.86929E-1 3.9000E-1 1.5000E-1 0.0000E+0 2.5000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 5 keff % dk/k 6.07454E-1 4.0000E-1 2.1000E-1 2.5000E-1 3.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 6 keff % dk/k 5.57225E-1 0.0000E+0 2.1000E-1 3.0000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 7 keff % dk/k 6.18142E-1 2.2000E-1 2.2000E-1 3.3000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 8 keff % dk/k 6.65582E-1 2.2000E-1 2.5000E-1 3.6000E-1 4.5000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 9 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 10 keff % dk/k 5.73062E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.8000E-1 5.0000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 11 keff % dk/k 4.45533E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 3.1000E-1 3.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 12 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 13 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 14 keff % dk/k 2.45967E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 1.1000E-1 2.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 15 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 16 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 17 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 18 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 19 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 20 keff % dk/k 4.69574E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 4.2000E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 21 keff % dk/k 6.97782E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4.2000E-1 2.1000E-1 3.0000E-1 4.2000E-1 22 keff % dk/k 7.30685E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4.5000E-1 2.5000E-1 3.3000E-1 4.0000E-1 23 keff % dk/k 6.99071E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 3.9000E-1 2.1000E-1 3.0000E-1 4.5000E-1 24 keff % dk/k 5.95651E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4.2000E-1 3.0000E-1 2.0000E-1 2.2000E-1 25 keff % dk/k 6.97782E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4.2000E-1 2.1000E-1 3.0000E-1 4.2000E-1 26 keff % dk/k 6.58863E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.2000E-1 3.1000E-1 4.0000E-1 3.6000E-1 27 keff % dk/k 6.97782E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4.2000E-1 2.1000E-1 3.0000E-1 4.2000E-1 28 keff % dk/k 6.71863E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 3.2000E-1 2.1000E-1 4.5000E-1 3.2000E-1 29 keff % dk/k 6.77126E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4.2000E-1 2.5000E-1 3.0000E-1 3.6000E-1 30 keff % dk/k 5.57225E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 3.0000E-1 4.2000E-1 31 keff % dk/k 6.02080E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 3.1000E-1 3.0000E-1 4.2000E-1 32 keff % dk/k 0.00000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 33 keff % dk/k 6.32851E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 3.1000E-1 4.0000E-1 3.8000E-1 34 keff % dk/k 5.57225E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 3.0000E-1 4.2000E-1 35 keff % dk/k 6.42336E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.5000E-1 4.0000E-1 4.3600E-1 36 keff % dk/k 6.32851E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 3.1000E-1 4.0000E-1 3.8000E-1 37 keff % dk/k 5.57225E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 3.0000E-1 4.2000E-1 38 keff % dk/k 5.57225E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 4.1000E-1 2.0000E-1 3.2000E-1 39 keff % dk/k 5.57225E-1 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 2.1000E-1 3.0000E-1 4.2000E-1 40 keff % dk/k 0.00000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 0.0000E+0 .. raw:: latex \end{landscape} .. code-block:: none :name: list6-6-6 :caption: Chi-squared analysis edit. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: CHI-SQUARED ANALYSIS FOR INCLUDED EXPERIMENTS INITIAL CHI-SQUARE PER DEGR. OF FREEDOM = 2.781E-01 TARGET CHI-SQUARE PER DEGR. OF FREEDOM = 3.000E+00 FINAL CHI-SQUARE PER DEGR. OF FREEDOM = 2.781E-01 * final number of degrees of freedom 37 * final chi-square per degr. of freedom.... diagonal contribution 4.556E-01 * final chi-square per degr. of freedom.... without correlations 1.561E-01 :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: .. code-block:: none :caption: Prior experiment correlation matrix edit. :name: list6-6-7 :class: long Prior Experimental-Response Correlation Matrix resp resp 1 resp 2 resp 3 resp 4 resp 5 resp 6 resp 7 resp 8 1 1.000E+00 1.556E-01 0.000E+00 3.696E-01 2.824E-01 1.134E-01 1.461E-01 2.234E-01 2 1.556E-01 1.000E+00 0.000E+00 3.216E-01 2.521E-01 1.211E-01 1.452E-01 2.086E-01 3 0.000E+00 0.000E+00 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 4 3.696E-01 3.216E-01 0.000E+00 1.000E+00 5.426E-01 3.018E-01 3.718E-01 3.638E-01 5 2.824E-01 2.521E-01 0.000E+00 5.426E-01 1.000E+00 4.219E-01 1.431E-01 4.222E-01 6 1.134E-01 1.211E-01 0.000E+00 3.018E-01 4.219E-01 1.000E+00 5.170E-01 3.907E-01 7 1.461E-01 1.452E-01 0.000E+00 3.718E-01 1.431E-01 5.170E-01 1.000E+00 4.264E-01 8 2.234E-01 2.086E-01 0.000E+00 3.638E-01 4.222E-01 3.907E-01 4.264E-01 1.000E+00 9 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 resp 10 thru resp 40 same as above resp resp 9 resp 10 resp 11 resp 12 resp 13 resp 14 resp 15 resp 16 1 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 resp 2 thru resp 8 same as above 9 1.000E+00 1.998E-01 1.907E-01 2.000E-01 2.000E-01 2.000E-01 2.000E-01 2.000E-01 10 1.998E-01 1.000E+00 1.933E-01 1.998E-01 1.998E-01 1.998E-01 1.998E-01 1.998E-01 11 1.907E-01 1.933E-01 1.000E+00 1.907E-01 1.907E-01 1.907E-01 1.907E-01 1.907E-01 12 2.000E-01 1.998E-01 1.907E-01 1.000E+00 2.000E-01 2.000E-01 2.000E-01 2.000E-01 13 2.000E-01 1.998E-01 1.907E-01 2.000E-01 1.000E+00 2.000E-01 2.000E-01 2.000E-01 14 2.000E-01 1.998E-01 1.907E-01 2.000E-01 2.000E-01 1.000E+00 2.000E-01 2.000E-01 15 2.000E-01 1.998E-01 1.907E-01 2.000E-01 2.000E-01 2.000E-01 1.000E+00 2.000E-01 16 2.000E-01 1.998E-01 1.907E-01 2.000E-01 2.000E-01 2.000E-01 2.000E-01 1.000E+00 17 2.000E-01 1.998E-01 1.907E-01 2.000E-01 2.000E-01 2.000E-01 2.000E-01 2.000E-01 resp 18 thru resp 20 same as above 21 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 resp 22 thru resp 40 same as above resp resp 17 resp 18 resp 19 resp 20 resp 21 resp 22 resp 23 resp 24 1 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 resp 2 thru resp 8 same as above 9 2.000E-01 2.000E-01 2.000E-01 2.000E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 10 1.998E-01 1.998E-01 1.998E-01 1.998E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 11 1.907E-01 1.907E-01 1.907E-01 1.907E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 12 2.000E-01 2.000E-01 2.000E-01 2.000E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 resp 13 thru resp 16 same as above 17 1.000E+00 2.000E-01 2.000E-01 2.000E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 18 2.000E-01 1.000E+00 2.000E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 19 2.000E-01 2.000E-01 1.000E+00 2.000E-01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 20 2.000E-01 0.000E+00 2.000E-01 1.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 21 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.000E+00 2.059E-02 1.808E-02 3.032E-02 ... .. code-block:: none :caption: Prior calculated response correlation matrix edit. :name: list6-6-8 :class: long Prior Calculated-Response Correlation Matrix (omitted responses are 0) resp resp 1 resp 2 resp 3 resp 4 resp 5 resp 6 resp 7 resp 8 1 1.000E+00 9.941E-01 9.988E-01 9.922E-01 9.918E-01 9.932E-01 9.988E-01 9.988E-01 2 9.941E-01 1.000E+00 9.925E-01 9.987E-01 9.768E-01 9.790E-01 9.921E-01 9.928E-01 3 9.988E-01 9.925E-01 1.000E+00 9.931E-01 9.931E-01 9.945E-01 1.000E+00 1.000E+00 4 9.922E-01 9.987E-01 9.931E-01 1.000E+00 9.769E-01 9.791E-01 9.926E-01 9.933E-01 5 9.918E-01 9.768E-01 9.931E-01 9.769E-01 1.000E+00 9.999E-01 9.936E-01 9.928E-01 6 9.932E-01 9.790E-01 9.945E-01 9.791E-01 9.999E-01 1.000E+00 9.950E-01 9.943E-01 7 9.988E-01 9.921E-01 1.000E+00 9.926E-01 9.936E-01 9.950E-01 1.000E+00 1.000E+00 8 9.988E-01 9.928E-01 1.000E+00 9.933E-01 9.928E-01 9.943E-01 1.000E+00 1.000E+00 9 9.922E-01 9.986E-01 9.931E-01 1.000E+00 9.770E-01 9.792E-01 9.927E-01 9.934E-01 10 9.934E-01 9.793E-01 9.947E-01 9.794E-01 9.998E-01 1.000E+00 9.951E-01 9.944E-01 11 9.923E-01 9.847E-01 9.927E-01 9.836E-01 9.883E-01 9.896E-01 9.926E-01 9.927E-01 12 9.943E-01 9.992E-01 9.923E-01 9.977E-01 9.786E-01 9.805E-01 9.920E-01 9.925E-01 13 9.901E-01 9.748E-01 9.913E-01 9.750E-01 9.995E-01 9.992E-01 9.919E-01 9.909E-01 14 9.995E-01 9.929E-01 9.983E-01 9.909E-01 9.935E-01 9.946E-01 9.984E-01 9.982E-01 15 9.985E-01 9.918E-01 9.997E-01 9.924E-01 9.945E-01 9.957E-01 9.998E-01 9.996E-01 16 9.932E-01 9.983E-01 9.942E-01 9.994E-01 9.805E-01 9.825E-01 9.938E-01 9.944E-01 17 3.190E-01 3.558E-01 3.167E-01 3.588E-01 3.060E-01 3.064E-01 3.164E-01 3.175E-01 18 9.745E-01 9.771E-01 9.756E-01 9.771E-01 9.692E-01 9.705E-01 9.754E-01 9.757E-01 19 9.782E-01 9.598E-01 9.774E-01 9.579E-01 9.925E-01 9.915E-01 9.783E-01 9.771E-01 20 9.804E-01 9.626E-01 9.785E-01 9.592E-01 9.923E-01 9.915E-01 9.793E-01 9.781E-01 21 9.860E-01 9.715E-01 9.831E-01 9.670E-01 9.909E-01 9.907E-01 9.836E-01 9.829E-01 22 9.994E-01 9.918E-01 9.979E-01 9.893E-01 9.939E-01 9.950E-01 9.980E-01 9.978E-01 23 9.715E-01 9.696E-01 9.715E-01 9.684E-01 9.659E-01 9.673E-01 9.713E-01 9.717E-01 24 9.738E-01 9.709E-01 9.735E-01 9.697E-01 9.723E-01 9.731E-01 9.735E-01 9.736E-01 25 9.697E-01 9.666E-01 9.694E-01 9.654E-01 9.696E-01 9.702E-01 9.695E-01 9.695E-01 26 9.581E-01 9.543E-01 9.576E-01 9.531E-01 9.621E-01 9.622E-01 9.579E-01 9.576E-01 27 9.653E-01 9.621E-01 9.647E-01 9.609E-01 9.675E-01 9.678E-01 9.650E-01 9.647E-01 28 9.756E-01 9.732E-01 9.748E-01 9.719E-01 9.750E-01 9.756E-01 9.750E-01 9.749E-01 29 9.833E-01 9.815E-01 9.826E-01 9.803E-01 9.798E-01 9.807E-01 9.826E-01 9.826E-01 30 9.828E-01 9.674E-01 9.828E-01 9.659E-01 9.914E-01 9.912E-01 9.833E-01 9.826E-01 31 9.663E-01 9.478E-01 9.663E-01 9.461E-01 9.828E-01 9.817E-01 9.671E-01 9.659E-01 32 9.511E-01 9.310E-01 9.511E-01 9.293E-01 9.717E-01 9.702E-01 9.521E-01 9.507E-01 33 9.875E-01 9.864E-01 9.870E-01 9.854E-01 9.796E-01 9.810E-01 9.868E-01 9.871E-01 34 9.830E-01 9.809E-01 9.825E-01 9.798E-01 9.788E-01 9.799E-01 9.825E-01 9.826E-01 35 9.773E-01 9.749E-01 9.770E-01 9.738E-01 9.750E-01 9.759E-01 9.770E-01 9.771E-01 36 9.680E-01 9.649E-01 9.676E-01 9.637E-01 9.692E-01 9.697E-01 9.678E-01 9.676E-01 37 9.738E-01 9.711E-01 9.736E-01 9.701E-01 9.725E-01 9.733E-01 9.736E-01 9.737E-01 .. code-block:: none :caption: Adjusted response correlation matrix edit. :name: list6-6-9 :class: long Adjusted-Response Correlation Matrix resp resp 1 resp 2 resp 3 resp 4 resp 5 resp 6 resp 7 resp 8 1 1.000E+00 8.916E-01 9.629E-01 8.530E-01 8.277E-01 8.526E-01 9.625E-01 9.634E-01 2 8.916E-01 1.000E+00 8.490E-01 9.648E-01 5.905E-01 6.214E-01 8.413E-01 8.529E-01 3 9.629E-01 8.490E-01 1.000E+00 8.796E-01 8.663E-01 8.917E-01 9.997E-01 9.998E-01 4 8.530E-01 9.648E-01 8.796E-01 1.000E+00 6.189E-01 6.501E-01 8.721E-01 8.838E-01 5 8.277E-01 5.905E-01 8.663E-01 6.189E-01 1.000E+00 9.985E-01 8.765E-01 8.606E-01 6 8.526E-01 6.214E-01 8.917E-01 6.501E-01 9.985E-01 1.000E+00 9.009E-01 8.865E-01 7 9.625E-01 8.413E-01 9.997E-01 8.721E-01 8.765E-01 9.009E-01 1.000E+00 9.992E-01 8 9.634E-01 8.529E-01 9.998E-01 8.838E-01 8.606E-01 8.865E-01 9.992E-01 1.000E+00 9 8.534E-01 9.646E-01 8.799E-01 9.999E-01 6.208E-01 6.518E-01 8.726E-01 8.842E-01 10 8.541E-01 6.235E-01 8.941E-01 6.520E-01 9.968E-01 9.990E-01 9.033E-01 8.888E-01 11 8.150E-01 6.935E-01 8.277E-01 6.829E-01 7.351E-01 7.573E-01 8.238E-01 8.291E-01 12 8.904E-01 9.808E-01 8.354E-01 9.423E-01 6.150E-01 6.419E-01 8.309E-01 8.392E-01 13 7.950E-01 5.588E-01 8.308E-01 5.896E-01 9.877E-01 9.833E-01 8.433E-01 8.244E-01 14 9.869E-01 8.703E-01 9.488E-01 8.280E-01 8.583E-01 8.786E-01 9.509E-01 9.466E-01 15 9.537E-01 8.341E-01 9.916E-01 8.669E-01 8.944E-01 9.158E-01 9.936E-01 9.896E-01 16 8.567E-01 9.549E-01 8.878E-01 9.898E-01 6.606E-01 6.891E-01 8.817E-01 8.902E-01 17 1.856E-01 2.321E-01 1.808E-01 2.363E-01 1.675E-01 1.688E-01 1.812E-01 1.819E-01 18 5.222E-01 5.967E-01 5.551E-01 6.082E-01 4.698E-01 4.846E-01 5.491E-01 5.553E-01 19 6.311E-01 3.866E-01 6.173E-01 3.713E-01 8.690E-01 8.530E-01 6.318E-01 6.111E-01 20 6.622E-01 4.169E-01 6.154E-01 3.653E-01 8.540E-01 8.398E-01 6.286E-01 6.100E-01 21 7.295E-01 5.167E-01 6.495E-01 4.285E-01 7.865E-01 7.834E-01 6.582E-01 6.456E-01 22 9.864E-01 8.495E-01 9.378E-01 7.951E-01 8.603E-01 8.801E-01 9.396E-01 9.365E-01 23 4.880E-01 4.903E-01 4.928E-01 4.794E-01 3.958E-01 4.132E-01 4.863E-01 4.972E-01 24 5.166E-01 5.003E-01 5.130E-01 4.885E-01 5.022E-01 5.113E-01 5.123E-01 5.148E-01 25 4.678E-01 4.510E-01 4.650E-01 4.406E-01 4.786E-01 4.847E-01 4.654E-01 4.662E-01 26 3.542E-01 3.359E-01 3.504E-01 3.294E-01 4.347E-01 4.321E-01 3.551E-01 3.493E-01 27 4.191E-01 4.060E-01 4.111E-01 3.969E-01 4.747E-01 4.745E-01 4.154E-01 4.104E-01 28 5.372E-01 5.292E-01 5.228E-01 5.150E-01 5.446E-01 5.493E-01 5.259E-01 5.229E-01 29 6.543E-01 6.511E-01 6.374E-01 6.342E-01 6.022E-01 6.136E-01 6.379E-01 6.391E-01 30 6.667E-01 4.381E-01 6.730E-01 4.322E-01 8.187E-01 8.159E-01 6.816E-01 6.694E-01 31 4.770E-01 2.395E-01 4.821E-01 2.341E-01 7.442E-01 7.284E-01 4.955E-01 4.767E-01 32 3.553E-01 1.294E-01 3.622E-01 1.263E-01 6.524E-01 6.326E-01 3.763E-01 3.564E-01 33 7.158E-01 7.218E-01 7.060E-01 7.095E-01 5.640E-01 5.864E-01 7.012E-01 7.099E-01 34 6.492E-01 6.403E-01 6.387E-01 6.257E-01 5.781E-01 5.927E-01 6.370E-01 6.413E-01 35 5.624E-01 5.510E-01 5.579E-01 5.404E-01 5.327E-01 5.433E-01 5.572E-01 5.598E-01 36 4.467E-01 4.327E-01 4.414E-01 4.238E-01 4.848E-01 4.872E-01 4.443E-01 4.412E-01 .. code-block:: none :caption: Bias convergence edit. :name: list6-6-10 :class: long *** Cumulative Conv. of (A-C)/C For Application *** =>Edited for reference response: 40 =>Based on similarity parameter : Ck =>Minimum similarity included : 0.300 =>Similarity bin width for edit : 0.010 CUM RANGE MAX. SIM NO. EXP %(A-C)/C NO. INCLUDED INCLUDED 1 0.390 1 -0.057 2 0.400 1 -0.057 3 0.410 1 -0.057 4 0.420 1 -0.057 5 0.430 1 -0.057 6 0.440 1 -0.057 7 0.450 1 -0.057 8 0.460 1 -0.057 9 0.470 1 -0.057 10 0.480 1 -0.057 11 0.490 1 -0.057 12 0.500 1 -0.057 13 0.510 1 -0.057 14 0.520 1 -0.057 15 0.530 1 -0.057 16 0.540 1 -0.057 17 0.550 1 -0.057 18 0.560 1 -0.057 19 0.570 1 -0.057 20 0.580 1 -0.057 21 0.590 1 -0.057 22 0.600 1 -0.057 23 0.610 1 -0.057 24 0.620 1 -0.057 25 0.630 1 -0.057 26 0.640 1 -0.057 27 0.650 1 -0.057 28 0.660 1 -0.057 29 0.670 1 -0.057 30 0.680 1 -0.057 31 0.690 1 -0.057 32 0.700 1 -0.057 33 0.710 1 -0.057 34 0.720 1 -0.057 35 0.730 1 -0.057 36 0.740 1 -0.057 37 0.750 1 -0.057 38 0.760 1 -0.057 39 0.770 1 -0.057 40 0.780 1 -0.057 41 0.790 1 -0.057 42 0.800 1 -0.057 43 0.810 1 -0.057 44 0.820 1 -0.057 45 0.830 1 -0.057 46 0.840 1 -0.057 47 0.850 1 -0.057 48 0.860 1 -0.057 49 0.870 1 -0.057 50 0.880 1 -0.057 51 0.890 1 -0.057 52 0.900 1 -0.057 53 0.910 1 -0.057 54 0.920 1 -0.057 55 0.930 1 -0.057 56 0.940 1 -0.057 57 0.950 1 -0.057 58 0.960 7 0.063 59 0.970 16 -0.076 60 0.980 21 -0.133 61 0.990 25 -0.170 62 1.000 37 -0.246 .. raw:: latex \begin{landscape} .. code-block:: none :caption: Adjustment summary edit. :name: list6-6-11 ___________________________________________________ | | | NOTATION: calc = prior calculated value | | exp = prior experimental value | | adj = adjusted calculated value | | = adjusted experimental value | |_________________________________________________| _________________________________________________________________________________________________________________________________ | | | | | | | | | | | |%(calc-exp)|%(adj- exp)|%(adj-calc)| %st.dev | %st.dev | %st.dev | indepndnt | diagonal | |++ R E S P O N S E ++ | /calc | /exp | /calc | exp | old calc | new adj | chi-sq. | chi-sq | |_______________________________|___________|___________|___________|___________|___________|___________|___________|___________| | EXPT keff 1_hst001-1 | 1.4808E-01|-7.8217E-02|-2.2618E-01| 6.9778E-01| 9.2613E-01| 1.6039E-01| 1.6325E-02| 5.2540E-02| | EXPT keff 2_hst001-1 |-1.4852E-01|-3.5635E-01|-2.0836E-01| 6.5368E-01| 9.4664E-01| 1.7812E-01| 1.6652E-02| 5.6775E-02| |*APPL keff 3_hst001-1 | NA | NA |-2.2029E-01| NA | 9.2936E-01| 1.6290E-01| NA | NA | | EXPT keff 4_hst001-1 | 7.4944E-02|-1.2371E-01|-1.9856E-01| 4.8693E-01| 9.4552E-01| 1.8134E-01| 4.9671E-03| 3.9194E-02| | EXPT keff 5_hst001-1 | 1.1098E-01|-1.1748E-01|-2.2833E-01| 6.0745E-01| 8.2656E-01| 1.5674E-01| 1.1714E-02| 6.0017E-02| | EXPT keff 6_hst001-1 | 4.5304E-01| 2.2589E-01|-2.2818E-01| 5.5723E-01| 8.3667E-01| 1.5545E-01| 2.0367E-01| 1.1014E+00| | EXPT keff 7_hst001-1 |-5.7003E-03|-2.2571E-01|-2.2002E-01| 6.1814E-01| 9.2336E-01| 1.6172E-01| 2.6316E-05| 1.4614E-04| | EXPT keff 8_hst001-1 |-2.4606E-02|-2.4519E-01|-2.2065E-01| 6.6558E-01| 9.3166E-01| 1.6335E-01| 4.6176E-04| 2.0174E-03| | EXPT keff 9_hst001-1 |-3.6745E-01|-5.6401E-01|-1.9864E-01| 4.6957E-01| 9.4345E-01| 1.8067E-01| 1.2139E-01| 6.6398E-01| | EXPT keff 10_hst001-1 |-4.8080E-01|-7.0685E-01|-2.2945E-01| 5.7306E-01| 8.2869E-01| 1.5179E-01| 2.2702E-01| 8.0333E-01| | EXPT keff 11_hst001-1 | 4.5571E-01| 1.8845E-01|-2.6812E-01| 4.4553E-01| 8.7207E-01| 1.5382E-01| 2.1696E-01| 1.1550E+00| | EXPT keff 12_hst001-1 | 1.0180E-01|-1.0271E-01|-2.0440E-01| 4.6957E-01| 8.9651E-01| 1.6755E-01| 1.0122E-02| 5.2290E-02| | EXPT keff 13_hst001-1 | 9.1516E-02|-1.2988E-01|-2.2127E-01| 4.6957E-01| 7.9206E-01| 1.5393E-01| 9.8828E-03| 4.2829E-02| | EXPT keff 14_hst001-1 | 1.9343E-01|-2.9197E-02|-2.2257E-01| 2.4597E-01| 8.8292E-01| 1.5096E-01| 4.4550E-02| 5.6670E-01| | EXPT keff 15_hst001-1 | 4.8573E-01| 2.7272E-01|-2.1434E-01| 4.6957E-01| 8.9044E-01| 1.5510E-01| 2.3331E-01| 1.2327E+00| | EXPT keff 16_hst001-1 |-1.1253E-01|-3.1683E-01|-2.0466E-01| 4.6957E-01| 8.9137E-01| 1.6221E-01| 1.2468E-02| 6.4284E-02| | EXPT keff 17_hst001-1 | 3.0586E-01|-5.2167E-02|-3.5787E-01| 4.6957E-01| 1.0537E+00| 4.0553E-01| 7.0374E-02| 8.2612E-02| | EXPT keff 18_hst001-1 |-3.8014E-02|-2.9288E-01|-2.5497E-01| 4.6957E-01| 7.4743E-01| 1.5210E-01| 1.8543E-03| 7.2266E-03| | EXPT keff 19_hst001-1 | 2.2897E-01|-2.3461E-02|-2.5238E-01| 4.6957E-01| 7.3482E-01| 1.5593E-01| 6.9036E-02| 2.6221E-01| | EXPT keff 20_hst001-1 | 3.1242E-01| 5.1984E-02|-2.6060E-01| 4.6957E-01| 7.4192E-01| 1.5158E-01| 1.2683E-01| 4.9994E-01| | EXPT keff 21_hst001-1 | 5.9168E-01| 3.2443E-01|-2.6916E-01| 6.9778E-01| 7.7116E-01| 1.4698E-01| 3.2540E-01| 7.1086E-01| | EXPT keff 22_hst001-1 |-1.5504E-01|-3.8713E-01|-2.3269E-01| 7.3068E-01| 8.7855E-01| 1.4818E-01| 1.8385E-02| 4.3978E-02| | EXPT keff 23_hst001-1 | 4.9385E-01| 1.9665E-01|-2.9817E-01| 6.9907E-01| 7.8336E-01| 1.6584E-01| 2.2221E-01| 4.8883E-01| ... .. raw:: latex \end{landscape} .. code-block:: none :caption: Adjusted response edit. :name: list6-6-12 :class: long ** Description of Adjusted Responses ** RESP.# EXPERIMENT NAME SENS. TITLE USE TYPE ADJUSTED RESPONSE 1 1_hst001-1 r1 expt keff 9.9922E-01 +/- 1.6063E-03 2 2_hst001-1 r2 expt keff 9.9644E-01 +/- 1.7785E-03 3 3_hst001-1 r3 appl keff 1.0002E+00 +/- 1.6330E-03 4 4_hst001-1 r4 expt keff 9.9876E-01 +/- 1.8147E-03 5 5_hst001-1 r5 expt keff 9.9883E-01 +/- 1.5691E-03 6 6_hst001-1 r6 expt keff 1.0023E+00 +/- 1.5616E-03 7 7_hst001-1 r7 expt keff 9.9774E-01 +/- 1.6171E-03 8 8_hst001-1 r8 expt keff 9.9755E-01 +/- 1.6331E-03 9 9_hst001-1 r9 expt keff 9.9436E-01 +/- 1.8001E-03 10 10_hst001-1 r10 expt keff 9.9293E-01 +/- 1.5106E-03 11 11_hst001-1 rot2 tank in cen expt keff 1.0019E+00 +/- 1.5453E-03 12 12_hst001-1 rot7 tank in cen expt keff 9.9897E-01 +/- 1.6772E-03 13 13_hst001-1 rot38 tank in ce expt keff 9.9870E-01 +/- 1.5407E-03 14 14_hst001-1 rot4 tank in cen expt keff 9.9971E-01 +/- 1.5125E-03 15 15_hst001-1 rot14 tank in ce expt keff 1.0027E+00 +/- 1.5586E-03 16 16_hst001-1 rot29 tank in ce expt keff 9.9683E-01 +/- 1.6202E-03 17 17_hst001-1 ol3ne 15.5 in. s expt keff 9.9948E-01 +/- 4.0678E-03 18 18_hst001-1 case 30 experime expt keff 9.9707E-01 +/- 1.5204E-03 19 19_hst001-1 heu-sol-therm-02 expt keff 9.9977E-01 +/- 1.5629E-03 20 20_hst001-1 heu-sol-therm-02 expt keff 1.0005E+00 +/- 1.5205E-03 21 21_hst001-1 heu-sol-therm-02 expt keff 1.0032E+00 +/- 1.4786E-03 22 22_hst001-1 heu-sol-therm-02 expt keff 9.9613E-01 +/- 1.4795E-03 23 23_hst001-1 heu-sol-therm-02 expt keff 1.0020E+00 +/- 1.6666E-03 24 24_hst001-1 heu-sol-therm-02 expt keff 1.0059E+00 +/- 1.3850E-03 25 25_hst001-1 heu-sol-therm-02 expt keff 9.9870E-01 +/- 1.3849E-03 26 26_hst001-1 heu-sol-therm-02 expt keff 9.9596E-01 +/- 1.4012E-03 27 27_hst001-1 heu-sol-therm-02 expt keff 1.0019E+00 +/- 1.3609E-03 ... .. code-block:: none :name: list6-6-13 :caption: Application bias summary edit. :class: long _________________________________________________________________________________ | | | | | APPLICATION AND BIAS SUMMARY | | | |_______________________________________________________________________________| APPLICATION TYPE CALC VALUE PRIOR UNC % REL. BIAS % BIAS ADJ VALUE ADJ UNC % ------------------------ ---- ----------- ----------- ----------- ----------- ----------- ----------- 3_hst001-1 keff 1.0024E+00 9.2936E-01 2.2029E-01 2.2083E-03 1.0002E+00 1.6290E-01 32_hst001-1 keff 9.9990E-01 6.2270E-01 2.7119E-01 2.7116E-03 9.9719E-01 1.5301E-01 40_hst001-1 keff 1.0017E+00 6.0802E-01 2.4635E-01 2.4677E-03 9.9922E-01 1.3718E-01 NOTE: The relative bias and uncertainty values are normalized to the calculated response value. ________________________________________________________________________________ | | | | | CONTRIBUTION TO THE REFERENCE APPLICATION BIAS FROM NUCLIDE-REACTION PAIRS | | | |______________________________________________________________________________| CONTRIB. TO BIAS FRACTION OF BIAS NUCLIDE REACTION % dk/k L1-NORM ------------ --------------- ------------------ ---------------- u-235 nubar 1.2213E-01 3.1407E-01 u-235 chi 1.0203E-01 2.6237E-01 u-235 n,gamma 5.6672E-02 1.4647E-01 h-1 elastic -4.9761E-02 1.2798E-01 u-235 fission 2.3867E-02 6.1390E-02 o-16 elastic -2.0793E-02 5.3522E-02 fe n,gamma 6.2393E-03 1.6044E-02 fe elastic 2.1543E-03 5.5536E-03 h-1 n,gamma 1.0344E-03 2.6600E-03 fe n,n' 9.3138E-04 2.3950E-03 cr n,gamma 7.2797E-04 1.8720E-03 ni n,gamma 3.7890E-04 9.7433E-04 u-234 n,gamma -3.0719E-04 8.0394E-04 Figure 6.6.13. Application bias summary edit cr elastic 2.4825E-04 6.3853E-04 u-238 n,gamma 2.4072E-04 6.1902E-04 b-10 n,alpha 2.1488E-04 5.5255E-04 n-14 n,p 2.1113E-04 5.4291E-04 u-235 n,n' -1.5322E-04 3.9401E-04 ni elastic 1.0020E-04 2.5830E-04 mn-55 n,gamma 8.5311E-05 2.1938E-04 cr n,n' 4.9517E-05 1.2733E-04 n-14 elastic -4.0198E-05 1.0883E-04 u-235 elastic 4.1221E-06 9.1845E-05 o-16 n,alpha 3.5175E-05 9.0453E-05 u-238 elastic 8.6693E-06 4.1429E-05 b-11 elastic 1.3516E-05 3.6278E-05 ni n,n' 1.0318E-05 2.6532E-05 mn-55 elastic 1.0216E-05 2.6270E-05 u-234 fission -7.9124E-06 2.0356E-05 u-236 n,gamma -5.3796E-06 1.4106E-05 u-238 n,n' 4.7750E-06 1.2292E-05 n-14 n,gamma 4.2428E-06 1.0911E-05 b-10 n,n' 3.5966E-06 9.5464E-06 n-14 n,alpha 3.5931E-06 9.2396E-06 n-14 n,n' 5.9234E-07 7.7528E-06 c elastic 2.2321E-06 6.1840E-06 b-10 elastic -8.9644E-07 6.1541E-06 b-10 n,p 1.5897E-06 4.0880E-06 ni n,p 1.3670E-06 3.5153E-06 c n,n' -1.0409E-06 2.6767E-06 u-234 n,n' -8.8825E-07 2.4901E-06 o-16 n,n' -7.8209E-07 2.3129E-06 u-236 fission -7.4671E-07 1.9550E-06 si elastic 6.9545E-07 1.8014E-06 u-234 nubar -6.3015E-07 1.6282E-06 u-235 n,2n -4.4596E-07 1.1468E-06 mn-55 n,n' 4.2551E-07 1.0942E-06 u-238 nubar 3.9169E-07 1.0072E-06 fe n,p 3.7711E-07 9.6973E-07 ti elastic 2.3945E-07 6.1575E-07 ti n,gamma 1.6831E-07 4.3281E-07 b-11 n,n' 1.0274E-07 2.6419E-07 o-16 n,gamma 1.0048E-07 2.5839E-07 ti n,n' 7.4571E-08 1.9176E-07 si n,n' 5.2998E-08 1.3628E-07 u-236 nubar -5.0705E-08 1.3039E-07 ... .. code-block:: none :name: list6-6-14 :caption: Cross-section adjustment edit. :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Multigroup Cross Section Changes Inferred from GLSS Adjustment :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: **************** u-235 chi group delta-XS st. dev. st. dev. rel sen contribution to fraction of number (%) old(%) new(%) coeff. appl. bias (%) bias - L1-norm 1 8.798E-06 8.125E-01 8.125E-01 1.585E-04 -1.395E-09 3.586E-09 2 -1.194E-04 2.310E-01 2.309E-01 3.486E-04 4.160E-08 1.070E-07 3 -1.044E-04 2.811E-01 2.811E-01 6.741E-04 7.035E-08 1.809E-07 4 -1.079E-04 2.298E-01 2.298E-01 2.170E-03 2.342E-07 6.023E-07 5 -9.178E-05 3.071E-01 3.071E-01 1.384E-03 1.270E-07 3.265E-07 6 -5.405E-05 2.888E-01 2.887E-01 4.064E-04 2.197E-08 5.648E-08 7 -5.449E-05 2.887E-01 2.886E-01 1.693E-03 9.228E-08 2.373E-07 8 -8.681E-05 2.444E-01 2.443E-01 1.896E-03 1.646E-07 4.232E-07 9 -1.706E-04 2.544E-01 2.543E-01 2.392E-03 4.082E-07 1.050E-06 10 -2.719E-04 1.946E-01 1.945E-01 3.378E-03 9.187E-07 2.362E-06 11 -3.341E-04 1.828E-01 1.827E-01 3.793E-03 1.267E-06 3.259E-06 12 -4.506E-04 2.503E-01 2.502E-01 2.920E-03 1.316E-06 3.384E-06 13 -1.078E-03 3.416E-01 3.415E-01 8.418E-04 9.071E-07 2.333E-06 14 -1.230E-03 2.896E-01 2.894E-01 5.316E-03 6.538E-06 1.681E-05 15 -1.187E-03 2.036E-01 2.034E-01 1.087E-02 1.289E-05 3.316E-05 16 -3.654E-03 4.411E-01 4.393E-01 2.317E-02 8.467E-05 2.177E-04 17 -4.655E-02 3.057E-01 2.995E-01 3.238E-02 1.507E-03 3.875E-03 18 -1.238E-01 3.167E-01 2.829E-01 2.907E-02 3.598E-03 9.251E-03 19 -1.238E-01 3.167E-01 2.829E-01 1.322E-02 1.636E-03 4.208E-03 20 -1.238E-01 3.167E-01 2.829E-01 4.597E-03 5.689E-04 1.463E-03 21 -1.257E-01 3.109E-01 2.752E-01 1.735E-03 2.182E-04 5.610E-04 22 -1.337E-01 3.108E-01 2.703E-01 5.343E-03 7.141E-04 1.836E-03 23 -1.337E-01 3.108E-01 2.703E-01 4.306E-03 5.755E-04 1.480E-03 24 -1.337E-01 3.108E-01 2.703E-01 1.447E-02 1.934E-03 4.973E-03 25 -1.337E-01 3.108E-01 2.703E-01 1.678E-02 2.242E-03 5.766E-03 26 -1.337E-01 3.108E-01 2.703E-01 2.346E-02 3.136E-03 8.065E-03 27 -1.337E-01 3.108E-01 2.703E-01 4.807E-03 6.424E-04 1.652E-03 28 -1.337E-01 3.108E-01 2.703E-01 5.858E-03 7.830E-04 2.013E-03 29 -1.337E-01 3.108E-01 2.703E-01 7.113E-03 9.507E-04 2.445E-03 30 -1.337E-01 3.108E-01 2.703E-01 1.844E-02 2.464E-03 6.337E-03 31 -1.337E-01 3.108E-01 2.703E-01 1.076E-02 1.438E-03 3.699E-03 32 -1.337E-01 3.108E-01 2.703E-01 1.164E-02 1.555E-03 3.999E-03 33 -1.337E-01 3.108E-01 2.703E-01 1.316E-02 1.759E-03 4.524E-03 34 -1.337E-01 3.108E-01 2.703E-01 3.669E-02 4.904E-03 1.261E-02 35 -1.337E-01 3.108E-01 2.703E-01 7.544E-02 1.008E-02 2.593E-02 36 -1.337E-01 3.108E-01 2.703E-01 9.663E-02 1.292E-02 3.321E-02 37 -1.337E-01 3.108E-01 2.703E-01 1.070E-01 1.430E-02 3.678E-02 38 -1.337E-01 3.108E-01 2.703E-01 7.069E-02 9.448E-03 2.430E-02 39 -1.337E-01 3.108E-01 2.703E-01 8.255E-02 1.103E-02 2.837E-02 40 -1.337E-01 3.108E-01 2.703E-01 4.197E-02 5.609E-03 1.442E-02 41 -1.336E-01 3.106E-01 2.700E-01 1.461E-01 1.951E-02 5.017E-02 42 -1.331E-01 3.108E-01 2.706E-01 2.156E-02 2.871E-03 7.383E-03 43 -1.331E-01 3.108E-01 2.706E-01 3.154E-02 4.199E-03 1.080E-02 44 -1.331E-01 3.108E-01 2.706E-01 1.072E-02 1.427E-03 3.670E-03 Total -3.625E+00 ---------- ---------- 9.994E-01 1.221E-01 3.141E-01 NOTE: The contribution to the application bias is tabulated in units of % dk/k. The contribution to the application bias is normalized to the calculated response value. NOTE: The fraction of the bias L1-norm is equal to the absolute value of the bias contribution divided by the sum of the absolute value of all groupwise bias contributions. **************** u-235 chi group delta-XS st. dev. st. dev. rel sen contribution to fraction of number (%) old(%) new(%) coeff. appl. bias (%) bias - L1-norm 1 3.011E+00 1.165E+01 8.267E+00 -2.143E-03 6.453E-03 1.659E-02 2 2.165E+00 8.336E+00 5.877E+00 -5.079E-03 1.100E-02 2.828E-02 3 1.528E+00 5.869E+00 4.122E+00 -1.255E-02 1.917E-02 4.930E-02 4 8.165E-01 3.138E+00 2.200E+00 -2.584E-02 2.110E-02 5.426E-02 5 3.741E-01 1.451E+00 1.024E+00 -8.856E-03 3.313E-03 8.518E-03 6 2.299E-01 9.050E-01 6.464E-01 -1.880E-03 4.321E-04 1.111E-03 7 7.697E-02 3.463E-01 2.717E-01 -3.955E-03 3.044E-04 7.827E-04 ... <<<<< NORMAL END OF CALCULATION >>>>> HTML Output ~~~~~~~~~~~ The input file for the TSURFER sample problem is named tsurfer.inp. In this case, the HTML-formatted output is stored in a file called tsurfer.html and additional resources are stored in directories called tsurfer.htmd and applet_resources. This section contains example TSURFER HTML-formatted output only for demonstration of the interface. When tsurfer.html is opened in a web browser, the information shown in :numref:`fig6-6-15` is displayed. The title of the input file is displayed between the two SCALE logos. Because this SCALE input file only executed TSURFER, only a single-output listing is available. The text "1. TSURFER" is a hyperlink to view the output from TSURFER. Clicking on the "1. TSURFER" hyperlink will bring up the information shown in :numref:`fig6-6-16`. Clicking on the SCALE logos will link the user to the SCALE website, if external internet access is available. .. figure:: figs/TSURFER/fig15.png :align: center :width: 600 :name: fig6-6-15 Initial screen from TSURFER HTML output. The initial page of output from TSURFER is shown in :numref:`fig6-6-16`. Program verification information is shown in the table under the TSURFER logo. This table includes information about the code that was executed and the date and time it was run. The menu on the left side of the screen contains hyperlinks to specific portions of the code output. Echoes of the input data are available in the Input Data section. Any errors or warning messages are available in the Messages sections. Results from the code execution are shown in the results section. .. figure:: figs/TSURFER/fig16.png :align: center :width: 600 :name: fig6-6-16 Program verification screen from TSURFER HTML output. Selecting Input Parameters will reveal the menu of available input data. Selecting Input Parameters causes the table shown in :numref:`fig6-6-17` to be displayed. Other input data can also be displayed by selecting the desired data from the menu. .. figure:: figs/TSURFER/fig17.png :align: center :width: 600 :name: fig6-6-17 Input parameters from TSURFER HTML output. Selecting Messages will reveal a menu of available messages. Selecting Warning Messages from the Messages section of the menu causes the information shown in :numref:`fig6-6-18` to appear. The Warning Messages edit contains all warning messages that were generated during the execution of the code. If errors were encountered in the code execution, an Error Messages item would have also been available in the menu under Messages. .. figure:: figs/TSURFER/fig18.png :align: center :width: 600 :name: fig6-6-18 Warning messages from TSURFER HTML output. Selecting Results causes a menu of available results to be revealed. From this menu, selecting Cross-Section Adjustments causes a menu on the right to be revealed. From this menu, nuclide-reaction pairs can be selected to visualize their cross-section adjustments in tabular format. The U-235 nubar adjustments are shown in :numref:`fig6-6-19`. .. figure:: figs/TSURFER/fig19.png :align: center :width: 600 :name: fig6-6-19 Cross-section adjustments from TSURFER HTML output. Various plots can also be viewed in the TSURFER HTML output. Selecting "Plots" in the Results menu brings up a submenu of various TSURFER plots. The correlation matrices may be viewed by selecting "Correlation Matrices" in the Plots submenu. A Java applet version of Javapeño will appear in the browser window with the correlation matrices preloaded. Data can be added to the plot by double-clicking on the list of available data on the right side of Javapeño. The plot shown in :numref:`fig6-6-20` was produced with this procedure. .. figure:: figs/TSURFER/fig20.png :align: center :width: 600 :name: fig6-6-20 Three-dimensional plot of correlation matrix in TSURFER HTML output. .. only:: html .. rubric:: References .. bibliography:: zSCALE.bib :cited: :keyprefix: TSURFER- :labelprefix: TSURFER- Appendices ---------- .. toctree:: tsurfer-appA