determination of colorimetric uncertainties in the spectrophotometric measurement of colour

Determination of colorimetric uncertainties in thespectrophotometric measurement of colour

P.J. Clarke*, A.R. Hanson, J.F. Verrill

Centre for Optical and Environmental Metrology, National Physical Laboratory (NPL), Queen's Road, Teddington, Middlesex TW11 0LW, UK

Received 30 June 1998; accepted 8 July 1998

Abstract

A method for the determination of colorimetric uncertainties has been developed in order to meet the requirements for

accreditation by the UK accreditation service (UKAS), which include a statement of uncertainty for all certi®ed quantities.

The values of the principal sources of spectrophotometric uncertainty are ®rst determined and are used to calculate

corresponding components of colorimetric uncertainty using a simple model. The components of uncertainty analysed are

100% level (diffuse re¯ectance), photometric non-linearity, dark level and wavelength error. Bandwidth error is not signi®cant

for NPL surface colour standards because a small bandwidth is always used.

The gloss trap error and specular beam error are determined and corrected so that only the uncertainties after correction

need be considered. These can be treated as components of dark level uncertainty. The uncertainties are determined for the

following colour data: x, y, Y, u0, v0, L*, a* and b* for the CIE 108 Standard Observer and the CIE Standard Illuminant D65 for

three geometries: specular included, specular excluded and 08/458. These are now quoted routinely on NPL certi®cates for

ceramic colour standards, white and black ceramic tile standards and Russian opal standards. # 1999 The National Physical

Laboratory Published by Elsevier Science B.V. All rights reserved.

Keywords: Colorimetry; Re¯ectance; 08/458 radiance factor; Ceramic colour standards; White and black tiles; Uncertainty

1. Introduction

With the increasing demands of measurement

accreditation and quality systems there is a need to

quote uncertainties on all certi®ed quantities. Mea-

surements, if they are to be meaningful, need a state-

ment of their associated uncertainty. The sources of

uncertainty need to be identi®ed and their contribu-

tions to the overall uncertainty of the certi®ed quantity

assessed.

An uncertainty is the range of values within which

the true value is expected to lie. It is always stated at a

given con®dence level. For example, `̀ The value of

re¯ectance is R��R, where �R is the uncertainty at

the 95% con®dence level'', means that there is a 95%

probability that R lies between R��R and Rÿ�R.

Uncertainty should not be confused with error, which

is the difference between the measured value and the

true value. (The colorimetric signi®cance of spectro-

photometric errors has been considered by several

authors [1±10].) In some cases the error is known

approximately. If it is not corrected, then it is often

appropriate to consider the value of the uncertainty to

Analytica Chimica Acta 380 (1999) 277±284

*Corresponding author. Fax: +44-181-943-6283; e-mail:

[email protected]

0003-2670/99/$ ± see front matter # 1999 The National Physical Laboratory Published by Elsevier Science B.V. All rights reserved.

P I I : S 0 0 0 3 - 2 6 7 0 ( 9 8 ) 0 0 4 7 9 - 6

be the same as the error. If it is corrected, then the

uncertainty will be a lower value which must be

estimated after correction.

For measurements made at the NPL the principal

sources of spectrophotometric uncertainty are: abso-

lute scales of diffuse re¯ectance and 08/458 radiance

factor, photometric non-linearity, dark level and wave-

length error. Bandwidth error is not signi®cant for

NPL surface colour standards because a small band-

width is always used.

Previously at the NPL, certi®cates for diffuse

re¯ectance and 08/458 radiance factor calibrations

of ceramic colour standards and white and black

ceramic tiles only quoted uncertainties for the

spectral data and not for the colour data. The colour

data are derived from and dependent in a complex way

on the spectral data. A computer model has been

developed at the NPL to determine the uncertainty

in colour data from the spectrophotometric uncertain-

ties.

2. Sources of uncertainty

Uncertainties can be split into Type A and Type B

standard uncertainties as de®ned by the UK Accredi-

tation Service (UKAS) and by ISO [11,12]. Type A are

those evaluated by statistical methods and Type B are

those evaluated by other means. For re¯ectance and

08/458 radiance factor measurements made at the

NPL, the only Type A uncertainty is the repeatability.

All the others are Type B uncertainties. The Type B

uncertainties for ceramic colour standards, black and

white tiles and Russian opals are

1. Uncertainty in the level of the absolute scales of

diffuse re¯ectance and radiance factor

2. Uncertainty in the spectral slope of the scales (skew

uncertainty)

3. Uncertainty in the transfer of the absolute scales to

working standards

4. Dark uncertainty

5. Linearity uncertainty

6. Wavelength scale uncertainty

7. Thermochromism uncertainty

8. Glossy to matt ratio uncertainty

9. Specular beam uncertainty (specular included geo-

metry only)

10. Gloss trap uncertainty (specular excluded

geometry only).

2.1. Repeatability

The repeatability is determined by making several

measurement scans on white, grey and black tiles and

averaging them. The standard deviation of the popula-

tion is then used in the calculation of the standard

uncertainty.

2.2. Uncertainty in the level of the absolute scales of

diffuse reflectance and radiance factor

The 100% levels come from the traceability to the

absolute scale. This is taken from a certi®cate of

calibration for a white standard. The uncertainty has

a normal distribution and is usually quoted with a

coverage factor of k�2, which provides a con®dence

level of approximately 95%. For the NPL, at present,

traceability is taken from PTB (Germany) for 08/458radiance factor and from the mean of the PTB and

NRC (Canada) diffuse re¯ectance scales for specular

included and specular excluded geometries. (The NPL

realised its own scales of diffuse re¯ectance and

08/458 radiance factor in March 1998. However no

change in the scales used for re¯ectance and radiance

factor calibrations is planned until after the forth-

coming CCPR intercomparison.) The uncertainty

limits �a in the level of the scale are shown in the

upper part of Fig. 1.

2.3. Uncertainty in the spectral slope of the scales

(skew uncertainty)

The uncertainty in the absolute scale of diffuse

re¯ectance or radiance factor may change with wave-

length. The scale may have a slope on it within the

uncertainty limits. This effect is assessed using a skew,

of half the scale uncertainty, on the 100% data, with

both positive and negative skew slopes being consid-

ered. The skew uncertainty is shown in the lower part

of Fig. 1.

2.4. Dark uncertainty

The dark uncertainty is a combination of the uncer-

tainty in the electronic zero offset of the instrument

278 P.J. Clarke et al. / Analytica Chimica Acta 380 (1999) 277±284

and, for diffuse re¯ectance, an optical offset due to a

halo of scattered light surrounding the sample beam

and falling on the integrating sphere wall. Offset errors

can be measured by placing a wedge consisting of two

plates of black glass at the sample port of integrating

sphere instruments [5], or by placing a black glass

plate at the sample port for 08/458 collectors. Blocking

the beam does not give a true offset reading, because it

does not quantify the effects of the halo of stray light.

If the dark reading is corrected for in re¯ectance

calculations the uncertainty is that after correction.

If it is not corrected for, the uncertainty is the error

itself.

The absolute re¯ectance (Ra) of a sample measured

against a master standard and corrected for dark error

reading is

Ra � �a ÿ �k

�t ÿ �k

Rt � �a 1ÿ ��k=�a�� t 1ÿ ��k=�t�� Rt

� �a

�t

1ÿ �k

�a

� �k

�t

� �Rt � �a

�t

Rt � K1�a

�t

ÿ 1

� �;

where K1 � ��k=�t�Rt, �t is the master standard read-

ing, �a the sample reading, �k the dark cavity reading,

and Rt is the absolute re¯ectance of the master stan-

dard.

2.5. Photometric non-linearity

This is non-linearity of the detector system. It can

be assessed from measurements on the spectrophot-

ometer using a white standard at the sample port and

neutral density ®lters, traceable to the NPL Reference

Spectrophotometer, at the entrance port to the inte-

grating sphere or in the incident beam for 08/458geometry. Non-linearity error is modelled using a

polynomial and is set to zero at 0% and 100%.

2.6. Wavelength scale uncertainty

Wavelength errors cause changes in re¯ectance in

regions of spectral slopes, where re¯ectance changes

with wavelength. They can be measured using a

wavelength standard or a known spectral line from

a source such as a deuterium lamp. These errors can be

corrected by using the formula

Rc � Ru � dR

d��;

where Rc is the corrected re¯ectance, Ru the uncor-

rected re¯ectance, dR/d� the spectral slope and �� the

wavelength error. When the errors are corrected, the

uncertainty is the error in the corrected value. Where

this is not done, the uncertainty is equal to the wave-

length error. The wavelength scale uncertainty will be

the quadratic sum of two components, the uncertainty

in the wavelength of the spectral feature such as the

absorption band or emission line which is used for

calibration and the uncertainty in the process of

calibration of the instrument. The latter will be related

to drift of the wavelength scale with time after the

instrument is switched on.

2.7. Thermochromism uncertainty

Using calibrated standards at a different tempera-

ture from that at which they were calibrated may cause

an uncertainty due to thermochromism. This is sig-

ni®cant for coloured (non-neutral) samples. A full

investigation of this has been carried out on the

Fig. 1. Level and skew uncertainties as a function of wavelength.

P.J. Clarke et al. / Analytica Chimica Acta 380 (1999) 277±284 279

ceramic colour standards [13]. Thermochromism

causes spectral data to shift in a similar way to a

wavelength error. Fig. 2 illustrates the effect.

2.8. Glossy to matt ratio uncertainty

Errors in the glossy to matt ratio are due to differ-

ences in ef®ciency of collection of the integrating

sphere with angle of re¯ectance. Freeman [14] has

used a laser scanning device to show that the integrat-

ing sphere ef®ciency varies throughout the sphere.

The errors have been determined by measuring glossy

samples against matt samples with different integrat-

ing spheres. The uncertainty is calculated from the

difference between the glossy to matt ratio from

measurements of re¯ectance using different integrat-

ing spheres. (Three spheres were used.)

2.9. Specular beam uncertainty

In the specular included geometry, the specularly

re¯ected light may not be collected with the same

ef®ciency as the diffusely re¯ected light. The error can

be determined using a mirror and a calibrated matt

white as described in [5,10]. If a correction is made the

uncertainty is that after correction. If a correction is

not made the uncertainty is equal to the uncorrected

error. For matt samples calibrated against a matt

standard this correction is zero. For the case of a

glossy sample calibrated against a glossy standard,

this uncertainty behaves in the same way as a dark

uncertainty.

2.10. Gloss trap uncertainty

The error due to incomplete absorption of the

specular beam in the gloss trap is known as the gloss

trap error. This error applies only to the specular

excluded geometry. The error can be determined

using a mirror and a calibrated matt white standard

as described in [5,10]. If a correction is made the

uncertainty is that after correction. If a correction is

not made the uncertainty is equal to the uncorrected

error.

Fig. 2. Diffuse reflectance vs wavelength at temperatures T1 and T2.


3. Total uncertainties

The uncertainties from each source need to be

combined together to give a total uncertainty. The

uncertainties all need to be quoted in terms of a

standard uncertainty at the 65% con®dence level,

where the coverage factor k is equal to 1. Each

uncertainty is characterised by the assessed probabil-

ity distribution for the uncertainty. The distributions

can be normal (Gaussian) or rectangular. Each

component of uncertainty is then divided by a

constant, (1 or 2, depending on the coverage factor,

in the case of a Gaussian distribution,��3p

in the case

of a rectangular distribution), to give the standard

uncertainty. Components of uncertainty are combined

in quadrature to give the total uncertainty��a21 � a2

2 � a23 � a2

4 � � � ��p

which is then multiplied

by a coverage factor, which is usually 2 to give a total

uncertainty at the 95% con®dence level.

The component uncertainties are assessed from

measurements at three wavelengths in the visible

spectrum: 400, 500 and 700 nm. The total re¯ectance

or radiance factor uncertainty and the total wavelength

uncertainty are calculated for the master NPL ceramic

tile standards measured against a white tile re¯ectance

standard. The ceramic tile standards for customers are

then calibrated against the corresponding master cera-

mic tile standard. The uncertainties for the tiles sup-

plied to customers are a combination of the

uncertainty in the master standard and the uncertainty

in the process of calibration against the master stan-

dard [11,12].

3.1. Total reflectance or radiance factor uncertainty

The Type A repeatability uncertainty is assessed

from six repeated measurements of re¯ectance or

radiance factor using the white, pale grey, mid grey,

deep grey and black tiles. The Type B uncertainties for

these tiles are also assessed. The linearity is measured

at two re¯ectance or radiance factor levels of approxi-

mately 73% and 32% and uncertainties for the white,

grey and black tiles calculated. Corrections are made

for the dark error, gloss trap error and specular beam

error, so the corresponding uncertainties are after

correction. The absolute scale uncertainty comes from

the standards calibrated by the PTB and NRC and

includes an allowance for the difference between the

PTB and NRC re¯ectance scales. The skew uncer-

tainty is assessed as half the absolute scale uncertainty.

The results of this calculation give the total uncer-

tainty of the white, grey and black master ceramic tile

standards used by the NPL. Ceramic tiles calibrated by

the NPL are calibrated against the corresponding

master white tile and therefore the uncertainties asso-

ciated with these tiles are a Type B uncertainty from

the master tile calibration and a Type A uncertainty

due to repeatability. These are added in quadrature to

give the total re¯ectance or radiance factor uncertain-

ties for the NPL calibrated ceramic tiles.

The total re¯ectance and radiance factor uncertain-

ties for NPL calibrated white, pale grey, mid grey,

deep grey and black NPL ceramic tile standards are

subject to a least squares ®t to give a linear relationship

to re¯ectance value. (A higher order polynomial

would give a better ®t. The slope of the linear relation-

ship is increased so that the straight line lies above the

curved line, thus increasing the uncertainty. This is

done in order to keep the statement of uncertainty on

the certi®cate as simple as possible by excluding

higher order terms). The values from the ®t are then

adjusted so that uncertainties generated from the

equation are the same or greater than the total uncer-

tainties from the uncertainty assessment. Fig. 3 shows

the relationship of the calculated uncertainties and the

®tted uncertainties to the re¯ectance value. This

allows a re¯ectance uncertainty to be calculated for

any re¯ectance value. The resulting uncertainties in

the re¯ectance or radiance factor for NPL white and

black tiles and ceramic colour standards at the 95%

con®dence level (k�2) are given in Table 1.

3.2. Total thermochromism uncertainty

In order to calibrate the uncertainty due to thermo-

chromism the spectral re¯ectance must be measured at

two temperatures, usually with a difference of at least

58C. The shift of the spectrum along the wavelength

scale is noted at the spectral slopes and a thermo-

chromism coef®cient ct��/dT determined.

Master ceramic colour standards are calibrated

against matt white re¯ectance standards. If the tem-

perature uncertainty is ��T18C then the correspond-

ing wavelength uncertainty is �ct��T1 nm.

Ceramic colour standards issued to customers are

usually calibrated against a master ceramic colour


standard of the same colour. If the uncertainty in

temperature equality between the readings from the

sample and the readings from the master is ��T28Cthen the corresponding wavelength uncertainty is

�ct��T2 nm.For ceramic colour standards calibrated for custo-

mers the total thermochromism uncertainty is

��tc � ct

��T2

1 ��T22

ÿ �q.

3.3. Total wavelength uncertainties

The total wavelength uncertainties for the NPL

master ceramic tiles are a combination of the uncer-

tainties due to wavelength scale (2.6) and thermo-

chromism. The wavelength scale uncertainty and the

total thermochromism uncertainty are combined in

quadrature. For the neutral tiles the thermochromism

Fig. 3. Reflectance uncertainties at 400 nm 08/458 geometry.

Table 1

Reflectance/radiance factor uncertainties for NPL white and black tiles and ceramic colour standards

Standard Specular excluded and specular included geometries 08/458 geometry

Wavelength range (nm)

320±460 465±780

Glossy white �0.6% �0.4% �0.65%

Matt white �0.55% �0.35% �0.65%

Glossy black �0.1% �0.1% �0.1%

Matt black �0.1% �0.1% �0.1%

Glossy coloured �(0.05�0.006R)% �(0.05�0.004R)% �(0.05�0.007R)%

Matt coloured �(0.05�0.0055R)% �(0.05�0.0035R)% �(0.05�0.007R)%

R is the reflectance or radiance factor.


is not signi®cant and the total wavelength uncertainty

is only that in the wavelength scale.

3.4. Total colour uncertainties

The uncertainties are determined for ceramic tiles

calibrated against a corresponding NPL master cera-

mic tile. Uncertainties are calculated for the following

colour data: x, y, Y, u0, v0, L*, a* and b* for the CIE 108Standard Observer and the CIE Standard Illuminant

D65 for three geometries: specular included, specular

excluded and 08/458 [15]. Calculations have been

limited to the CIE 108 Standard Observer and CIE

Standard Illuminant D65 due to the large amount of

data involved.

A computer program, which assumes the uncertain-

ties are independent, has been written to assess the

effect of the spectrophotometric uncertainties on the

colorimetric data. It uses the re¯ectance or 08/458radiance factor uncertainties and the wavelength

uncertainties to calculate the colour uncertainties

using the mathematical relationships below

The level uncertainty UL is input as

UL � a

100R��;

where R(�) is the re¯ectance or 08/458 radiance factor

and a is the estimated uncertainty in the re¯ectance or

08/458 radiance factor scale level at 100% re¯ectance

or 08/458 radiance factor.

The skew uncertainty is

US � a

100� �ÿ 580

400R��: (1)

Level and skew uncertainties are combined in quad-

rature.

The dark uncertainty is

UD � bR

100ÿ 1

� �; (2)

where b is the dark uncertainty for a black sample.

The linearity uncertainty is of the form

Ul � �R� �R2; (3)

where �, � are constants. A non-linearity error falls to

zero at 0% and 100% by de®nition, hence the linearity

uncertainty also falls to zero at 0% and 100%. There-

fore

Ul � �R 1ÿ R

100

� �:

Errors in linearity can be corrected using an appro-

priate polynomial. However, if the error is too small to

need a correction, the uncertainty can be equated with

the error as measured by a calibration ®lter, or re¯ec-

tance standard. If the difference between the calibrated

value of the ®lter and the instrument reading is �R,

and R is the calibrated value of the ®lter (or re¯ectance

standard) then � can be determined.

Colorimetric uncertainties are determined for each

component of spectrophotometric uncertainty and

then combined in quadrature.

The changes in the colour data between the mea-

sured re¯ectance or radiance factor values and the

same values plus the uncertainties are calculated for

each coef®cient. The program combines the compo-

nent uncertainties in quadrature [11] to give total

uncertainties for the colorimetric data.

Table 2

Uncertainties in L*, a* and b* for NPL glossy white and black tiles and glossy ceramic colour standards

Standard Specular excluded and specular included geometries 08/458 geometry

��L* ��a* ��b* ��L* ��a* ��b*

White, pale grey 0.25 0.10 0.10 0.30 0.10 0.10

Mid grey, difference grey, deep grey 0.20 0.10 0.10 0.20 0.10 0.10

Black 0.45 0.10 0.10 0.45 0.10 0.10

Deep pink 0.15 0.15 0.15 0.20 0.15 0.15

Red 0.20 0.25 0.50 0.25 0.25 0.50

Orange 0.20 0.20 0.30 0.25 0.20 0.30

Bright yellow 0.25 0.15 0.30 0.25 0.15 0.30

Green, difference green 0.20 0.15 0.20 0.20 0.15 0.20

Cyan 0.20 0.20 0.15 0.20 0.20 0.20

Deep blue 0.40 0.50 0.50 0.40 0.50 0.50


NPL uncertainties for the CIE 108 Standard Obser-

ver and CIE Standard Illuminant D65 are given in

Tables 2 and 3 for glossy white and black tiles and the

glossy ceramic colour standards. Uncertainties are

those currently used in NPL certi®cates and are sub-

ject to change.

4. Conclusions

Re¯ectance and wavelength uncertainties have

been described. These uncertainties have a complex

relationship with the colorimetric uncertainties. A

method for determining colorimetric uncertainties

has been described.

References

[1] A.R. Robertson, J. Opt. Soc. Am. 57(5) (1967) 691.

[2] V.I. Lagutin, Measurement Techniques 25 (1982) 404.



[5] F.J.J. Clarke, J.A. Compton, Color Res. Appl. 11 (1986) 253.

[6] R.S. Berns, K.H. Petersen, Color Res. Appl. 13 (1988) 243.

[7] R.S. Berns, L. Reniff, Color Res. Appl. 22 (1997) 51.

[8] D.C. Rich, AIC Color 97, The Color Science Association of

Japan, Tokyo, 1997, p. 395.

[9] J.F. Verrill, AIC Color 97, The Color Science Association of

Japan, Tokyo, 1997, p. 42.

[10] J.F. Verrill, P.J. Clarke, J. O'Halloran, NPL Report COEM 2,

ISSN 1369-6807, NPL Teddington, 1997.

[11] M3003 The Expression of Uncertainty and Confidence in

Measurement, UKAS, Feltham, 1997.

[12] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, Guide to the

Expression of Uncertainty in Measurement, ISBN 92-67-

10188-9, International Organisation for Standardization,

Geneva, 1993.

[13] F. Malkin, J.A. Larkin, J.F. Verrill, R.H. Wardman, J. Soc.

Dyers Colorists 113 (1997) 84.

[14] G.H.C. Freeman, in: C. Burgess, D.G. Jones (Eds.), Advances

in Luminescence and Color; Science and Compliance,

Elsevier, Amsterdam, 1995, pp. 71±96.

[15] CIE Publication 15.2, Colorimetry, CIE, Vienna, 1986.

Table 3

Uncertainties in x, y, Y, u0 and v0 for NPL glossy white and black tiles and glossy ceramic colour standards

Type of standard All geometries Specular excluded and specular 08/458 geometry,

��x ��y ��u0 ��v0included geometries, (��Y) (��Y)

White 0.0002 0.0002 0.0002 0.0002 0.55 0.65

Black 0.0003 0.0002 0.0002 0.0002 0.10 0.10

Pale grey 0.0002 0.0002 0.0002 0.0002 0.40 0.45

Mid grey, difference grey 0.0003 0.0002 0.0002 0.0002 0.20 0.20

Deep grey 0.0003 0.0002 0.0002 0.0002 0.10 0.10

Deep pink 0.0006 0.0003 0.0004 0.0003 0.15 0.15

Red 0.0019 0.0002 0.0016 0.0004 0.15 0.15

Orange 0.0005 0.0004 0.0005 0.0002 0.25 0.30

Bright yellow 0.0004 0.0002 0.0002 0.0002 0.40 0.45

Green, difference green 0.0004 0.0006 0.0002 0.0002 0.15 0.20

Cyan 0.0002 0.0005 0.0002 0.0004 0.20 0.20

Deep blue 0.0019 0.0032 0.0032 0.0049 0.10 0.10


determination of colorimetric uncertainties in the spectrophotometric measurement of colour

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