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The CCPR K2.c key comparison of spectral responsivity from 200 nm to 400 nm
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2014 Metrologia 51 S336
(http://iopscience.iop.org/0026-1394/51/6/S336)
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Bureau International des Poids et Mesures
Metrologia
Metrologia 51 (2014) S336–S343
doi:10.1088/0026-1394/51/6/S336
The CCPR K2.c key comparison of spectral
responsivity from 200 nm to 400 nm
Lutz Werner
Physikalisch-Technische Bundesanstalt, Berlin, Germany
E-mail: [email protected]
Received 9 June 2014, revised 25 September 2014
Accepted for publication 30 September 2014
Published 20 November 2014
Abstract
The CCPR K2.c key comparison of spectral responsivity from 200 nm to 400 nm was carried
out in the framework of the CIPM Mutual Recognition Arrangement by 14 participating
national metrology institutes. The key comparison was piloted by the Physikalisch-Technische
Bundesanstalt (PTB). The comparison principle, measurements, analysis and results including
a procedure to deal with the issue of inconsistent data are described.
Keywords: key comparison, radiometry, detectors, spectral responsivity
(Some figures may appear in colour only in the online journal)
1. Introduction
The Mutual Recognition Arrangement (MRA) for national
measurement standards and for calibration and measurement
certificates issued by the national metrology institutes (NMIs)
was signed in 1999. The objectives of the MRA were to
establish the degree of equivalence of national measurement
standards maintained by NMIs and to provide for the mutual
recognition of calibration and measurement certificates issued
by NMIs. Under the MRA the metrological equivalence of
national measurement standards is determined by a set of
key comparisons chosen and organized by the Consultative
Committees of the CIPM working closely with the Regional
Metrology Organizations (RMOs). At its meeting in March
1997, the Consultative Committee for Photometry and
Radiometry (CCPR) identified several key comparisons in
the field of optical radiation metrology. One of these was the
CCPR K2.c key comparison of spectral responsivity in the
spectral range from 200 nm to 400 nm.
The Physikalisch-Technische Bundesanstalt (PTB), the
NMI of Germany, was asked to be the pilot of this key comparison. The technical protocol of the key comparison was
drawn up by a small working group comprising PTB (convenor), BIPM, MSL, NPL, NIST, and NRC.
Laboratories from 14 NMIs including the pilot laboratory took part in this comparison. These were LNE-Cnam
(France), NMIA (Australia), MIKES (Finland), IO-CSIC
(Spain), MSL (New Zealand), NIM (China), NIST (United
0026-1394/14/06S336+8$33.00
States of America), NMIJ AIST (Japan), VSL (Netherlands),
NPL (United Kingdom), NRC (Canada), PTB (Germany) pilot, NMC-A*STAR (Singapore), and VNIIOFI (Russian
Federation).
2. Comparison artefacts and principle of the
comparison
Three types of transfer detectors based on two types of photodiodes were used as comparison artefacts: (i) single element photodiode detectors based on windowless SUV100
PtSi/n-Si Schottky photodiodes manufactured by the
Swiss Federal Institute of Technology (ETH) in Zurich,
Switzerland, (ii) single element photodiode detectors and
(iii) three element reflection trap detectors, both based on
windowless Hamamatsu S5227 1010 Si pn junction photodiodes. The latter two detector types, henceforth referred to as
Si-photodiode-based detectors, were only used in the wavelength region from 250 nm to 400 nm because their spectral
responsivity can alter due to exposure in the short-wavelength
ultraviolet (UV) spectral range [1–3]. The PtSi-photodiodebased detectors were used in the whole wavelength region of
this comparison ranging from 200 nm to 400 nm as they were
thought to be less sensitive to UV exposure [4]. However,
because of the poorer results of the PtSi-photodiode-based
detectors in comparison with those of the Si-photodiodebased detectors, the former detectors were only used in the
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© 2014 BIPM & IOP Publishing Ltd Printed in the UK
L Werner
Metrologia 51 (2014) S336
wavelength range from 200 nm to 240 nm to calculate the
degrees of equivalence (DoE) and the key comparison reference value (KCRV).
Each detector was equipped with a Pt100 temperature
sensor to monitor the detector temperature during the measurements. Because of the large temperature coefficient of the
PtSi photodiodes and the possibility of large differences in
laboratory temperature between the participants and the pilot,
these photodiodes were mounted on a brass block which could
be warmed or cooled by water. Thus, the temperature of the
PtSi photodiodes could be set independently of the laboratory
temperature.
The comparison was organized in a star form, carried out
in three rounds with three groups of participants and using
sets of transfer detectors each consisting of three PtSi photodiodes, three Si photodiodes and one Si-photodiode-based
trap detector. The PtSi photodiodes showed a severe malfunction twice at NIM after transportation, thus, a fourth
round of the comparison was added to allow NIM to repeat
their measurements as the only participant in this round. To
check the stability of the transfer detectors and to provide
the linkage between the participants, the pilot performed
its measurements in each round before the transfer detectors were sent to the participants, and in addition, after the
last return of the transfer detectors. The final measurements
of the pilot form an additional round comprising only these
measurements of the pilot without any measurements of
participants.
3. Measurements
The measurements were performed from 2004 to 2007. Each
participating NMI i determined the spectral responsivity si, j
of their transfer detectors j in round k in order to perform
the comparison on the spectral responsivity. The mean relative standard uncertainty u rel,cal, i of the spectral responsivity
of the transfer detectors as stated by the participants is shown
in figure 1. For reasons of clarity, only the average over the
values for the PtSi photodiodes in the wavelength range from
200 nm to 240 nm and over the values for the Si-photodiodebased detectors in the wavelength range from 250 nm to
400 nm is given for each participant.
The transfer detectors were compared by the pilot with
reference detectors from the pilot at each wavelength before
they were sent to the participants in each round and after
their last return. For this, the ratios R of the photocurrent of
the pilot’s reference detectors and the photocurrent of each
transfer detector j when irradiated by the same source at the
same wavelength were measured by the pilot delivering ratio
Rj, k. These photocurrent ratios are equal to the ratios of the
spectral responsivity of the corresponding detectors at the corresponding time. They were therefore used to calculate the
spectral responsivity of the reference detector
based on the results of participant i obtained in round k and the
results of the pilot which were obtained before the measurements of the participant (i.e. in round k) and after the measurements of the participant (i.e. in round k + 1).
Because of the large differences between PtSi photodiodes
and Si photodiodes, different reference detectors were used:
a PtSi photodiode as reference detector for PtSi-photodiodebased transfer detectors and a Si-photodiode-based trap
detector as reference detector for Si-photodiode-based
transfer detectors. The reference detectors were compared
against a group of secondary detector standards of PTB in
each phase of the comparison to check the stability of their
spectral responsivity. It turned out that the detectors were
not stable. The time dependent spectral responsivity of the
Si-photodiode-based reference could be well fitted using
a linear drift model with a wavelength dependent relative
annual drift rate between −0.24% and −0.07%. The temporal change of the spectral responsivity of the PtSi photodiode reference was smaller and less regular and was
described by a more common approach. It is assumed that
the responsivity change of the reference detector during the
measurements of a participant lies with equal probability
somewhere in the interval limited by the values determined
by the surrounding measurements of the pilot. The responsivity change of the reference detector during the measurements of a participant is then the mean value of the limits of
this interval. As a consequence, this leads to a stepwise drift.
The stepwise drift between subsequent rounds depends on
the wavelength and the round and ranges between 0% and
0.13% with an overall mean of about 0.04%.
4. Analysis
The calculation of the DoE and the KCRV are performed in
agreement with the CCPR-G2 guidelines [5] and are based
on the method of the weighted mean (with cut-off) which has
been extended for non-stable artefacts. Due to the large differences between PtSi photodiodes and Si photodiodes regarding
the magnitude, spectral dependence, temperature dependence,
and temporal drift of the spectral responsivity, these two types
of photodiodes were treated separately within the data analysis. The differences between the single element detectors and
the trap detectors made up of Si photodiodes were rather small
compared to the differences between the two types of photodiodes. Therefore, the detectors made up of Si photodiodes
are summarised as Si-photodiode-based detectors and treated
together in the data analysis. The data analysis is, unless specified otherwise, identical for both types of detectors and is
briefly described below.
In the first step, the results on the spectral responsivity of
the reference detectors xi, j, k and xi, j, k + 1 are averaged over the
set of transfer detectors of each type:
M
xi, j, k = Rj, k si, j
xi, k = (1 / M )
and
∑x
j=1
xi, j, k + 1 = Rj, k + 1si, j
(1)
and
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i, j , k
L Werner
Metrologia 51 (2014) S336
12%
2.0%
LNE-Cnam
Relative standard uncertainty
IO-CSIC
10%
MIKES
1.5%
MSL
VSL
8%
NIM
6%
NIST
1.0%
NMIA
NMIJ
4%
NPL
0.5%
NRC
PTB
2%
NMC-A*STAR
0%
VNIIOFI
0.0%
200
250
200
250
300
350
400
Wavelength / nm
Figure 1. Mean relative standard uncertainty u rel,cal, i of the participants’ determination of the spectral responsivity of the PtSi-photodiode-
based and Si-photodiode-based transfer detectors in the wavelength ranges from 200 nm to 240 nm and from 250 nm to 400 nm, respectively.
The left part of the figure shows the wavelength range from 200 nm to 250 nm in an extended range of the relative standard uncertainty.
M
xi, k + 1 = (1 / M )
∑
xi, j , k + 1
(2)
j=1
M is the number of transfer detectors of one type, i.e. M is
equal 3 and 4 for the PtSi photodiodes and the Si-photodiodebased detectors, respectively.
It is assumed in the following that the temporal drift of
all detectors (i.e. reference and transfer detectors) of one type
is very similar. The spectral responsivity xi of the reference
detector as obtained by the transfer from the transfer detectors
calibrated by participant i at time ti is then
xi = (1 − τi, k ) xi, k + τi, kxi, k + 1
(3)
In case of the Si-photodiode-based detectors, drifting linearly with time, it is
τi, k = (ti − tkP ) / (tkP+ 1 − tkP )
(4)
with tkP and tkP+ 1 the time at which the pilot performed its measurements of the ratios Rj, k and Rj, k + 1 before and after the measurements of the participant in round k, i.e. in round k and k + 1,
respectively. In case of the PtSi-photodiode-based detectors,
described by a stepwise drift with time, xi is calculated as the
mean of xi, k and xi, k + 1 which corresponds to τi, k = 0.5.
Thus, N values of the spectral responsivity xi of the reference detector, derived from the N participants’ results, are
obtained and used to calculate a reference value. A separate
reference value is calculated for each wavelength. The relative
standard uncertainty of xi is given by
u rel (xi ) =
2
2
urel,cal,
i + u rel,T, i
(5)
with u rel,cal, i — the mean relative standard uncertainty of the
spectral responsivity of the transfer detectors determined by
participant i. u rel,T, i is the relative standard uncertainty of the
transfer comprising contributions arising from the measurements of the pilot, deviations from the assumed temporal
behaviour of the transfer detectors, and averaging over the
transfer detectors of one participant.
The measurement results on the Si-photodiode-based
and the PtSi-photodiode-based detectors are analysed using
the linear drift model and the stepwise drift model, respectively, which are described in the appendix. The estimate
yi (i = 1, … , N ) of the time dependent spectral responsivity
of the reference detector forms the time dependent KCRV.
The DoEs are expressed by the deviation di = xi − yi of the
ith participant’s result from the KCRV and the expanded
(k = 2) uncertainty of this deviation U (di ) = 2u(di ). In addition to the drift model based analysis, all results have been
analysed with the common method of the weighted mean
for comparison.
The use of the above models requires the consistency of
the set of data which was checked using the chi-squared test.
The chi-squared test failed at a considerable number of wavelengths. Therefore, a procedure to achieve consistency and to
exclude outlying results was developed by the pilot and discussed and agreed by all participants. The procedure is based
on the CCPR-G2 Guidelines [5] and uses the following steps
to achieve consistency:
(1)Perform the chi-squared test. The test fails if the observed
2 = ∑ N w d 2
chi-squared value χobs
i = 1 i i is greater than
2
χ(0.05)
(ν ) being the critical value of the chi-squared distribution, with the corresponding degrees of freedom ν, at
the 5 % significance level. (The weights wi are defined in
the Appendix.)
(2)If the chi-squared test fails and there are participants
with di / U (di ) >2, then exclude the corresponding results
as potential outliers from the calculation of the tentative KCRV.
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L Werner
Metrologia 51 (2014) S336
Relative deviation from KCRV
20%
2.5%
10%
LNE-Cnam
2.0%
IO-CSIC
1.5%
MIKES
MSL
1.0%
0%
VSL
NIM
0.5%
-10%
NIST
0.0%
-20%
NMIA
-0.5%
NMIJ
-1.0%
NPL
NRC
-1.5%
-30%
PTB
NMC-A*STAR
-2.0%
-40%
VNIIOFI
-2.5%
200
250
200
250
300
350
400
Wavelength / nm
Figure 2. Relative deviation Di from the KCRV. The left part of the figure shows the wavelength range from 200 nm to 250 nm in an
extended range of the relative deviation.
(3)If the chi-squared test fails and there are no participants
with di / U (di ) >2 or if the chi-squared test still fails after
the exclusion of potential outliers in step 2, then use
the Mandel-Paule method [6] by applying an additional
2 under the
‘interlaboratory variance’, that is by adding uadd
square root in equation (5), so that the data set will be
forced to pass the chi-squared test.
(4)If the Mandel-Paule method has been applied in step
3 after excluding potential outliers in step 2, check
whether these outliers can be reincluded after applying
the Mandel-Paule method. For this, check whether the
condition di / U (di ) >2 still remains when the additional
2 of step 3 is applied. If it
interlaboratory variance uadd
remains the potential outliers remain excluded, otherwise apply the Mandel-Paule method without excluding
these results. If the interlaboratory variance that has
to be applied now is not considerably larger than that
obtained in step 3, the potential outliers will not be
excluded and the latter interlaboratory variance has to
be applied. Otherwise, the potential outliers have to be
excluded and the interlaboratory variance of step 3 has
to be applied.
Because of the poorer results of the PtSi-photodiode-based
detectors in comparison with those of the Si-photodiodebased detectors, only the Si-photodiode-based detectors were
used to calculate the KCRVs and the DoEs in the wavelength
range from 250 nm to 400 nm. In this wavelength range, the
results of the PtSi-photodiode-based detectors are calculated
for information only. In the wavelength range from 200 nm
to 240 nm the KCRVs and the DoEs are based on the PtSiphotodiode-based detectors.
5. Results and discussion
In the wavelength region from 200 nm to 240 nm where
PtSi-photodiode-based detectors were used, an outlier was
excluded at two wavelengths, the Mandel-Paule method
had to be applied at another two wavelengths and the
Mandel-Paule method was applied after excluding one outlier at a fifth wavelength. In the wavelength region from
250 nm to 400 nm where Si-photodiode-based detectors
were used, five outliers were excluded at three wavelengths
in total and the Mandel-Paule method had to be applied at
a fourth wavelength. The results of the PtSi-photodiodebased detectors in the wavelength range from 250 nm to
400 nm which were not used for the calculation of the
KCRV and DoEs showed a much higher number of wavelengths with inconsistency than the Si-photodiode-based
detectors. Outliers were excluded at three wavelengths, the
Mandel-Paule method was applied at two wavelengths and
the Mandel-Paule method together with excluding outliers
was applied at four wavelengths, leaving only seven wavelengths with consistent data.
The linear drift model applied for the Si-photodiode2
based detectors reduces the observed chi-squared value χobs
by about 20% compared to that obtained with the common
weighted mean without the linear drift model and thus
assuming stable artefacts. This shows that the drift of the
Si-photodiode-based detectors is significant and well handled by the linear drift model. The effect of the stepwise
drift model which was used for the PtSi-photodiode-based
detectors is less significant. The stepwise drift model was
nevertheless used because otherwise the uncertainty contribution assigned to the stability of the detectors would have
increased.
The DoEs are a very important outcome of a key comparison. The relative deviation Di = di / xi from the KCRV
is shown in figure 2. It is obvious that both the relative differences Di and the participants’ reported relative standard
uncertainties u rel,cal, i shown in figure 1 strongly increase with
decreasing wavelength below 250 nm. This is a result of
increased experimental difficulties in the short-wavelength
UV, e.g. the small available radiant power associated with
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L Werner
Metrologia 51 (2014) S336
4.5
LNE-Cnam
IO-CSIC
4.0
MIKES
3.5
MSL
VSL
|di / U(di)|
3.0
NIM
2.5
NIST
NMIA
2.0
NMIJ
1.5
NPL
NRC
1.0
PTB
0.5
NMC-A*STAR
VNIIOFI
0.0
200
250
300
350
400
Wavelength / nm
2.0
LNE-Cnam
1.8
IO-CSIC
1.6
MIKES
MSL
|di / U(di)|
1.4
VSL
NIM
1.2
NIST
1.0
NMIA
0.8
NMIJ
0.6
NPL
NRC
0.4
PTB
NMC-A*STAR
0.2
VNIIOFI
0.0
200
250
300
350
400
Wavelength / nm
Figure 3. Ratio di / U (di ) of the deviation from the KCRV and its expanded uncertainty (k = 2) displayed over its full range (top) and a
reduced range (bottom).
the small responsivity of the PtSi-photodiode-based detectors
which are used in this wavelength range.
The uncertainties claimed by a participant are considered
to be supported by the key comparison if the ratio di / U (di )
of the deviation from the KCRV and its expanded uncertainty
(k = 2) is smaller than 1. Figure 3 shows that 239 of the 277
values in total of this ratio are smaller than 1, whereas, 15
values are larger than 1.5 indicating the existence of underestimated uncertainties or overlooked biases in the measurement
results of the participants.
Figure 4 shows the relative standard deviation of the differences di of the participant’s results from the KCRV for
this key comparison. The strong increase of the standard
deviation with a decreasing wavelength below 250 nm is
once more an evidence of the increased experimental difficulties in the short-wavelength UV already mentioned
above. Figure 4 shows that the standard deviation of
the PtSi-photodiode-based detectors in the wavelength
range from 250 nm to 400 nm is larger than that of the
Si-photodiode-based detectors by a factor of up to 3 in this
key comparison. This and the large number of wavelengths
with outliers and inconsistencies requiring the application of
the Mandel-Paule method support the decision to use only
the Si-photodiode-based detectors for the calculation of the
KCRV and the DoEs and to exclude the PtSi-photodiodebased detectors in the spectral region from 250 nm to 400 nm.
Since the previous key comparison CCPR-K2.b of the spectral responsivity in the wavelength range from 300 nm to
1 000 nm was performed with Si-photodiode-based single
element detectors and trap detectors, both key comparisons
can be compared well in the overlapping spectral region.
Figure 4 shows that the standard deviation of CCPR-K2.c
is between 20% and 50% smaller than that of CCPR-K2.b
which indicates a considerable improvement of the calibration capabilities in this spectral region since the time of the
measurements for CCPR-K2.b in the years 2000 and 2001.
The issue of inconsistent data and outliers has been
addressed by the particular procedure described above.
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L Werner
Metrologia 51 (2014) S336
1.6%
K2.c (PtSi)
14%
1.4%
K2.c
(PtSi, not used)
12%
1.2%
K2.b (Si)
10%
1.0%
8%
0.8%
6%
0.6%
K2.c (Si)
4%
0.4%
2%
0%
Relative standard deviation
Relative standard deviation
16%
0.2%
200
250
300
350
400
0.0%
Wavelength / nm
Figure 4. Relative standard deviation of the differences of the participant’s results from the KCRV for the PtSi-photodiode-based detectors
in the spectral region from 200 nm to 240 nm (full squares) and the Si-photodiode-based detectors in the spectral region from 250 nm to
400 nm (full circles). For comparison with the preceding CCPR-K2.b, the same relative standard deviation for the Si-photodiode-based
detectors in the CCPR-K2.b (full triangles) is shown in the spectral region from 300 nm to 400 nm. In addition, the relative standard
deviation of the differences from the reference value based on the PtSi-photodiode-based detectors of the CCPR-K2.c in the spectral region
from 250 nm to 400 nm (blank squares) is shown. These latter results are not used for the calculation of the KCRV and DoEs and are given
for information only. (The data in the wavelength range from 250 nm to 400 nm are plotted for the right vertical axis.)
The approach is not based on a statistical model but rather
includes several steps and procedures. Nonetheless, all cases
of inconsistent data and potentially outlying results in this
key comparison were handled by this procedure as consistently as possible. The obtained results were also compared
to those of a Bayesian method based on a statistical model
[7]. This comparison indicates that the procedure applied in
this analysis excluded only the strongest outliers and that
in some cases additional results might have been identified
as being not in accordance with their quoted uncertainties.
However, the applied procedure had been set up with the
intention of avoiding any possibility of excluding a nonoutlying result.
6. Conclusions
Laboratories of 14 NMIs took part in the key comparison of
the spectral power responsivity of detectors in the spectral
range from 200 nm to 400 nm. The comparison was carried
out using three types of transfer detectors based on PtSi/n-Si
Schottky photodiodes and Si pn junction photodiodes showing
different temporal drift of the spectral responsivity. The calculation of the DoE and the KCRV were based on the method
of the weighted mean (with cut-off) which was extended for
non-stable artefacts drifting stepwise or linear with time. A
comparison with the preceding key comparison in the overlapping spectral range from 300 nm to 400 nm showed a considerable improvement of the calibration capabilities in this
spectral region.
The inconsistency of the data at some wavelengths required
the development and application of a particular procedure to
achieve consistency. Though all cases of inconsistency and
potentially outlying results could be handled as consistently
as possible, alternative approaches might have been applied as
well, leading to different conclusions in some cases. Hence,
we also conclude that there is an urgent need for a harmonized, statistically well-founded treatment of inconsistent data
in key comparisons.
Acknowledgment
The author would like to thank Clemens Elster and Alfred
Link for their helpful short lectures and discussions on statistics and the introduction to Bayesian model averaging, Alfred
Link for the data analysis with a Bayesian method based on a
statistical model which served as a check for the results on the
exclusion of outliers, Peter Meindl for encouraging and helpful discussions, and Katrin Vogel, Peter Meindl, Josef Breuer,
Marianne Fleischer-Bartsch, and Sylvia Ludwig for performing the large number of measurements PTB had to perform as
the pilot of this comparison.
The author would like to thank all participants for their
cooperation and their patience.
Appendix
The model for linear drift of the spectral responsivity of a
detector is x = a + bt with c = (a, b )T being the unknown
quantities of the linear trend in vector form, i.e., a is the value
of the spectral responsivity at time t = 0 and b is the rate of
its change.
The following N + 1 equations are available to estimate the
unknown quantities:
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xi = a + bti (i = 1, … , N ) )
L Werner
Metrologia 51 (2014) S336
xN + 1 = b
(A.1)
ti is the time at which the ith participant determined the spectral responsivity of its transfer detectors, and xN + 1 is the rate of
change of the spectral responsivity of the reference detector as
determined by the pilot.
These equations can be written compactly in matrix form
x = Ac
(A.2)
x = (x1, … , xN , xN + 1)T
(A.3)
and
⎛1
⎜⋮
A =⎜
⎜1
⎝0
t1 ⎞
⋮⎟
⎟
tN ⎟
1⎠
(A.4)
xi = a + Bk i = 1, … , N
xN + k = Bk k = 1, … , 4
x = (x1, … , xN , xN + 1, … , xN + 4 )T
(A.6)
c = (a, B1, B2, B3, B4 )T
(A.7)
and
1 0
1 0
0 1
0
⋮
0
1
0
0
0
0
⋮
0
0
1
0
0
0 0⎞
0 0⎟
⎟
0 0⎟
1 3 ⎟
4 4⎟
⋮ ⋮⎟
1 0⎟
0 0⎟
0 0 ⎟⎟
1 0⎟
0 1⎠
(A.10)
u rel,adj (xi ) =
2
2
u rel,cal,adj,
i + u rel,T, i
(A.11)
u rel,adj (xi ) differs from u rel (xi ) defined in equation (5) by the
replacement of u rel,cal, i by u rel,cal,adj, i taking into account the
cut-off in the uncertainty according to the policy of the CCPR
Working Group on Key Comparisons (WG-KC) to limit the
weight of the measurement results of an individual participant
[5]. The solution is
c = (AT WA)−1 AT Wx
(A.12)
with the covariance matrix
The index k in the first of the upper two equations denotes
the round of the measurements of participant i. The xN + k are
the shift of the spectral responsivity during the jth round of
measurements of participants with respect to its value at time
t = t0 as determined by the pilot. Again these equations can be
written compactly in matrix form (A.2) with
⎛1
⎜1
⎜
⎜1
⎜1
⎜
A = ⎜⋮
⎜1
⎜0
⎜0
⎜
⎜0
⎝0
−2
wi = (xiu rel,adj (xi ) )
with
(A.5)
(A.9)
The weighting W is a diagonal matrix with the diagonal
elements
The model for stepwise drift of the spectral responsivity of a
detector is y = a + Bk. a is the value of the spectral responsivity
at time t = t0 and Bk is the shift of the spectral responsivity
during the kth round of measurements of participants with
respect to its value at time t = t0. The N + 4 equations available to determine the unknown quantities a and Bk are
χ 2 = (x − Ac)T W (x − Ac)
with
in the third round. The fourth participant took part with one
detector in the third round and with three detectors in the
fourth round as it is in the case of NIM that had to repeat parts
of its measurements because of broken detectors.
The estimation of the unknown quantities was obtained
by applying weighted least square fit, which means, the estimation minimizes the sum of squared weighted differences
between data and model:
T
(c) = (AT WA)−1 AT WUx[(AT WA)−1 AT W]
cov
(A.13)
The covariance matrix Ux has the diagonal elements
u2 (xi ) = (xiu rel (xi ) )2 with u rel (xi ) defined in equation (5). The
off-diagonal elements are zero since the measurement results of
the participants are not correlated in this comparison. The estimates for the spectral responsivity of the reference detector
yi = y(ti ) at time ti (i = 1, … , N ) and for the drift are summarized
in y = (y1, … , yN , … , yN + k )T with K equal 1 or 4 in case of the
linear drift or the stepwise drift, respectively. The yi with i > N give
the estimates of the drift parameters b or Bk. The yi (i = 1, … , N )
form the time dependent KCRV. Vector y is given by
y = Αc
(A.14)
The corresponding covariance matrix is
(A.8)
Uy = Acov (c) AT .
(A.15)
The DoE di = xi − yi and U (di ) = 2u(di ) are taken from the
vector d = (d1, … , dN , … , dN + K )T and the matrix Ud with
the elements u2(di ) at the diagonal. It is
The example given for A indicates that the first two participants performed their measurements in the first round, the
third participant in the second round, and the last participant
d = x − y,
(A.16)
Ud = Ux + Uy − Ux, y − Uy, x
(A.17)
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L Werner
Metrologia 51 (2014) S336
and
Ux, y = Uy, x = A(AT WA)−1 AT WUx .
(A.18)
If a result xi is excluded from the calculation of the fit, then
the standard uncertainty u (di ) of the difference di = xi − yi is
calculated according to
u2 (di ) = u2 (xi ) + u2 (yi )
(A.19)
References
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low-power ultraviolet radiation on silicon photodiodes
Metrologia 32 515–8
[2] Durant N M and Fox N P 1995/1996 Evaluation of solid-state
detectors for ultraviolet radiometric applications Metrologia
32 505–8
[3] Werner L 1998 Ultraviolet stability of silicon photodiodes
Metrologia 35 407–11
[4] Kuschnerus P Rabus H Richter M Scholze F Werner L and
Ulm G 1998 Characterization of photodiodes as transfer
detector standards in the 120 nm to 600 nm spectral range
Metrologia 35 355–62
[5] Guidelines for CCPR Comparison Report Preparation 2013
CCPR-G2 Rev. 3 July 1 (www.bipm.org/utils/common/pdf/
CC/CCPR/CCPR-G2.pdf)
[6] Paule R C and Mandel J 1982 Consensus values and weighting
factors J. Res. NBS 87 377–385
[7] Elster C and Toman B 2010 Analysis of key comparisons:
estimating laboratories’ biases by a fixed effects model
using Bayesian model averaging Metrologia 47 113–9
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