.. _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