NMI TR 9
Uncertainties in Photometric Integrals
James L. Gardner
First edition — November 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
This document was originally prepared for the Korean Institute of
Standards and Science in 2003 while James Gardner was there as a
guest researcher. It had minor revisions up to November 2003.
It was subsequently modified in December 2003 and circulated as
Revision 2.2. This document is Revision 3.1. It contains significant
changes to the method for f1.
CONTENTS
1
Photometric Integrals ............................................................................................................. 1
1.1 Photometric Response ................................................................................................... 1
1.2 Spectral Mismatch Factor.............................................................................................. 1
1.3 Quality Factor of a Photometer ..................................................................................... 2
2
Propagation of Uncertainty .................................................................................................... 2
3
Uncertainty Propagation by Component of the Spectral Measurement ................................. 3
4
Spectral Measurement as a Transfer ...................................................................................... 5
5
Uncertainty Components in Spectral Measurement ............................................................... 5
5.1 Base Uncertainty of the Reference Standard ................................................................. 5
5.2 Offset Components of the Transfer Ratio ..................................................................... 6
5.3 Scaling Components of the Transfer Ratio ................................................................... 7
5.4 Wavelength Components of the Transfer...................................................................... 8
5.5 Multiple Transfers from the Reference ......................................................................... 9
6
Uncertainty Estimates for the Photometric Integrals ............................................................. 9
6.1 Photometric Response for Illuminant A ...................................................................... 10
6.2 Spectral Mismatch Factor............................................................................................ 11
6.3 Quality Factor of a Photometer. .................................................................................. 12
7
Conclusion ............................................................................................................................ 14
8
References ............................................................................................................................ 14
iii
SUMMARY
Integration of spectral measurements of photometer response and source power distributions
is used to calculate illuminance response, spectral mismatch factors and photometer quality
factors. Methods are given to estimate uncertainties in these various quantities. The spectral
measurements are treated as a transfer from a reference detector or source. Uncertainties are
then calculated for effects arising in either the base reference or in the transfer measurement.
Uncertainties of the spectral distributions are propagated through the integration process.
Various systematic and random effects contributing to the uncertainty are treated separately
and then combined into the final uncertainty.
Photometric integrals are simple convolutions of two spectral distributions. Many systematic
effects can be described by a single parameter. Then for fully-correlated effects, the
uncertainty of each integral value is a linear sum of the signed spectral uncertainties, weighted
by the appropriate convolving function. Further, pairs of similar integral values are also fully
correlated and their covariance is simply obtained. For effects random between wavelengths,
the variance of a spectral integral is a linear sum of the spectral variances, weighted by the
convolving function. The covariance of integral pairs is the product of the linear sums used to
calculate the uncertainty of systematic effects. The uncertainty of combinations of the spectral
integrals is simply propagated from their variances and covariances.
Representative examples are given for the luminous response of a reference photometer, the
spectral-mismatch factor and the photometer quality factor f1'.
iv
1
PHOTOMETRIC INTEGRALS
While photometric quantities [1] are defined for different geometries, photometers are
most commonly used to measure illuminance, and discussion here is in terms of
illuminance response, calculated from spectral distributions. The principles used are
directly applicable to other luminous quantities provided the response function is
measured for the appropriate geometry. The integrals we deal with are photometric
response, spectral mismatch factor, and quality factor of a photometer.
1.1
Photometric Response
Suppose we have a photometer whose responsivity at wavelength  is R(). The
relative spectral response is a close match to V(, the relative spectral luminous
efficacy function shown in Figure 1. The response of the photometer to a source
whose spectral irradiance is E() is:
RV  k  R( ) E ( )d 
(1)
where k is a normalising factor. Most often the response is measured as a power
response, not irradiance, and the factor k includes the aperture area. In many instances
the response is measured only in relative terms, and k then includes various
normalising constants. In discussing uncertainties here, we are concerned only with
that of the integral sum itself and not the scaling factors. Components common to all
wavelengths affect the absolute value of the integral sum and its uncertainty. Some
common factors may be included in the scaling factor k and care must be taken not to
include them twice in a complete response uncertainty analysis.
The integral in Eq. (1) is important when determining the photometric response of a
primary reference standard. This is a ratio of integrated response to the illuminance
calculated as:
EV  km  V ( ) E ( )d 
(2)
where V( is normalised to 1 at 555 nm and km = 683 lm·W-1. If the photometer
spectral response does not exactly match that of V(, the photometric response will
depend on the source distribution. Hence most calibrations are given for response to
the defined source CIE Illuminant A. This source has a relative power distribution of
a black-body at a temperature of 2856 K, shown in Figure 1.
Relative strength
1.2
1.0
0.8
0.6
Response
Reference
Transfer ratio
Illuminant A
0.4
0.2
0.0
400
500
600
700
800
Wavelength /nm
Figure1. Transfer calibration of a photometer against a silicon photodiode detector.
The response function of the photometer is the V() spectral luminous efficacy
function. Also shown is the spectral source distribution CIE Illuminant A
NMI TR 9
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1.2
Spectral Mismatch Factor
Most photometers are calibrated by comparison to a primary reference using a source
whose spectral energy distribution approximates that of CIE Illuminant A. They may
then be used to measure the illuminance of sources with different spectral power
distributions. If the photometer spectral response is not an exact match to V(, the
illuminance measured by a photometer calibrated for illuminance response to CIE
Illuminant A must be corrected by multiplying by the spectral mismatch factor. This
is given by:
E A ( )R( )d   E ( )V ( )d i

A
(3)
F ( E, E ) 
A
E
(

)
V
(

)
d

E
(

)
R
(

)
d



The integrals in the numerator and denominator depending on the common values of
measured spectral response R() are correlated, as are those depending on the
common source spectral distribution measurements E(). These correlations must be
considered when estimating the uncertainty of the spectral mismatch factor. This
factor is also known as the colour-correction factor. While we have retained the
symbol E for the source, denoting irradiance, it is only the relative spectral irradiance
or spectral power distribution that is important in calculating correction factors.
1.3
Quality Factor of a Photometer
This is calculated from the relative spectral response of the photometer and used to
indicate how well the photometer matches the V( luminous efficacy distribution. It
is defined as:
R * ( )  V ( ) d 

'
f1 
(4)
V
(

)
d


R * ( ) 
where
2
E
E
A
( )V ( )d 
A
( ) R ( ) d 
R ( )
(5)
PROPAGATION OF UNCERTAINTY
We wish to estimate the collective effect that the uncertainty of a particular measured
spectral value has on the uncertainty of each of the calculated quantities. Uncertainty
propagation is described in detail in the ISO Guide to the Expression of Uncertainty
in Measurement [2]. The uncertainty in a quantity X formed by combining measured
quantities xi through the relationship X  f ( x1 , x2 ,..xN ) is given by:
2
N 1 N
 f  2
f f
u (X )  
u ( xi , x j )
(6)
 u ( xi )  2 
i 1  xi 
i 1 j i 1 xi x j
where u ( xi ) is the uncertainty in xi and u ( xi , x j ) is the covariance of xi and x j . For
uncorrelated input quantities, the covariance of pairs of variables is zero and Eq. (6)
reduces to:
N
2
2
 f  2
u (X )  
 u ( xi )
i 1  xi 
the ‘sum of squares’ commonly applied. The derivatives f xi are sensitivity
coefficients for the dependence of X on the various measured quantities.
N
2
NMI TR 9
(7)
2
If we form another quantity Y by combining the measured quantities xi through the
relationship Y  g ( x1 , x2 ,..xN ) , the uncertainty in Y is given by an expression similar
to that of Eq. (6), but now the quantities X and Y are correlated through dependence
on the common set xi. The covariance between X and Y is given by:
N N
f g
u ( X , Y )  
u ( xi , x j )
(8)
i 1 j 1 xi x j
Correlation coefficients are normalised covariance values, defined as:
u( X , Y )
(9)
r( X ,Y ) 
u ( X )u (Y )
We note that if r ( xi , x j )  1 for all pairs, Eq. (6) reduces to:
f
(10)
u( xi )
i 1 xi
independently of the sign of the sensitivity coefficients. However, while uncertainty
components of a spectral measurement, systematic across wavelengths, are fully
correlated, the correlations may have mixed signs for different wavelength pairs; this
is dealt with in the next section.
N
u( X )  
Just as relative uncertainty is given as a ratio of the uncertainty of a quantity to its
value, relative covariance is given by:
u( X , Y )
urel ( X , Y ) 
(11)
XY
3
UNCERTAINTY PROPAGATION BY COMPONENT OF THE SPECTRAL
MEASUREMENT
Individual sources of uncertainty are generally independent. Table 1 shows the two
ways that uncertainties in spectral values due to these individual effects can be
combined when the spectral values themselves are combined. Because the individual
effects are not correlated, we can form the uncertainties of the combined effects at a
given wavelength by sum-of-squares, Eq. (7). However, these spectral values are now
most often partially correlated, and we must use the full expression of Eq. (6) to
calculate the final combined uncertainty, u c .
Table 1. Uncertainties, ui , j , in spectral values, Si , due to different effects, j, combined
to estimate the uncertainty, u c , in the value of a colour quantity. The total uncertainty
of Si at wavelength i is u ic and the uncertainty in the colour value due to effect j is u c | j
Wavelength
Effect 1
Effect 2
Effect m
Combined
1
u1,1
u1,2
u1,m
u1c
2
u2,1
u2,2
u2,m
u 2c
n
un ,1
un ,2
un ,m
u nc
Combined
u c |1
u c |2
u c |m
uc
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The final combined uncertainty u c is also found by combining the u c | j values by
sum-of-squares. The advantage of this form of combination is that many of the
systematic effects that can contribute to uncertainty across wavelengths are either
uncorrelated (random) -which means that the uncertainty for the combination across
wavelengths is found by sum-of-squares- or fully correlated, for which simple linear
sums can be used. It also allows estimation of the contribution of the individual
effects to the final uncertainty.
For random components, we have that the variance of a combination X of the spectral
values ( xi  Si ) is given by Eq. (7). The covariance of the spectral values at different
wavelengths is zero, and Eq. (8) for the covariance between two spectral
combinations X and Y reduces to:
N
X Y 2
(12)
u( X , Y )  
u ( Si )
i 1 Si Si
Even though the spectral values are uncorrelated, the common dependence on the one
set of spectral values correlates the integrals.
Uncertainty components of the spectral measurement systematic across wavelengths
are fully correlated between wavelengths, but we have to consider the sign of the
correlation. Most uncertainty components (whether random or correlated between
wavelengths) can each be described by a single parameter p as:
(13)
Si  Si ( p)
Components may have a wavelength dependence, but in the case of systematic
effects, uncertainties at different wavelengths scale with the parameter p. Noting that
uncertainties are always positive, the uncertainty of Si due to this effect alone is:
u ( Si )| p 
Si
u ( p)
p
S S
 sgn( i ) i u ( p)
p p
where sgn( ) is the sign function and Si/p is the sensitivity coefficient of the
spectral value for the effect.
(14)
By substituting these forms into Eq. (6) we have:
N
X Si
(15)
u ( X )| p  
u ( p)
i 1 Si p
N
X
or
(16)
us ( X )| p  
us ( Si )
i 1 Si
where the s subscript indicates that the uncertainty carries the sign of the effect.
Similarly, by substituting into Eq. (8), the covariance between two combinations X
and Y is given by:
N
N
X
Y
(17)
u( X , Y )| p  
u s ( Si )
u s ( Sj )
i 1 Si
j 1 Si
Note that combinations of fully correlated components of the spectral values are
themselves fully correlated. The covariance values are a product of terms already
calculated for the (signed) uncertainties of the individual combinations.
NMI TR 9
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The variances and covariances of the integrals for all effects are found as sums of
those of the independent individual effects.
4
SPECTRAL MEASUREMENT AS A TRANSFER
Spectral responsivity and spectral power distribution are measured by comparison to
reference standards. Hence the spectral value Si can be written as a transfer from that
of a reference standard, S iRef :
Si  ti SiRef
(18)
Uncertainties in the spectral value Si arise both from those of the reference value and
those introduced by the spectral transfer. Photometric integrals are formed by
combining the spectral measurements at different wavelengths. Systematic effects in
a spectral measurement have a cooperative relationship between wavelengths, that is,
the values at different wavelengths are correlated. Correlations also exist between the
reference values at different wavelengths, arising from systematic effects in the
methods used to derive them.
For uncertainty components of the reference spectrum, from Eq. (18) we have:
u ( Si )  ti u ( SiRef )
(19)
whereas for components in the transfer we have:
u ( Si )  SiRef u (ti )
(20)
In both cases we assume that the system efficiency is constant and calculate the
transfer value as the ratio, ti  Si SiRef .
5
UNCERTAINTY COMPONENTS IN SPECTRAL MEASUREMENT
For each independent component we calculate the transfer ratio at each wavelength
and hence the spectral uncertainties using Eq. 20 or 19. We then propagate those
uncertainties to the integral values, and from those to the desired integral
combination. Discussions here are given in terms of a general spectral quantity Si at
the ith wavelength. Uncertainty contributions arise from the uncertainty of the
reference values, from offsets in the transfer, scaling effects in the transfer and
wavelength effects in the transfer. Each of these effects can have random and
systematic causes. Corrections are applied for measurable effects, but these
corrections carry uncertainty. Other effects may be considered too small for
correction, but there is still an uncertainty in their estimated value.
5.1
Base Uncertainty of the Reference Standard
The spectral reference standard is likely to be measured at a limited number of
wavelengths, with some systematic errors in the process. The reference values then
will be at least partially correlated. In this case the full expressions of Eq. (6) must be
used to calculate the integral uncertainties and correlations. Often the contribution of
the reference standard can be calculated by taking the correlation coefficient as
constant, at its average through the visible range [3]. Then this calculation can be split
into fully-correlated and uncorrelated parts and the simpler expressions of Eqs (7) and
(10) used to calculate the variance. At the highest levels of accuracy, systematic
effects dominate the reference spectra uncertainties and they are highly correlated.
NMI TR 9
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The reference spectrum and the measured spectrum generally are defined at different
wavelengths. To avoid introducing correlations through interpolation procedures [4],
we use different procedures for reference and transfer uncertainty components. When
dealing with reference uncertainty components, we interpolate the measured spectrum
(which has no uncertainty if we are only considering a single component of the
reference spectrum) to the wavelengths of the reference spectrum. When dealing with
transfer uncertainty components, we interpolate the reference spectrum (which has no
uncertainty if we are only considering a single component of the transfer) to the
wavelengths of the measured spectrum.
An alternative to dealing with different wavelength intervals in reference and
measured spectra when the correlation coefficient for the reference uncertainties is
taken as constant is to interpolate the reference spectrum and its uncertainties to the
measurement wavelengths. Propagation of the systematic part of the reference
uncertainties will be correct, but propagation of the random part will underestimate
the reference contribution by the square root of the ratio of the data wavelength
intervals; a correction factor is simply applied for this effect.
5.2
Offset Components of the Transfer Ratio
Offsets in signals are additive effects. Hence:
S p
ti  Refi
(21)
Si  pR
with p, pR  0 but u ( p), u ( pR )  0 . It follows that for offsets in the signal channel:
u s ( Si ) p  u ( p )
and for offsets in the reference channel:
u s ( Si )
pR
(22)
 ti u ( p R )
(23)
Offsets random across wavelengths arise from background noise in the sample and
reference signal channels, which can be related to electronic noise in the
measurements.
5.2.1 Combining Random Offset Components across Wavelengths
Random offsets in the sample and reference channels can be treated separately, or
combined in quadrature. The contribution of the two effects to the variance of a
combination X is:
2
 X  2
2 2
u ( X ) p , pR   
 u ( p )  ti u ( p R )

S
i 1 
i 
and the contribution to the covariance between combinations X and Y is:
N
X Y 2
u( X , Y ) p, pR  
u ( p)  ti2u 2 ( pR )
i 1 Si Si
N
2




(24)
(25)
5.2.2 Combining Systematic Offset Components across Wavelengths
Scattered light in either reference or sample beam can cause an offset systematic
across wavelengths, as can electronic offset in an amplifier. Such offsets have
positive correlation between wavelengths, in a given signal channel. Offset effects in
the reference and sample beams are not equivalent. For systematic offsets in the
signal channel:
NMI TR 9
6
X
(26)
u ( p)
i 1 Si
and for systematic offsets in the reference channel:
N
X
(27)
us ( X )| pR  
ti u( pR )
i 1 Si
For each channel, the covariance between two combinations X and Y is given by the
product of their signed uncertainties.
N
us ( X )| p  
Most systematic offset effects are uncorrelated between the sample and reference
channels, and the variance of the two terms are separately added. However, an offset
that is common to both channels is correlated. An example is incorrect background
subtraction of room light reaching a detector from outside the signal paths. In such a
case the (signed) uncertainties of Eqs (26) and (27) are fully correlated with a
correlation coefficient of +1; then the two terms are added, and the result squared to
obtain the variance in a combination due to this effect.
5.3
Scaling Components of the Transfer Ratio
Scaling may arise from a number of instrumental effects. Whether the effect is in the
reference or measured spectrum, we have:
Si  pSi'
(28)
where Si' is the uncorrected value. We have p  1 (if it is not a correction must be
applied) and we estimate its uncertainty u(p), usually in relative terms. Hence:
us (Si )  Siurel ( p)
(29)
5.3.1 Combining Random Scaling Components across Wavelengths
Random noise in the magnitude of the system transfer function is a scaling term. The
noise arises from source fluctuations if comparing detectors, response fluctuations if
comparing sources and fluctuations in the transmission of the optical system in both
cases. For random scaling components, the variance of a combination X is given by:
2
 X  2 2
u (X ) p  
 Si urel ( p )
i 1  Si 
and the covariance between two combinations X and Y is given by:
N
X Y 2 2
u( X , Y ) p  
Si urel ( p)
i 1 Si Si
N
2
(30)
(31)
5.3.2 Combining Systematic Scaling Components across Wavelengths
Clipping of beams reaching reference and sample detectors through different optical
paths, or variation in the distance setting of the reference and sample sources are
examples systematic across wavelengths. Non-linearity is a form of scaling
systematic over wavelengths, where the scaling factor depends on the spectral value.
Corrections must be applied for the linear effects when the absolute value of the
transferred value is important. Scaling factors affect the calibration of photometric
response, which is an absolute value, but not the uncertainty of ratios such as the
spectral mismatch factor. Correlations for systematic scaling factors are generally
positive, although it is possible to conceive of non-linearity corrections that may vary
NMI TR 9
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in sign for different wavelength pairs. For scaling effects systematic across
wavelengths, the (signed) uncertainty in a combination X is given by:
N
X
(32)
us ( X )| p  
Si urel ( p)
i 1 Si
and the covariance between two combinations X and Y is given by the product of their
signed uncertainties.
5.4
Wavelength Components of the Transfer
The uncertainty in spectral value arising from an uncertainty in the wavelength setting
u(p) in the sample channel is:
S
u s ( Si )  i u ( p )
(33)

and in the reference channel is:
S Ref
us ( Si )  ti i u ( p)
(34)

The derivatives are calculated numerically.
5.4.1 Combining Random Wavelength Offsets across Wavelengths
Random errors in the wavelength setting arise from errors in the wavelength selection
mechanism, from calibration procedures used to determine line centres, or
imprecision in the knowledge of the calibration wavelengths themselves. The
treatment of random wavelength effects depends on the method used to record
spectra.
In step-and-compare mode, the wavelength uncertainties at the one wavelength are
correlated between channels; hence:
 S
S Ref 
(35)
us ( Si )   i  ti i  u ( p)
 
 
If the reference and sample spectra are recorded separately over a range of
wavelengths, the uncertainties at the one wavelength are uncorrelated; hence:
2
Ref
 S   S 
(36)
us ( Si )   i    ti i  u ( p)
     
For random wavelength offsets, the variance of a combination X is given by Eq. (7)
and the covariance between two combinations X and Y is given by Eq. (12), each with
the appropriate form of u(Si).
2
5.4.2 Combining Systematic Wavelength Offsets across Wavelengths
We may also have an offset in the wavelength setting. This can arise if the spectral
lamp(s) used for calibrating the wavelength scale have different alignment to the
broad-spectrum lamp used for measurement, or as an offset from the calibration
process. The uncertainty treatment for each spectral value is identical to that for the
random setting but now the uncertainties of different wavelengths are fully correlated.
Note that the signs can vary throughout the spectrum, and the correlation between
different wavelength pairs can be positive or negative.
NMI TR 9
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For scaling effects systematic across wavelengths, the (signed) uncertainty in a
combination X is given by:
N
SiRef 
X  Si
(37)
us ( X )  
 ti

 u ( p)
 
i 1 Si  
and the covariance between two combinations X and Y is given by the product of their
signed uncertainties.
5.5
Multiple Transfers from the Reference
The spectral result of one measurement may be the reference for a subsequent one, as
we progress from base standards through various levels of working standards. Here
the base reference spectrum may be used to calculate uncertainties provided the
transfer uncertainty components are combined for all the transfers. The usual rules for
such combination apply; if the uncertainties for a given component are uncorrelated,
the combined transfer uncertainty is found by sum-of-squares, Eq. (7). If the effects
are correlated, they are added linearly, Eq. (10). Care must be taken here. For
example, if the wavelength scale was recalibrated between transfers, and the lamp
repositioned, the offset applicable for the second transfer is not correlated to that of
the first, whereas it is if the system is undisturbed between transfers.
6
UNCERTAINTY ESTIMATES FOR THE PHOTOMETRIC INTEGRALS
Uncertainties were estimated for the photometric integrals listed in section 1. The
values of the various independent uncertainty components for detector and source
measurements are shown in Table 2. These values were chosen as reasonable
representation for typical measurements. Figure 1 shows the various distributions
used. Transfer measurements were calculated at 5 nm intervals. Offsets in the transfer
were set as fractions of the maximum signal in each channel, here taken as the
maxima of the reference and sample spectra.
Table 2. Responsivity uncertainty components
used to calculate the representative examples
Component
Base reference – detector
Reference signal offset noise
Reference signal background offset
(drift)
Test signal offset noise
Test signal background offset
Source noise
(combined over both channels)
Absolute scaling
Non-linearity
Wavelength setting
Wavelength offset
NMI TR 9
Type
Random
Uncertainty
Ideal photodetector, 0.1% relative
uncertainty with a correlation coefficient
of 0.3 between wavelengths
0.1%, relative to max. signal
Systematic
0.01%, relative to max. signal
Random
Systematic
0.1%, relative to max. signal
0.01%, relative to max. signal
Random
0.1%, relative
Systematic
Systematic
Random
Systematic
0.1%,relative
0.01%, relative to max. signal
0.03 nm
0.1 nm
Random and
systematic
9
Reference standards usually have relatively smooth spectral distributions and they are
derived at limited wavelengths. The results shown here assume reference standards
derived at 20 nm intervals.
True integrals are required for data recorded at discrete intervals. While end-point
integral effects can be ignored because the convolving function tends to zero, not all
data are presented at uniform wavelength spacing. Hence in converting from integrals
to sum forms we include a weighting i for each term. For N spectral points and
trapezoidal integration:
1  (2  1 ) / 2
 N  (N  N 1 ) / 2
i  (i 1  i 1 ) / 2
6.1
(38)
i  1 or N
Photometric Response for Illuminant A
The integral part of the response of a photometer to illuminant A is:
(39)
RV   i Ri Ei
Here we are concerned only with the contribution of the spectral measurement to the
relative value of the integral. Because we are dealing with a broad-band source we
use the V( distribution itself for the photometer spectral response The effect of the
various independent components is shown in Table 3, along with their combined
total. Refer to Table 2 for a more complete description of the uncertainty settings.
Table 3. Relative uncertainty in response of a typical photometer to
CIE Illuminant A arising from the components listed in Table 2
Component
uncertainty
Relative
uncertainty (%)
0.1% (r = 0.3)
0.063
Reference signal offset noise
0.1%
0.028
Reference signal background offset (drift)
0.1%
0.015
Test signal offset noise
0.1%
0.055
Test signal background offset
0.01%
0.042
Source noise (combined over both channels)
0.1%
0.019
Absolute scaling
0.1%
0.100
Non-linearity
0.01%
0.010
Wavelength setting
0.03 nm
0.009
Wavelength offset
0.1 nm
0.051
Component
Base reference - detector
Total:
NMI TR 9
0.151
10
6.2
Spectral Mismatch Factor
For a perfect photometer the spectral mismatch factor is 1, independent of the source.
Practical photometers can have a response significantly different from ideal at
wavelengths where the photometer response is small. Also accurate photometric
measurement of narrow-band sources such as quasi-monochromatic LEDs in those
spectral regions may require smaller bandwidth and more closely spaced spectral
measurements than broad-band sources – errors due to such effects are not considered
here.
To estimate this quantity and its uncertainty we need to know both the photometer
spectral response and the source spectral power distribution. It is sufficient to have
these in relative terms only, although in propagating uncertainties from an assumed
reference standard we generate the uncertainty for the absolute spectral quantity, and
the correlations between quantities at different wavelengths. We treat the
measurements of spectral response and spectral power distribution as independent and
separately calculate the uncertainty of the two ratios:
 i EiA Ri
(40)
FR 
 i Ei Ri
and
 i EV
i i
(41)
FS 
 i Ei Ri
For each ratio, we estimate the uncertainties u ( N ) and u ( D) of the sums in the
numerator and denominator, respectively, and their covariance u ( N , D ) . The relative
uncertainty of the ratio is then given as:
N
2
2
2
urel
( )  urel
( N )  urel
( D)  2urel ( N , D)
(42)
D
Because the integrals in the ratios are similar, correlations between the numerator and
denominator are strong. This is particularly true for the FS ratio of Eq. (41) because
the relative spectral response of any practical photometer is close to a V()
distribution, and as a consequence the contribution from source uncertainties is often
negligible. Source distributions can be quite different, however, and correlations
between the numerator and denominator of the FR ratio of Eq. (40) are reduced
compared to those for FS.
As the uncertainties of the source and response spectra are independent, the relative
uncertainty in the spectral mismatch factor is found combining the relative
uncertainties of the two ratios in quadrature.
Again we can use V() as a representative photometer spectral response distribution
when calculating uncertainties (but not the mismatch value, of course, as this depends
on the response distribution itself). Table 4 shows results for uncertainties the spectral
mismatch factor of a D65 source for a photometer calibrated against a CIE Source A.
The uncertainty treatment is general; it can be applied to any two source distributions
used for calibration and subsequent measurement.
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Table 4. Spectral mismatch factor uncertainty, by component, for a practical
photometer and a source distribution matching CIE Illuminant D
Component
Component
uncertainty
Relative
uncertainty (%)
0.1% (r = 0.3)
0.008
Reference signal offset noise
0.1%
0.007
Reference signal background offset (drift)
0.1%
0.000
Test signal offset noise
0.1%
0.036
Test signal background offset
0.01%
0.009
Source noise (combined over both channels)
0.1%
0.005
Absolute scaling
0.1%
0.000
Non-linearity
0.01%
0.000
Wavelength setting
0.03 nm
0.004
Wavelength offset
0.1 nm
0.088
Base reference - detector
Source contribution
(negligible)
Total:
6.3
0.096
Quality Factor of a Photometer
In practical terms, the quality factor of a photometer is an indicator, rather than an
important photometric quantity. Propagation of the uncertainty for the quality factor
is complicated first by the renormalisation of the reponsivity of Eq. (44) and secondly
by the absolute value of the integrand in Eq. (4). A complete uncertainty propagation
is presented here more out of interest than need.
In sum terms we have:
'
1
f
  R V

 V
*
i
i
i


i
i
Ri*  Vi
i
VS
(43)
with the renormalisation given as:
*
i
R
 E V

 E R
A
i i
A
i
i
i
i
Ri
(44)
Sensitivity coefficients for f1’ in terms of Ri are:
A
f1'
f1' Ri*  i   i Ei Vi


Ri Ri* Ri VS   i EiA Ri

 E AR 
*
1  i i Ai  , Ri  Vi
   i Ei Ri 
 i   i EiAVi 
 i EiA Ri 
*
1

(45)

 , Ri  Vi
A
A
VS   i Ei Ri    i Ei Ri 
 0, Ri*  Vi
Note that uncertainty at those wavelengths for which the normalised response equals
V() do not contribute to uncertainty in f1' – this is confirmed by calculating
sensitivity coefficients directly for positive and negative offsets in responsivity,

NMI TR 9
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where it is seen that the sensitivity changes sign with the offset and the net
contribution is zero. This seems a non-physical result at first thought, because it
implies that even for a large uncertainty in response, the contribution to uncertainty in
f1' can be small. However, the larger the uncertainty, the less likely that the
normalised response will equal V().
The sensitivity coefficient in fact changes sign within the uncertainty band for values
of normalised responsivity close to V(). We allow for this effect by calculating an
average sensitivity coefficient through the uncertainty band. This calculation is
simplified by noting that the negative term in the brackets of Eq. (45) is small
compared to 1 and can be ignored, and by assuming a rectangular probability
distribution within the uncertainty band for a given component of the spectral
responsivity values, at each wavelength. The average sensitivity coefficients then
become:
A
f1'  i   i Ei Vi
 i EiA Ri V  u ( R )

,
R

i
i
i i
Ri VS   i EiA Ri
 i EiAVi
  E R V  u ( R )
V  E R
 E V

  E R V 
 R 
  E V 
  E V  , otherwise

V  E R
u(R )

 i   i EiAVi
S
i
A
i
, Ri 
i
i
i
A
i
i
A
i i
i
i
i
S
i
A
i i
A
i
i
A
i
i
A
i i
i
i
i
i
(46)
i
i
By directly applying systematic positive and negative offsets to the responsivity data,
we can show also that the shift in value of f1' is generally non-symmetric for
systematic effects; this asymmetry is ignored here.
The photometer response cannot be set to V() to estimate uncertainties, as the
sensitivity coefficients all become zero. Table 5 shows the relative uncertainty in f1'
calculated for a typical photometer with good matching at red wavelengths, relaxed at
blue wavelengths, for which the f1' value was 2.99%.
Table 5. Uncertainty in f1' for a typical photometer with f1' = 2.99%
arising from the various components
Component
Base reference - detector
Reference signal offset noise
Reference signal background offset (drift)
Test signal offset noise
Test signal background offset
Source noise (combined over both channels)
Absolute scaling
Non-linearity
Wavelength setting
Wavelength offset
NMI TR 9
Component
uncertainty
0.1% (r = 0.3)
0.1%
0.1%
0.1%
0.01%
0.1%
0.1%
0.01%
0.03 nm
0.1 nm
Total:
Uncertainty in f1'
(%)
0.17
0.13
0.02
0.17
0.04
0.09
0.00
0.02
0.05
0.01
0.30
13
It can be seen from Eq. (44) that the f1' dependence on systematic scaling of the
responsivity values is zero. This in fact can be used to check the accuracy of the
uncertainty calculation. For the photometer used here, the propagated uncertainty was
0.1%; this component has been set to zero in the table.
7
CONCLUSION
Systematic effects in the measurement system generally dominate the uncertainties of
photometric integrals. Calculation of the uncertainty in for combinations of spectral
integrals due to the various systematic effects is relatively straightforward. The
principles explained here are also useful for estimating uncertainties in the various
colour quantities [5]. Individual component uncertainties of combinations scale with
their input values. The representative examples given here can be used as a guide to
uncertainties to be expected from given experimental conditions.
The progression of examples given here also demonstrates the principles of
uncertainty propagation in increasing complexity. Photometric response is a single
integral and only its uncertainty is required. The spectral mismatch factor requires
correlations between integrals to be known. The sensitivity coefficients required to
calculate f1' are complex due to taking the absolute value inside an integral. The
various uncertainty components introduced by both systematic and random
measurement processes can be treated in a consistent manner by applying the
methods described in the ISO Guide; however, the f1' example is one where the firstorder calculation of the Guide is only approximate.
8
REFERENCES
[1]
The Basis of Physical Photometry (1983) Publication CIE 18.2, International
Commission on Illumination, Vienna
Guide to the Expression of Uncertainty in Measurement (1993) International
Organisation for Standardisation
JL Gardner (2003) Correlations in Primary Spectral Standards
Metrologia 40, S167–S171
JL Gardner (2003) Uncertainties in Interpolated Spectral Data
J. Res. Natl Inst. Stand. Technol. 108, 69–78
JL Gardner (2005) NMI TR 8 Uncertainties in Colour Measurements,
National Measurement Institute
[2]
[3]
[4]
[5]
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