NMI TR 2
Uncertainty Analysis for the Derivation of the
NML2003 Spectral Irradiance Scales
Jim Gardner
First edition — August 2005
Bradfield Road, Lindfield, NSW 2070
PO Box 264, Lindfield, NSW 2070
Telephone: (61 2) 8467 3600
Facsimile: (61 2) 8467 3610
Web page: http://www.measurement.gov.au
© Commonwealth of Australia 2005
CONTENTS
1
Introduction ............................................................................................................ 1
2
Propagation of Ratio Uncertainty Components ..................................................... 2
2.1 Measured Signal Ratio ................................................................................. 2
2.2 Ratio Calculated from the Spectral Components ......................................... 2
2.2.1
Offset Components ....................................................................... 4
2.2.2
Scaling Components ..................................................................... 4
2.2.3
Wavelength Components .............................................................. 4
3
Uncertainty Propagation for the Radiometer Measurements ................................. 4
3.1 McPherson Measurement System ................................................................ 4
3.2 Transfer between Lamp and Blackbody  ( ) ............................................ 5
3.3 Window Transmittance ................................................................................ 6
3.4 Emissivity .................................................................................................... 6
3.5 Radiometer Responsivity ............................................................................. 7
4
Results .................................................................................................................... 8
iii
PREFACE
This document propagates uncertainties through the spectral integrals used to
establish the temperature of the NMI blackbody and hence the NML scales of spectral
irradiance in 2003, rigorously treating the random and systematic components of the
measurement.
The experimental details involved in the derivation of the scales are described in
NMI TR 1. Basis of NML2003 Scale of Relative Spectral Irradiance. The uncertainty
analysis in that document has a more complete treatment of the overall uncertainty,
but a less rigorous treatment of the uncertainties in the spectral sums used to establish
the temperature of the blackbody. Hence the analysis is presented here; the final result
then adopts the remaining components from NMI TR 1 and compares the two
analyses.
iv
1
INTRODUCTION
Details of the experimental method and the filter radiometers are given in NMI TR 1.
Each filter radiometer signal is
S     ( ) ( ) ( ) L( , T ) R( )
(1)
where  ( ) is the ratio of lamp spectral irradiance to the blackbody spectral
irradiance (a measured quantity)
 ( ) is the blackbody window transmittance (a measured quantity)
 ( ) is the blackbody (BB) spectral emissivity (estimated)
L( , T ) is the calculated blackbody radiation at temperature T
R( ) is the spectral responsivity of the filter radiometer (a measured quantity)
Given the ratio of measured signals S1 and S 2 from two channels, we solve
R 
S1

S2
1  i i i Li (T ) R1,i
2   j j j L j (T ) R2, j

F1 (T )
F2 (T )
(2)
for the temperature of the blackbody. Subscripts i and j are used to indicate
wavelength. Note that the spectral interval may be different for the two channels,
although we have assumed that the interval in a given channel is fixed. Now we need
to estimate the uncertainty in that temperature.
We can use implicit differentiation to solve Eq. (2) for the sensitivity of the
temperature directly in terms of each measured or estimated spectral value
(responsivity, window transmittance, emissivity, spectral quotients). However, it is
much simpler to deal in terms of the spectral ratios. Once we have solved Eq. (2), we
numerically calculate the sensitivity coefficient for temperature on that spectral ratio
dT
 R (T  T )  R (T ) 
1
(3)
T
dR
Alternatively the sensitivity coefficient can be calculated from the two-colour
effective wavelength. This sensitivity is different for each chosen pair of filters.
Then for each independent component contributing to the temperature uncertainty
dT
u (T ) 
u (R )
(4)
dR
Total uncertainty is then found by adding these independent components in quadrature.
The two-colour effective wavelength  is given by
1
1
1


 e ,1 e,2
where e is the effective wavelength of the respective filter. The temperature
uncertainty for a given filter combination is then approximately
T 2
u (T ) 
urel (R )
c2
where c2 = 1.44 107 nm K is Plancks second radiation constant.
NMI TR 2
(5)
(6)
1
Calculated uncertainties taking the peak centre as the effective wavelength are shown in
Table 1 for a temperature of 3000 K and an uncertainty in the spectral ratio of 0.1%.
Table 1. Uncertainty in temperature for 0.1% relative uncertainty
in signal ratio at 3000 K (bold indicates the filter combinations used)
Radiometer
e (nm)
Combination
Λe
U(T)
#4
340
4,3
1391
0.87
#3
450
3,11
2475
1.55
#11
550
11,6
2567
1.60
#6N
700
3,6N
1260
0.79
#8
940
6N,8
2742
1.71
#1
1300
8,1
3394
2.12
#9
1540
1,9
8342
5.21
6N,1
1283
0.80
8,9
2413
1.51
2
PROPAGATION OF RATIO UNCERTAINTY COMPONENTS
2.1
Measured Signal Ratio
The uncertainty in the measured ratio is simply that of the signals after correction for
background and any non-linearity in the amplifier. This component is generally small.
2.2
Ratio Calculated from the Spectral Components
The uncertainty in the ratio R is first estimated for each of the constituent quantities
F1 and F2 ; these uncertainties are then combined into those of the ratio, taking into
account the correlation between the numerator and denominator. The correlations are
different for systematic and random components:


For components random between wavelengths
The filter combinations used show little or no overlap between wavelengths.
Hence the sum terms in the numerator and denominator of Eq. (2) have no or
negligible common components. It follows that for random effects (ie random
between wavelengths), the correlation between denominator and numerator is
zero, and
2
2
2
urel
(R )  urel
( F1 )  urel
( F2 )
(7)
For components systematic between wavelengths
Uncertainties in linear combinations of correlated components are fully
correlated. The correlation can be positive or negative – we need to consider
signed uncertainties (in practice only for systematic wavelength effects). From
Eq. (2) and the law of propagation of uncertainties, we have
2
2
 1 
 F 
1 F1
u (R )    u 2 ( F1 )    12  u 2 ( F1 )  2r
u ( F1 )u ( F2 )
F2 F22
 F2 
 F2 
where r = ±1 is the correlation coefficient between the fully-correlated
numerator and denominator.
2
NMI TR 2
(8)
2
Hence
urel (R )  urel ( F1 )  rurel ( F2 )
(9)
Now we consider the four different measured or estimated spectral terms in Eq. (1).
For each taken in turn as pi , we can write
F   pi qi Li (T )
(10)
where qi is the product of the other three terms.
In all cases we calculate relative uncertainties in the integrals, using only the central
significant range of the spectral responsivity data, for which the wavelength spacing is
constant for a given radiometer, and hence have dropped the  term. If the spectral
quantity for two radiometers are measured independently (in this case it means with
different wavelength offsets due to different alignments), they are uncorrelated, and
for each quantity in turn, we estimate the uncertainty in the spectral sums F1 and F2
for the two filter radiometers and hence in their ratio for each independent component
of pi . The sensitivity coefficient for each pi is
dF
 qi Li (T )
(11)
dpi
We then estimate us ( pi ) , where the s subscript indicates a signed uncertainty
(important for determining the correlation for systematic effects), for each
independent component of pi .
For each measured quantity, the independent uncertainty components can be generally
treated as three types, offsets, scaling and wavelength, each with systematic and
random divisions. For random components
2
u 2 ( F )    qi Li (T )  us2 ( pi )
(12)
and there is no correlation between the numerator and denominator of the ratio.
For systematic components
us ( F )   pi qi Li (T )us ( pi )
and the correlation coefficient between two filter signals is
r  sgn(us ( F1 ))sgn(us ( F1 ))
where sgn( ) is the sign function.
(13)
(14)
Note that for uncertainties in the measured lamp-BB ratios, we need the full form of
Eq. (1) at the solved BB temperature; for the other components, Eq. (1) reduces to
S   E ( ) ( ) ( ) R( )
(15)
where E ( ) is the spectral irradiance of the lamp, and Eq. (11) becomes
dF
 qi
(16)
dpi
where qi contains Ei in place of i Li (T ) . Lamp FEL4 is taken as representative for
the uncertainty calculations here.
NMI TR 2
3
2.2.1
Offset Components
These are typically constant and able to be expressed as a constant fraction of the
maximum signal, although corrected signals may have a variable uncertainty
estimated at each wavelength. For these, us ( pi ) is known directly.
2.2.2
Scaling Components
Systematic scaling components are not important here if constant, as they affect only
the relative value of the signals and hence cancel in the ratio. Of course, any scaling
of one filter relative to another, eg due to drift in one and not the other, directly affects
the ratio and hence the estimated temperature. Random noise in the relative value will
contribute to the overall uncertainty.
2.2.3
Wavelength Components
Here
dpi
u ( )
(17)
d
dependent on the slope of the measured quantity. This is most important for the
radiometer responsivity functions which vary much more rapidly with wavelength
than the other terms. Because all radiometers were calibrated at the one time, any
wavelength offsets are fully correlated between the radiometers in a pairing. Note also
that the radiometer responsivity is measured as a transfer against a reference device,
and the slope of the reference responsivity should be accounted for, as shown below,
although the dominant effect will be due to the slope of the filter transmittance.
us ( pi ) 
3
UNCERTAINTY PROPAGATION FOR THE RADIOMETER
MEASUREMENTS
Now we are in a position to estimate actual uncertainties for the radiometer pairs. In
all these estimates, the temperature of the blackbody is taken as an average of the
estimated value, 2920 K for measurements below 700 nm, 2830 K above.
In each case we avoid correlations introduced through interpolation by interpolating
the components other than the one under study to the wavelengths at which the
studied component was measured.
3.1
McPherson Measurement System
Spectral comparisons with the McPherson monochromator system are made in blocks
over a limited wavelength range to reduce effects of drift during the measurement
time. Reference and zero measurements are made at the start, middle and end of each
block, for both the reference artefact and the test artefact at a reference wavelength
where signals for both are strong (generally near the centre of the response band for
measurements with the filter radiometers). Each measurement is the mean of up to 50
separate readings. Reported results include the standard deviation of the mean value.
Hence we can readily estimate uncertainty components required for propagation
through the spectral sums.

The relative random uncertainty in each measurement sequence is taken as a
typical value for the standard deviation of the mean at the reference wavelength
across the measurement blocks, divided by the signal at the reference
wavelength.
NMI TR 2
4

The random uncertainty of the offset in the measurement is taken as a typical
standard deviation of the mean for the measurement zeroes across the
measurement blocks.

The systematic offset is estimated from a typical range of zero measurements
within the measurement blocks. The uncertainty is taken as the half-range
divided by √3.
For each of these components, the reference and test items are treated separately.
Small numbers of repeat runs were averaged – no account is taken of the averaging
here.
3.2
Transfer between Lamp and Blackbody  ( )
Three different lamp types were used, with different spectral characteristics in the UV
region. Lamp FEL4 is taken as representative. The sensitivity coefficient at each
wavelength is
dF
  i i Li (T ) Ri
(18)
dpi
We take the measured signals at each wavelength as the irradiance values themselves
(this ignores monochromator gain differences)
E
i  i
(19)
Li (T )
Random noise in the measurements is uncorrelated between measurements and
wavelengths; for the component proportional to the signal (eg current noise, gain
noise)
urel ( i )  urel 2 ( Ei ) 2  urel 2 ( Li (T ))
2
 E   L (T ) 
 k  i   i

 Ei   Li (T ) 
2
(20)
 2k
where k is the relative uncertainty in the signals (assumed the same for both sources),
and
(21)
u( i )  iurel ( i )  2k i
Offsets are treated as a fractions k of the signal at some reference wavelength.
For the lamp signal, from Eq. (19)
u ( Ei ) k1 ER

Li (T ) Li (T )
For the blackbody signal, from Eq. (19)
E u ( L (T ))
k L (T ) i
u L ( i )   i i 2   2 R
Li (T )
 Li (T ) 
u E ( i ) 
(22)
(23)
Random offsets are due to electronic background noise. We separately propagate
random offsets in the lamp and blackbody signals, then combine them in quadrature.
Systematic offsets are correlated across wavelengths for the blackbody and for the
lamp.
NMI TR 2
5
Systematic offsets due to scatter are correlated for those radiometers measured in the
one setup, ie filters in the visible range and in the infrared range. Hence systematic
effects are summed linearly for the lamp and for the blackbody offsets. The total
effect due to offsets is then found by adding the lamp and blackbody results for a
given filter ratio in quadrature.
For wavelength uncertainty in the lamp signal
u ( Ei )
k dEi
u E ( i ) 

Li (T ) Li (T ) d 
and in the blackbody signal
u L ( i )  
Ei u ( Li (T ))
 Li (T ) 
2

k i dLi (T )
Li (T ) d 
(24)
(25)
where k is the uncertainty (in nm where the derivative has units of responsivity per
nm). Random wavelength settings are correlated between lamp and blackbody.
Systematic wavelength uncertainties are fully correlated across wavelengths for the
blackbody and the lamp signals.
3.3
Window Transmittance
The sensitivity coefficient at each wavelength is
dF
 i i Li (T ) Ri  Ei i Ri
(26)
dpi
For random relative measurement noise
u( i )  k i
(27)
For offsets
u( i )  k R
(28)
where the R subscript denotes the reference wavelength (not so important here as the
window transmittance is approximately constant).
The window transmittance was measured on the Cary spectrophotometer. Estimates of
the peak-to peak variation between adjacent points were used to derive the random
uncertainty of the measurement. Data were taken at 10 nm intervals, then interpolated
to 2 nm intervals for the calculation of the integrals. Hence the random uncertainty
estimate was multiplied by √5 to compensate for the effect of averaging random
effects over five times too many points. Uncertainty in a systematic offset in the
window transmittance was estimated from steps in the data over a wider range of
wavelengths.
3.4
Emissivity
Emissivity is treated separately, simply as a change in relative emission of the
blackbody at the centre of the radiometer response. This component is covered as an
average over the radiometer response range in the original analysis (refer to
NMI TR 1) and is not recalculated here.
NMI TR 2
6
3.5
Radiometer Responsivity
The sensitivity coefficient at each wavelength is
dF
 i i i Li (T )  Ei i i
(29)
dpi
The responsivity is a transfer from that of a reference detector, spectral responsivity
RR ,i ; as for the lamp-blackbody transfer, we take the signals to be proportional to the
responsivities themselves, and hence
Ri  ti RR ,i
(30)
with
R
(31)
ti  i
RR ,i
Uncertainties in spectral responsivity arise through the transfer process and from those
of the reference spectral responsivity.
For the reference detector (H8W taken in the visible range, TD1 in the IR range)
u ( Ri )  ti u ( RR ,i )
(32)
The uncertainties are correlated for near-neighbours, ie for wavelengths within the
pass-band of the filter radiometers, and we can estimate a constant correlation
coefficient and hence contributions from systematic and random components for a
given filter.
For random relative uncertainties in the transfer
u ( Ri )  RR ,i u (ti )  RR ,i kti  kRi
(33)
where k is the relative uncertainty in the transfer.
For offsets in the signal channel
u( Ri )  RR,iu(ti )  RR,i
u( Ri )
 kRr
RR,i
(34)
where kRr is the scaled uncertainty in terms of the responsivity at the reference
wavelength. For offsets in the reference channel
R
(35)
u( Ri )  RR,i u(ti )   RR ,i i 2 u( RR ,i )  kti RR ,r
RR,i
Offsets are correlated within the signal channel and within the reference channel, but
not between the channels.
Wavelength uncertainties in the signal and reference channels are fully correlated.
Hence
dt
u ( Ri )  i RR ,i u ( )
d
 1 dRi
R dR 

 i 2 R ,i  RR ,i u ( )
(36)
 R d R
d  
R ,i
 R ,i
dR 
 dR
  i  ti R , i  u (  )
d 
 d
NMI TR 2
7
4
RESULTS
The spreadsheet ‘BB uc analysis’ (found in … haea\Sp irradiance project\JG)
propagates uncertainties for all the above components. Lamp FEL4 is taken as
representative for calculating the blackbody to lamp transfer. Measurements in the
visible range assume a blackbody temperature of 2920 K, those in the infrared region
assume 2830 K. Two-colour effective wavelengths were calculated for these
temperatures.
Random wavelength uncertainties are assumed correlated between lamp and
blackbody in the BB-lamp transfer at each wavelength, similarly for the reference and
radiometer detectors in the responsivity determination. Wavelength offsets are
assumed correlated for all the visible measurements, and all the IR measurements, but
not for the visible-ir transfer using radiometers R6N and R1.
Systematic offsets are assumed uncorrelated between artefacts, but correlated for the
one artefact at all wavelengths.
Table 2 shows uncertainty values estimated for the measurement data assumed for the
various uncertainty components. Offset components for the spectral measurements are
relative to the signal strength at the centre of the response band. Table 3 shows the
equivalent temperature uncertainties. The spectral change in the sensitivity factors for
the various components give rise to quite different uncertainties for the different filters.
Table 2. Uncertainty components for each of the filter radiometers
(systematic offsets are quoted relative to maximum signal levels)
Uncertainty components
Responsivity
Reference, relative, random
Transfer, relative, random
Transfer, radiometer, offset, random
Transfer, radiometer, offset, systematic
Transfer, reference, offset, random
Transfer, reference, offset, systematic
Wavelength, random
Wavelength, systematic
Window
Relative, random
Offset, random
Offset, systematic
Wavelength, random
Wavelength, systematic
BB transfer
Offset, lamp, random
Offset, lamp, systematic
Offset, BB, random
Offset, BB, systematic
Relative, random
Wavelength, random
Wavelength, systematic
NMI TR 2
R3,vis
R6N,vis
R8,vis
R8,ir
R1,ir
R9,ir
1.00E-03
4.00E-05
1.00E-05
8.66E-05
1.00E-06
4.33E-06
0.03
0.1
1.00E-03
5.00E-05
2.70E-06
2.89E-05
7.00E-07
2.89E-07
0.03
0.1
1.00E-03
9.00E-05
1.30E-05
5.77E-05
1.00E-05
2.89E-07
0.03
0.1
1.00E-03
9.00E-05
1.30E-05
5.77E-05
1.00E-05
2.89E-07
0.03
0.2
1.00E-03
3.90E-05
4.50E-06
2.89E-04
3.30E-06
3.18E-05
0.03
0.2
1.00E-03
3.40E-05
2.80E-05
1.21E-03
2.70E-06
4.33E-06
0.03
0.2
3.47E-04 3.47E-04 3.47E-04 3.47E-04 3.47E-04 3.47E-04
0.001
0.001
0.001
0.001
0.001
0.001
1.34E-04 1.34E-04 1.34E-04 1.34E-04 1.34E-04 1.34E-04
0.03
0.03
0.03
0.03
0.03
0.03
0.1
0.1
0.1
0.2
0.2
0.2
2.00E-06
2.89E-06
2.00E-06
2.89E-06
2.30E-04
0.03
0.1
2.00E-06
2.89E-06
2.00E-06
2.89E-06
2.30E-04
0.03
0.1
1.30E-06
3.58E-04
1.80E-05
5.20E-04
6.00E-05
0.03
0.1
1.30E-06
3.58E-04
1.80E-05
5.20E-04
6.00E-05
0.03
0.2
1.30E-06
3.58E-04
1.80E-05
5.20E-04
6.00E-05
0.03
0.2
1.30E-06
3.58E-04
1.80E-05
5.20E-04
6.00E-05
0.03
0.2
8
Note that the values in Table 3 do not include an allowance for emissivity variations,
the integrated radiometer signal measurement uncertainties, drift components or shifts
in window transmittance due to contamination. These and similar effects directly
affect the filter ratio and are propagated from the sensitivity coefficient calculated
from the two-colour effective wavelength.
Table 3. Temperature uncertainties (K) for the component values shown in Table 2
Temperature uncertainties
R3 R6N
R6N R8
R6N R1
R8 R9
Reference, relative, random
0.208
0.410
0.266
0.183
Transfer, relative, random
0.009
0.029
0.012
0.013
Transfer, radiometer, offset, random
0.285
0.002
0.005
0.002
Transfer, radiometer, offset, systematic
0.226
0.119
0.220
1.164
Transfer, reference, offset, random
0.215
0.000
0.003
0.001
Transfer, reference, offset, systematic
0.109
0.003
0.001
0.029
Wavelength, random
0.312
0.506
0.489
0.238
Wavelength, systematic
0.771
0.487
0.313
0.195
Relative, random
0.129
0.238
0.092
0.141
Offset, random
0.488
0.971
0.464
0.405
Offset, systematic
0.142
0.309
0.171
0.151
Wavelength, random
0.000
0.000
0.000
0.000
Wavelength, systematic
0.000
0.003
0.002
0.000
Offset, lamp, random
0.001
0.001
0.001
0.001
Offset, lamp, systematic
0.007
0.350
0.399
0.503
Offset, BB, random
0.002
0.004
0.009
0.009
Offset, BB, systematic
0.005
0.300
0.173
0.234
Relative, random
1.457
3.293
1.822
0.502
Wavelength, random
0.111
0.028
0.016
0.034
Wavelength, systematic
0.893
0.166
0.275
0.072
Totals
1.989
3.591
2.367
1.020
Responsivity
Window
BB transfer
NMI TR 2
9
Table 4. Component uncertainties from NMI TR 1 and total uncertainties in
temperature (K) for the various radiometer combinations
Other component uncertaintiess
R3 R6N
R6N R8
R6N R1
R8 R9
Emissivity of blackbody
0.095
0.196
0.128
0.080
Integrated wing response
1.086
0.586
2.496
0.725
Sphere geometric factors
1.267
2.757
2.332
1.351
Temporal drift
0.484
1.054
1.158
0.516
Non-linearity of detector
0.528
1.149
0.576
0.890
Amplifier gain ratio
0.211
0.460
0.254
0.225
Totals, including Table 3 components
2.7
4.8
4.7
2.1
Values from NMI TR 1
1.9
3.4
3.8
3.7
Table 4 shows additional components propagated from the uncertainty values given in
NMI TR 1. The final combined uncertainty for each radiometer pair is also shown in
Table 4, along with the values estimated by other means in NMI TR 1. The final
temperature uncertainties estimated by this more complicated analysis are slightly
higher than those in NMI TR 1, except for the infrared radiometer combination. The
general conclusion is that the two analyses agree fairly well. Propagation from the
temperature uncertainties to those in the NMI spectral irradiance scales are discussed
in detail in NMI TR 1.
NMI TR 2
10