0:45 Surface Color

Maria E. Nadal 

Edward A. Early1

Robert R. Bousquet2

NIST Special Publication SP250-71
 

NIST Special Publication SP250-71

0:45 Surface Color

Maria E. Nadal 

Edward A. Early1

Robert R. Bousquet2

Optical Technology Division 

Physics Laboratory 

National Institute of Standards and Technology 

Gaithersburg, MD 20899-0001 

January 2008 

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 

National Institute of Standards and Technology 

James M. Turner, Acting Director 
 

1

AFRL/HEDO, 2650 Loiuse Bauer Drive, Brooks City-Base, TX 78235 2

Genesis Engineering Solution, 9811 Greenbelt Road, Lanham, MD, 20706 

Certain commercial entities, equipment, or materials may be identified in this 

document in order to describe an experimental procedure or concept adequately. Such 

identification is not intended to imply recommendation or endorsement by the 

National Institute of Standards and Technology, nor is it intended to imply that the 

entities, materials, or equipment are necessarily the best available for the purpose. 

National Institute of Standards and Technology Special Publication SP250-71 

Natl. Inst. Stand. Technol. Spec. Publ. SP250-71, 84 pages (January 2008) 

CODEN: NSPUE2 
 

ABSTRACT 

 This document, SP250-71 (2007), NIST Measurements Services: 0:45 Surface 

Color, describes the instrumentation, standards, and techniques used to measure the 

surface color of reflective, non-fluorescent materials at a 0:45 geometry, that is an 

irradiance angle of 0° and measuring angle of 45°. The organization of this document is 

as follows. Section 1 describes the motivation for establishing a 0:45 Reflectometer and 

associated calibration services that are available. Section 2 presents the theory relevant to 

the measurements described in this document. Section 3 describes the 0:45 reference 

standard. The NIST 0:45 Reflectometer including illuminator, sample wheel, and 

receiver and the associated characterization and validation of the instrument are described 

in Section 4. Section 5 presents the sources of uncertainty and the associated analysis. A 

sample calibration report and reference containing details of the uncertainty method are 

presented as appendices. 
 

TABLE OF CONTENTS 

1. Introduction ........................................................................................................1 

2. Theory ................................................................................................................2 

 2.1 Measurement Equations ...............................................................................2 

 2.2 Color Values ................................................................................................4 

 2.2.1 Tristimulus Values ...........................................................................4 

 2.2.2 Chromaticity Coordinates ................................................................5 

 2.2.3 CIE Color Spaces .............................................................................6 

3. Reference Standard ............................................................................................6 

 3.1 Method of Calibration ..................................................................................6 

 3.2 Uncertainty ...................................................................................................7 

 3.3 Maintenance of Reference Standards ...........................................................8 

4. Reference 0:45 Reflectometer ............................................................................8 

 4.1 Instrument Description and Properties.........................................................8 

 4.2 Characterization of Instrument ..................................................................11 

 4.3 Instrument Validation ................................................................................13 

 4.4 Calibration Services ...................................................................................16 

5. Uncertainty Analysis ........................................................................................16 

5.1 Sources of Uncertainty ...............................................................................17 

5.2 Uncertainty and Correlation of R(λ) ..........................................................21 

5.3 Uncertainties in L*, a*, and b* ..................................................................22 

Acknowledgements ................................................................................................28 

References ..............................................................................................................29 

Appendix A: Sample Report of Calibration for 0:45 Surface Color 

Appendix B: Uncertainty Analysis for Reflectance Colorimetry (Copy of Ref. 16) 
 

LIST OF FIGURES 

3.1 Typical 0:45 spectral reflectance factor of a Spectralon 

 reference standard ..............................................................................................8 

4.1 Schematic of the 0:45 reflectometer ..................................................................9 

4.2 Top view of the components and optical path of the illuminator ....................10 

4.3 Schematic of the optical path of the receiver ...................................................11 

4.4 Spectral line spread function measured by scanning over 

 the 546.074 nm Hg lamp emission line using variable exit 

 slit width as listed in the legend .......................................................................12 

4.5 Typical 0:45 spectral reflectance factor of cyan, deep gray, 

 and orange check standards .............................................................................14 

4.6 Control chart of the measured L* for the cyan tile. 

 The solid lines represent the tolerance condition. ............................................15 

5.1 An example of a bandpass function z0(λ) ........................................................19 

5.2 An example of the error due to a 5 nm FWHM triangular 

 bandpass of the monochromator for the measurements 

 of a yellow BCRA tile......................................................................................20 

5.3 Standard uncertainties in L*, a*, and b* ..........................................................27 
 

LIST OF TABLES 

4.1 Geometrical specifications of 0:45 reflectometer ............................................11 

4.2 Sources and wavelength of emission lines used 

 for the wavelength calibration. ........................................................................13 

4.3 Spectral specifications of 0:45 illuminator ......................................................13 

4.4 Comparison standards ......................................................................................15 

4.5 Acceptance criteria for 0:45 test samples ........................................................16 

5.1 Standard, u and expanded (k = 2), U uncertainties of the 

 CIELAB values for each source of uncertainty for the CIE 

 standard illuminant D65 and the CIE 1931 standard 

 observer for selected color tiles .......................................................................25 
1

1. Introduction 

 

 The National Institute of Standards and Technology (NIST) have a long history in 

color and appearance, with many important contributions to our present understanding of 

color measurements. Interest in colorimetry at NIST dates back to the 1930’s [1,2]. 

NIST continues to take an active role in the measurement of color with the development 

of the Reference 0:45 Reflectometer, henceforth referred to as the 0:45 reflectometer. 

The primary purpose of the 0:45 reflectometer is to disseminate color standards to the 

color and appearance community through a set of calibrated color tiles, British Ceramic 

Research (BCRA) series II*. These tiles are used to transfer the NIST 0:45 color scale to 

other instruments. Because color often plays a major role in the acceptability of a 

product, this calibration service is designed to meet a demand for improved 

measurements and standards to enhance the color matching of products. 

 The surface color of an object depends upon several factors – the spectral power 

distribution of the illuminant, the spectral sensitivity of the observer, the geometrical 

conditions of illumination and observation, and the optical properties of the object. Only 

the last factor depends upon the object, and is quantified by the spectral reflectance factor 

for opaque reflecting materials under the given geometrical conditions. Once the spectral 

reflectance factor is known, the tristimulus values are calculated for a given illuminant 

and observer, and from these values other quantities are derived, such as chromaticity 

coordinates and color space values. 

The 0:45 reflectometer was designed to provide calibrated reflectance color 

standards with a target of ΔE*ab < 0.5 as the primary design specification. The 

instrument specifications were determined by a series of simulation of errors caused by 

the measuring instrument on the calibrated spectral reflectance of color standard and 

calculated color values. Particular attention was given to the inherent properties of the 

instrument such as stray light, band pass error, random noise, and wavelength 

uncertainty. The calibrations are performed at a geometry of 0° for the irradiation angle 

and at a measuring angle of 45°. The calibration process is carried out by measuring the 

reflected flux for each sample and the reference standard for wavelength from 380 nm to 

780 nm in 5 nm increments. The measurements are performed for two orthogonal 

polarizations and averaged. Up to 20 samples can be measured automatically during a 

given scan. The spectral reflectance factor for each sample is determined by the ratio of 

the reflected radiant flux of the sample to the reflected radiant flux of a reference 

standard. Once the spectral reflectance factors have been determined, the color values 

along with their associated uncertainties can be determined for the samples. 

 

 

* Certain commercial equipment, instruments, or materials are identified in this paper in order to 

specify the experimental procedure adequately. Such identification is not intended to imply 

recommendation or endorsement by the National Institute of Standards and Technology, nor is it 

intended to imply that the materials or equipment identified are necessarily the best available for 

the purpose.
2

 The 0:45 reflectometer consists of a source section, a sample section and a 

detection section. The source section consists of a dual lamp housing, a double grating 

monochromator, and optics that direct the incident beam towards the sample. The sample 

section consists of a rotating sample holder which holds and positions the sample and 

standards in the incident beam. The detection section consists of silicon photodiodes that 

measure the reflected flux and monitor the incident radiation. Comparisons and check 

standards ensure that instrument is operating within its uncertainties on a measurement to 

measurement basis while periodic realizations of the scales are used to maintain the 

traceability of the instrument. 

 This document is intended to provide customers and potential customers of the 

0:45 surface color measurements with a detailed overview of the capabilities and 

procedures used in performing the measurements. The NIST Calibration Services 

Program offers Service ID Number 38091S, Special Test of 0:45 Surface Color, as listed 

in the Surface Color and Appearance section of the Optical Radiation Measurements 

Chapter of the NIST Calibration Services Users Guide [3]. The measurement service 

quality system is based on the International Organization for Standardization (ISO), 

17025 standard [4]. 

2. Theory 

 This section details the basic definitions and the relevant measurement equations 

to determine 0:45 reflectance factors and color values for a given measurement. The 

approach used in this section is based upon the concepts presented in Refs. 5-7. 

2.1 Measurement Equations 

According to the CIE international Lighting Vocabulary, reflectance factor is 

defined as the ratio of the radiant flux reflected (from a sample) in the directions 

delimited by a given cone angle to that reflected in the same directions by the perfect 

reflecting diffuser identically irradiated (same spectral composition, polarization, and 

geometrical distribution [8, 9]. If the cone angle is infinitely small, reflectance factor is 

equal to radiance factor. If the cone angle is a whole hemisphere, the reflectance factor is 

equal to reflectance. In this document, an infinitely small cone angle is always assumed 

for reflectance factor and the cone angle is not described hereinafter. The measurement 

of the spectral reflectance factor of a sample by the 0:45 reflectometer is performed by 

comparing signals from the sample and reference standard, under the same measurement 

conditions, at each wavelength and two orthogonal polarization of the incident radiant 

flux. This process is described by the measurement equation 

( ) ( )

( ) ( ) σλ

σλ

λ σ σλ , ,

, , s

s

x

 x i

i

i

i R

s

s R ⋅= (2.1) 

where λi and σ are the wavelength and polarization, respectively, of the incident radiation 

from the instrument illuminator, sx and ss are the measured signals from the sample and 

reference standard, respectively, and Rx and Rs are the reflectance factors for the sample 
3

and reference standard, respectively. The 0:45 signals from the sample and the reference 

standard are normalized by the signals from a monitor photodiode to account for drift in 

the radiant flux from the illuminator. Measurements are performed at polarizations of 0°

and 90° relative to the illumination plane, so the reflectance factor of the sample for 

unpolarized incident radiant flux is given by 

[ ])90,()0,( 2

1 )( x x x = °+° R λi R λi R λi . (2.2) 

A scan means measuring the 0:45 and monitoring signals for all samples and 

reference standards for the wavelength range of 380 nm to 780 nm at 5 nm increments 

and s and p polarization σ resulting in the reflectance factor for the samples. Μultiple 

scans are used to compute the final spectral reflectance factor of the samples. 

Let i index the scan, j index the reference standard, and k index the signal reading. 

The signals from the sample are sx(λ, σ, i, k), the signals from the reference standard are 

ss(λ, σ, i, j, k), the signals from the monitor photodiode while measuring the sample are 

sm,x(λ, σ, i, k), and the signals from the monitor photodiode while measuring the 

reference standard are sm,s(λ, σ, i, j, k). The average signal from the sample is 

∑=

=

l

k

kis

l

is

1

x x ),,,( 1 σλ ),,( σλ , (2.3) 

where l is the number of repetitions. The equation is applied similarly for the other 

signals ss, sm,x, and sm,s. The normalized signals N are obtained by dividing the average 

signals from the sample photodiode by those from the monitor photodiode, given by 

),,(

),,( ),,(

xm,

x

x is

is iN

σλ

λ σ σλ = and (2.4) 

),,,(

),,,( ),,,(

sm,

s

s jis

jis jiN σλ

λ σ σλ = . (2.5) 

The normalized signals are used to reduce the effect of changes in the output radiant flux 

from the illuminator. The ratio Qs of the reflectance factor of the reference standard to 

the normalized signal is given by 

),,,(

),,( ),,,(

s

s

s jiN

jR jiQ σλ

λ σ σλ = , (2.6) 

and the average of these ratios for all the reference standards is given by 

∑=

=

m

j

jiQ

m

iQ

1

s s ),,,( 1 σλ ),,( σλ . (2.7) 
4

The reflectance factor R of the sample, for each scan, wavelength, and polarization, is 

given by 

 ) ,,(),,(),,( x x s λ σ = λ σ ⋅ λ σ iQiNiR . (2.8) 

Averaging over polarizations, usually σ = 0° and 90° with respect to the plane of 

incidence, yields the reflectance factor for unpolarized incident radiant flux, given by 

[ ]),90,(),0,( 2

1 ),( x x x λ = λ + λ iRiRiR . (2.9) 

The final spectral reflectance factor for a sample is obtained by averaging over all scans, 

n, given by 

∑=

=

n

i

iR

n

R

1

 x x ),( 1 λ)( λ . (2.10) 

After the determination of the final reflectance factor, the methods specified in the next 

section are used to determine the color values for a specified observer and illuminant. 

Hereafter, for convenience, we will denote Rx(λ) by R(λ). 

2.2 Color Values 

 

 The color values determined by the 0:45 reflectometer consist of the Commission 

Internationale de l’Eclairage, (CIE) tristimulus values, chromaticity coordinates, and 

color-space values. The determination of each of these values is briefly discussed. 

Further details on the derivation and the application of these values can be found in 

reference [10, 11]. 

2.2.1 CIE Tristimulus values 

 The CIE tristimulus values can be determined by the integration of the spectral 

reflectance factor, R(λ), over the visible wavelength range with a set of spectral 

weighting factors, W(λ). 

() () X RW d x

λ

= λ λ λ ∫ , (2.11) 

() () Y RW d y

λ

= λ λ λ ∫ , 

() () Z RW d z

λ

= λ λ λ ∫ , 

with 
5

() ()() W kS x x λ = λ λ

() ()() W kS y y λ = λ λ

() ()() W kS z z λ = λ λ

where S(λ) is the relative spectral power of one of the CIE standard illuminants, x (λ), 

y (λ) and z (λ) are the color matching functions of the CIE colorimetric standard 

observer, and k is the normalizing factor for Y equaling 100 for the perfect diffuser. 

 Practically, the tristimulus values are determined by a set of summations over the 

wavelength range. Conversion of the 2.11 equations to summations results in the 

following equations, 

XkRW () ()

λ

= Δ ∑ λ λ λ , (2.12) 

Yk RW () ()

λ

= Δ ∑ λ λ λ , 

Zk RW () ()

λ

= Δ ∑ λ λ λ , 

with 

k Sy 100 / ( ) ( )

λ

= Δ ∑ λ λ λ . 

The CIE defines standard tables of color matching functions and illuminant spectral 

power distributions for the calculation of tristimulus values [12, 13]. 

2.2.2 CIE Chromaticity Coordinates 

 The relative amounts of the tristimulus values are determined by the ratio of each 

value to the sum of the values. The results are referred to as the chromaticity coordinates. 

x = ++ XXYZ /( ) (2.13) 

y = ++ YXYZ /( )

z ZXY Z = ++ /( ) , 

these coordinates satisfy the following condition 

x ++= y z 1. (2.14) 
6

The chromaticity coordinates are generally specified by x and y, since the two known 

coordinates can be used to calculate the third coordinate. The two-dimensional plot of (x, 

y) is called the CIE 1931 chromaticity diagram. 

2.2.3 CIE 1976 (L* a* b*) Color Space 

 The (x, y) chromaticity coordinates specify any color of light stimuli but not the 

color of an object surface, which has another dimension, lightness (black to white). CIE 

defines two object color spaces: L*a*b* and L*u*v* [11]. In this document, L*a*b* is 

used to specify object colors, as it is most commonly used in the color community. This 

object color space is produced by plotting, in three dimensional coordinates, the 

quantities L*, a*, and b* defined as follow: 

116* 16

3/1

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = Yn

Y L

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

3/1 3/1

500*

n Y n

Y

X

X

a (2.15) 

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

3/1 3/1

200*

n Zn

Z

Y

Y b

where X/Xn; Y/Yn; Z/Zn > 0.01 and Xn, Yn, and Zn define the tristimulus values of the 

reference white that is calculated from the reflectance from a perfect diffuser. 

The color difference * ΔEab between two color samples is determined from 

[( )( ) ( ) ] * 2 2 2 2/1 baLE *** ab Δ+Δ+Δ=Δ (2.16) 

where *

2

*

1

*

2

*

1

*

2

*

1 * and ,* ,* −=Δ−=Δ−=Δ bbbaaaLLL and the subscripts 1 and 2 refer to 

the two samples. 

3. Reference Standard 

3.1 Method of Calibration 

The 0:45 reflectometer measures the 0:45 spectral reflectance factor of a sample 

by comparing the signal from the sample and a reference standard, under the same 

measurement conditions, at each wavelength and two orthogonal polarizations of the 

incident radiant flux. The reference standards are 50.8 mm square sintered PTFE, 

Spectralon mounted in a 101.6 mm holder with the front of the sample parallel to the 

front of the sample holder. There are three reference standards denoted SP01, 02, and 03. 

These reference standards are also used as working standards for routine calibrations. 
7

The reference standards are calibrated on the Spectral Tri-function Automated 

Reference Reflectometer (STARR) [14]. The STARR instrument is the national 

reference instrument for spectral reflectance measurements of spectrally neutral, nonfluorescent samples at room temperature. STARR performs absolute measurements of 

bi-directional reflectance on samples with widths ranging from 50 mm (2 in) to 300 mm 

(12 in). The measurements are all in-plane, meaning the sample normal, illumination 

axis, and receiver axis, all lie within the same horizontal plane. 

The major components of the STARR instrument are a source, a sample holder, 

and a detector. The source system provides a collimated, monochromatic beam of light 

with a known polarization over the spectral region of 200 nm to 2500 nm. The 

goniometer is a monoplane gonio-instrument with a rotating sample table and a movable 

detector arm. The sample table and the detector arm can rotate at a constant distance 

independently of each other but normal to the plane of incidence. The goniometer varies 

the angle of incidence from 0o to 80o, and the viewing angle can be any value greater than 

5o relative to the incident beam. The absolute reflectance is calculated from 

measurements of the incident and the reflected radiant fluxes obtained by rotating the 

detector arm. The receiver is attached to the goniometer and has a precision aperture, a 

lens, and a photodiode detector, either Si, or a cooled extended InGaAs. The sampling 

aperture, defined by the illumination beam, has a diameter of 17 mm and is located at the 

center of the calibration item. The maximum deviation of any ray within the illumination 

beam from the illumination angle is 0.36o. The entrance pupil of the receiver has an area 

of 796.7 mm2

 and is located 671.4 mm from the sampling aperture, resulting in a solid 

angle of collection of 0.00177 sr. The maximum deviation of any ray within the receiver 

beam from the viewing angle is 1.9o. 

The fundamental quantity for spectral reflectance factor measurements is the bidirectional reflectance distribution function (BRDF). For a fixed linear polarization of 

the incident beam and incident and viewing directions in the same plane, the 

measurement equation for BRDF is given by 

( ) ( ) i r

rir

rir

;

);;,(

;;, Ω⋅ = σλ

θ θ λ σ σλθθ

s

s f (2.17) 

where θi and θr are the incident and reflected angles, respectively, λ is the wavelength, σ

is the polarization, sr and si are the signals recorded by the detector during the 

measurement of the reflected radiance and incident irradiance, respectively, and Ωr is the 

projected solid angle from the center of the area of illumination to the aperture stop of the 

detector. Reflectance factor, R is calculated from the BRDF by 

( ) θ φ θ φ λ;;,;, σ π (θ φ θ φ λ;;,;, σ ) rrii rrii R ⋅= f (2.18) 

3.2 Uncertainty 

Following the procedure for uncertainty analysis presented in Ref. [14], the 

relative expanded uncertainty (k=2) for the 0:45 spectral reflectance factor of Spectralon 
8

is 0.4 %. The components of uncertainly are divided into those arising from random and 

systematic effects. The random effects, which include signal noise, result in a relative 

expanded uncertainty of 0.02 %. The systematic effects, which include the wavelength 

uncertainty, stray light, angle of reflection, aperture stop area, and the distance from the 

sample to the aperture stop, result in a relative expanded uncertainty of 0.2 %. 

3.3 Maintenance of Reference Standards 

The reference standards are measured on STARR every year. The new reflectance 

factors are used in the 0:45 reflectometer measurements. The typical 0:45 spectral 

reflectance factor of a Spectralon reference standard is shown in Figure 3.1. 

300 400 500 600 700 800

0.975

0.980

0.985

0.990

0.995

1.000

1.005

1.010

1.015

1.020

Polarization [deg]

0/45 Reflectance Factor

Wavelength [nm]

 0

 90

Figure 3.1 Typical 0:45 spectral reflectance factor of a Spectralon reference standard. 

4. Reference 0:45 Reflectometer 

4.1 Instrument Description and Properties 

The 0:45 reflectometer consists of a monochromatic illuminator, sample wheel, 

and receiver on a vibration-isolation table and is described in details in ref. [5]. The 

illuminator and receiver are fixed while the sample wheel rotates holding up to 20 

samples. A computer located adjacent to the instrument provides fully automated system 

control, data acquisition, and analysis. A schematic of the 0:45 reflectometer indicating 

the major components is shown in Fig 4.1. 
9

Double Monochomator

Sample Wheel

Receiver

Monitor

Recevier

Lamp

Housing

Figure 4.1 Schematic of the 0:45 reflectometer consisting of a monochromatic 

illuminator, sample wheel, and receivers. 

The illuminator provides a monochromatic, polarized influx beam centered on the 

sample compartment of the sample wheel with a triangular spectral bandwidth of 5 nm at 

full width half maximum (FWHM). The illuminator consists of a dual-source 

illuminator, filter wheel, double monochromator, telecentric aperture, light chopper, 

ellipsoidal and flat mirrors, silica window, monitor diode, and polarizer. 

 The monochromator dual illuminator housing holds a 150 W Xe arc lamp and a 

100 W quartz-tungsten-halogen (QTH) incandescent lamp. The sources are selected by 

rotating a mirror. In general, the Xe lamp is used for wavelengths ≤ 460 nm and the QTH 

lamp is used for wavelengths > 460 nm. The filter wheel has four long-pass filters which 

are used for order-sorting to eliminate higher-order wavelengths from passing through the 

monochromator and is located between the source and entrance slit of the 

monochromator. 

The double additive monochromator has a focal length of half meter and f/# of 

7.3. The gratings are 110 mm square, 600 grooves/mm, and are blazed at 550 nm. The 

mirrors and gratings are coated with aluminum (Al) on a quartz substrate with an overcoating of magnesium fluoride (MgF2). The focal length of the exit mirror is 546 mm 

giving a mirror magnification ratio of 1.1. There is a slit height mask of 4 mm on the exit 

slit cemented to the inside of the slit assembly. A telecentric aperture of diameter 7.23 

mm is attached to the monochromator exit slit assembly at 75 mm ± 0.3 mm from the exit 

slit plane and defines the convergence angle to be ± 0.95°. In a telecentric configuration, 

the aperture stop is placed at the focal point of the lens therefore the central ray of each 

bundle is parallel to the lens axis providing uniform illumination on the sample. An 

optical light chopper is placed after the telecentric aperture. The chopper is used to 

square-wave modulate the intensity of the optical signal at a stable frequency which can 

be amplified easily. 

A 15° off-axis ellipsoidal mirror made of Al and Ni plated with a protective layer 

of MgF2, surface roughness of 2 nm, f1 = 300 mm, and f2 = 900 mm is used to relay the 

exit slit of the monochromator to the sample plane. This mirror has a 177.8 mm focal 
10

length mirror with a 3 times magnification which produces a 10 mm x 12 mm rectangular 

illumination area on the sample. A periscope follows the ellipsoidal mirror to steer the 

beam for alignment and to compensate for the difference in height between the exit slit of 

the monochromator and the sample plane. The polarizer is a Glan-Thompson polarizing 

prism with an extinction ratio of 1 x 10-6 at λ = 633 nm and is used to perform 

measurements for polarizations of the incident beam both parallel and perpendicular to 

the plane of incidence. The monitor diode is the same type as the signal diode described 

below. The purpose is to monitor the influx beam to correct for drift in the output of the 

instrument illuminator. Figure 4.2 shows a top view of the components and optical path 

of the illuminator. 

Double Monochromator

Grating Grating

100 W QTH 150 W Xe

Filter Wheel

Lamp Housing Ellipsoidal Mirror

Chopper

Flat

Mirror

Telecentric 

Aperture

Silica 

Window

Polarizer

Monitor

Diode

Flat

Mirror

Figure 4.2 Top view of the components and optical path of the illuminator. 

Motions of the monochromator components including the grating, port selection, 

and slit width, filter wheel, and source selector are driven by precision stepper motors. 

The grating position is read by an absolute rotary encoder attached to the end of the 

grating lead screw. 

The sample wheel is custom-made and is mounted on a computer control rotation 

stage. The wheel is 81 cm in diameter, fabricated from one piece of aluminum, and has 

20 sample compartments. Each sample compartment can be individually aligned. 

Samples with widths from 3 cm to 10 cm and heights from 3 cm to 20 cm can be 

accommodated. 

The receiver measures the efflux beam from the sample at an angle of 45° and 

consists of a bi-convex lens, a telecentric aperture stop, and a Si photodiode mounted in a 

tube. The receiver aperture is 3.55 mm in diameter and is located 3.5 mm from the 

receiver. The telecentric aperture is sized to provide a receiver half subtended angle of ± 
11

1.36°, to closely match the geometry of STARR. The detector is underfilled with the 

image of the illumination area at an exitance of 45° to the sample plane. A schematic of 

the optical path of the receiver is shown in Fig. 4.3. A summary of the geometrical 

specifications of the 0:45 reflectometer is given in Table 4.1. 

The receiver lens is an achromat mounted with an optimized meniscus lens with 

an effective focal length of 70.5 mm and coated for low reflectance in the visible 

wavelength region. The receiver optical detector is a Si photodiode with UV to visible 

sensitivity and suppressed IR sensitivity and an active area of 10 mm x 10 mm. The 

current from the Si photodiode is converted to voltage by a transimpedance amplifier and 

measured with a lock-in amplifier which auto-ranges. The signal from the test sample is 

normalized to the signal from the reference standard. A signal proportional to the 

reflected radiant flux is measured at each wavelength and each sample for one 

polarization. Then, the polarization is rotated and the measurements are repeated. 

300 mm

Sample

Bi-convex 

lens

Receiver 

aperture

stop Photodiode

70.5 mm

3.5 mm

Figure 4.3 Schematic of the optical path of the receiver. 

Table 4.1 Geometrical specifications of 0:45 reflectometer 

Illuminator Receiver 

Type of geometry Directional Directional (uniplanar) 

Direction of axis 0° ± 0.1° 45° ± 0.1° 

Area 10 mm by 12 mm 20 mm by 24 mm 

Cone half-angle 0.30° 1.36° 

4.2 Characterization of Instrument 

The line spectral function (LSF) was measured by scanning several lines of 

emission lamps with variable entrance and exit slit widths. An example is shown in Fig. 

4.4 of the 546.074 nm line of an Hg emission lamp with variable entrance and exit slit 

widths. An entrance slit width of 3.01 mm, exit slit width of 3.31 mm, and center slit 

width of 4 mm, results in a triangular spectral bandwidth at FWHM of 5 nm. 

The stray-light rejection level was measured using a multimode Ar+ laser 

illuminating an integrating sphere at the front entrance slit. The bandwidth was set to 5 
12

nm with no polarizer in the beam path and a 22.5° mirror placed in one of the sample 

compartments. The stray-light rejection level was determined to be 1 x 10-3 5 nm away 

from the set wavelength and 1 x 10-6 15 nm away from the set wavelength. 

The wavelength scale was calibrated using a 22.5° mirror at the sample position, the 

polarizer was removed from the beam path, and an integrating sphere illuminated with 

pen-ray lamps was placed at the front entrance port. The emission lamps and 

corresponding lines are listed in Table 4.2. These emission lines were selected for their 

signal strength and separation from other emission lines. The front entrance slit was set 

to a width of 1.73 mm, the front exit slit was set to 1.9 mm, and the center slit was set to 

4 mm. Wavelength scale calibration of the monochromator relates the motor step of the 

grating drive to wavelength. The calibration consists of scanning a set of emission lines, 

calculating the motor step centroids of the lines, and fitting the wavelengths of the lines 

to the centroids. Single fits were able to reproduce the wavelengths of the lines to within 

0.01 nm. However, the motor step centroids changed with each scan, and these changes 

were within 0.05 nm. Therefore, the standard uncertainty in wavelength scale is assigned 

to be u(λi) = 0.05 nm at all wavelengths. This provides a conservative estimate of the 

wavelength uncertainty without considering further details about possible wavelength 

dependencies. A summary of the spectral specifications of the illuminator is given in 

Table 4.3. 

Figure 4.4 Spectral line spread function measured by scanning over the 546.074 nm 

Hg lamp emission line using variable exit slit widths as listed in the 

legend. The entrance slit width was 0.909 of the exit slit width and the 

center slit width was 4 mm. 

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

540 542 544 546 548 550 552

Wavelength [nm]

Signal [ 

μV]

0.5 mm

1.1 mm

2.1 mm

3.1 mm

3.3 mm

4 x 105

0

3 x 105

2 x 105

1 x 105

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

540 542 544 546 548 550 552

Wavelength [nm]

Signal [ 

μV]

0.5 mm

1.1 mm

2.1 mm

3.1 mm

3.3 mm

0.E+00

1.E+05

2.E+05

3.E+05

4.E+05

540 542 544 546 548 550 552

Wavelength [nm]

Signal [ 

μV]

0.5 mm

1.1 mm

2.1 mm

3.1 mm

3.3 mm

4 x 105

0

3 x 105

2 x 105

1 x 105
13

Table 4.2 Sources and wavelength of emission lines used for the wavelength 

calibration. 

Source Wavelength [nm] 

Hg 435.834 

Hg 546.074 

Ne 692.947 

Ar 763.510 

Table 4.3 Spectral specifications of 0:45 illuminator 

Property Value 

Wavelength Range 380 nm to 780 nm/ 5nm interval 

Spectral Bandwidth 5 nm, triangular 

Stray-light Rejection less than 1.5 x 10-6

Wavelength Uncertainty 0.05 nm (standard uncertainty) 

Polarization 0o (parallel to plane of illumination) 

90o (perpendicular to plane of illumination) 

Each sample compartment was aligned by placing a mirror and adjusting the 

degrees of freedom so that the reflection was retro-reflected. Differences in the angles of 

illumination and viewing are less than 0.04°, based upon the angular alignment of each 

sample position. In addition, a set of color tiles was measured multiple times in random 

sample compartments of the sample wheel and no changes were found correlated with the 

sample compartments. 

The signals are assumed to be directly proportional to the radiant fluxes reflected 

by the sample and reference standard. The measured signal levels depend on the sample 

being measured. For dark colored samples such as black or deep gray the reflected flux 

is lower than for the reference standard. Therefore, several sensitivity ranges of the lockin-amplifiers are required to cover the dynamic range. The linearity of the photodiodeamplifier-lock-in-amplifier combination was measured at the sensitivity ranges used 

during data acquisition. The signal levels at the detector were adjusted by scanning the 

wavelength of the monochromator to yield radiant flux levels at the different sensitivity 

ranges used in the lock-in-amplifier during measurements. The measured relative 

responsivity of the photodiode-amplifier combination is linear within 0.2 % for the 

sensitivity ranges of 50 μV to 20 mV. 

4.3 Instrument Validation 

 The purpose of validation is to ensure proper operation of the 0:45 Reflectometer 

and is accomplished by repeated measurements of check standards. These measurements 

over time provide the control charts for the instrument to track its performance. The 
14

validation of the instrument includes measurements of reference, check, and comparison 

standards. 

The reference standards are measured on STARR every year for the measurement 

geometry of 0:45 and the wavelength range of 380 nm to 780 nm at 5 nm increments. 

The reference standards are 50.8 mm square Spectralon mounted in a 101.6 mm square 

holder with the front of the Spectralon parallel to the front of the sample holder. 

The check standards for the 0:45 reflectometer are measured every year, or after 

significant changes to the instrument, to verify proper operation of the 0:45 reflectometer. 

The check standards are all fourteen BCRA Series II tiles, serial number NISTCS-1. The 

CIE Lab color space values are determined and recorded in a spreadsheet to distinguish 

any changes in the measured values outside of the expanded uncertainty (k = 2). The 

0:45 spectral reflectance factors of selected check standards are measured during each 

calibration. The selected check standards are the cyan, deep grey and orange tiles of set 

NIST CS-1. The 0:45 spectral reflectance factors of the three selected check standards 

are shown in Fig. 4.5. The measured spectral reflectance factors of the tiles should be 

within the expanded uncertainties of the 0:45 reflectometer. Control charts are 

maintained for measured L, a, and b values of the check standard. Figure 4.6 is a control 

chart for the measured L for the cyan tile. The lines represent the tolerance condition 

which is based on a 95 % confidence interval of the instrument repeatability. 

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

380 430 480 530 580 630 680 730 780

Wavelength [nm]

0/45 Reflectance Factor

Cyan

Deep Gray

Orange

Figure 4.5 Typical 0:45 spectral reflectance factors of cyan, deep gray, and orange 

check standards. 
15

52.90

52.92

52.94

52.96

52.98

53.00

53.02

53.04

53.06

53.08

53.10

0 2 4 6 8 10 12

Time

Measured L*

Figure 4.6 Control chart of the measured L* for the cyan tile. The solid lines 

represent the tolerance condition. 

The comparison standards are measured as needed on both the 0:45 reflectometer 

and on STARR to verify equivalence between the two instruments for neutral colors. 

The comparison standards are five neutral, matte BCRA Series II tiles, serial number NS1, listed in Table 4.4. 

Table 4.4 Comparison Standards 

Sample 

White 

70 % Grey 

Mid Grey 

40 % Grey 

Black 

 The five tiles are placed in the sample compartments of the 0:45 reflectometer 

along with the reference standards. The spectral reflectance factors for each tile are 

determined and each value is compared to the values obtained with STARR. The spectral 

reflectance factors should be within the combined expanded uncertainties due to random 

effects. 
16

4.4 Calibration Services 

 The 0:45 measurements are available at cost on submitted samples or as a set of 

calibrated BCRA Series II color tiles available for purchase. A preliminary discussion 

with NIST staff is required before the samples are submitted. 

 Upon receipt of submitted samples, the samples are inspected and the condition is 

noted. Submitted samples that do not meet the minimum requirements listed in Table 4.5 

require additional arrangements before acceptance. The samples will not be cleaned with 

the exception of dust that can be removed with an air bulb blower. The samples are kept 

in the original measurement container provided by the customer, and powder free gloves 

are used while handling the sample to prevent contamination. 

Table 4.5 Acceptance criteria for 0:45 test samples 

Physical Damage Sample should be undamaged in the measurement area (No 

major scratches, chips, etc.) 

Uniformity The samples should be of uniform appearance in the 

measurement area (i.e. smooth, no stains and no fingerprints). 

Fluorescence Samples should not exhibit fluorescence upon radiation in the 

measurement spectrum (360 nm to 800 nm) 

Cleanliness 

The measurement surface of the sample should be clean. 

Foreign contaminants other than light dust will not be removed 

from the sample surface 

 The availability of BCRA Series II color tiles from NIST is dependent on the 

availability from the supplier. The color tile set consists of fourteen BCRA glossy color 

tiles manufactured by CERAM Research, 10 cm square. The set is divided in a wooden 

box containing twelve colored tiles and a twin box containing a black and a white tile. 

Each tile in the set is identified by its color designation. 

 An example of the calibration report for the set of BCRA Series II tiles is 

provided for reference in Appendix A of this document. This report provides a 

description of the calibration items, the calibration method, results, and general 

information on the calibration. 

5. Uncertainty Analysis 

 The purpose of the 0:45 reflectometer is to provide calibrations of color standards 

of the highest possible quality to establish traceability which requires an estimate of the 

uncertainty [15]. Therefore the uncertainties in the color standard samples provided by 

the NIST 0:45 reflectometer are carefully determined following international accepted 

protocols describe in the ISO Guide to the Expression of Uncertainty in Measurement

[16]. Recent publications have applied the principles of this Guide to uncertainty 

analysis for colorimetry, emphasizing the treatment of correlation [17,18] 
17

The sources of uncertainty associated with the 0:45 reflectometer were identified 

from the characterization of the instrument [5]. The components of uncertainties were 

characterized and analyzed in [19, 20]. The procedure for determining these uncertainties 

follows the approach given in [21], which presents a systematic, analytical method for 

calculating uncertainties. Two key points are emphasized in this method. First, the 

measurement equation, which relates the measured signals to the reflectance factor, is the 

starting point for the analysis (this has been developed in section 2 of this work). Since 

ratios of signals are present in the measurement equation, the spectral properties of the 

reference reflectance reflectometer are required. Second, the effects of correlations both 

between signals at the same wavelength and between reflectance factors at different 

wavelengths must be included in the analysis. Once the uncertainties of all components 

in these measurements have been established, the uncertainties in the other values may be 

determined by the propagation of uncertainties. 

 The systematic, analytical method for calculating uncertainties presented in 

reference [19] provides a step-by-step procedure for analyzing uncertainties, and this 

procedure is followed here. The steps are 

1. Identify all independent sources of uncertainty. 

2. Assign standard uncertainties to these sources of uncertainty. 

3. Calculate the standard uncertainty (called uncertainty contribution) in spectral 

reflectance factor, u(R(λi)), for each source of uncertainty, using the measurement 

equation and taking any correlations between signals at the same wavelength into 

account. 

4. Calculate the correlation coefficient between spectral reflectance factors at different 

wavelengths, r(R(λi), R(λj)), for each source of uncertainty. 

5. Calculate the sensitivity coefficient for each color space value Γ with respect to R(λi). 

6. Calculate the uncertainty contribution in each color space value, u(Γ), using the results 

from steps 3 and 4 for each source of uncertainty. 

7. Calculate the combined standard uncertainty, uc, in each color space value according 

to the uncertainty propagation principle. 

8. Calculate expanded uncertainty, U, with a coverage factor (k = 2). 

5.1 Sources of Uncertainty 

The sources of uncertainty associated with the 0:45 reflectometer are mainly the 

signals from the sample and reference standard, the wavelength of the monochromator, 

bandpass of the monochromator, and the reflectance factor of the reference standard. The 

uncertainties in L*, a*, and b* from these sources of uncertainty are evaluated for a set of 

color tiles, which are available as calibrated standards from NIST. Therefore, in the 

following the terms sample and color tile are synonymous. 

Uncertainties in the L*, a*, and b* values caused by random effects, primarily 

signal noise in both the sample and monitor receivers, are evaluated by Type A methods 

and are given by the standard deviations, u1, of the values from ten repeat measurements. 
18

Uncertainties in the signals caused by systematic effects are evaluated by Type B 

methods and are non-linearity of the signal from the sample relative to the signal from the 

reference standard, and the effect of stray-light on both signals. Mis-alignment of the 

specimen is a possible source of uncertainty and is a result of a difference in angle or 

location, with respect to the reference plane, between the specimen and reference 

standard. Differences in the angles of illumination and viewing are less than 0.04°, based 

upon the angular alignment of each sample position. The changes in reflectance factors 

resulting from these differences were evaluated for each color tile using the reference 

reflectometer, and found to be negligible. Because the receiver is a telecentric optical 

system, and the size of the sampling aperture is less than the field-of-view of the receiver 

at the reference plane, differences in location between the specimen and reference 

standard with respect to the reference plane have no effect on the measured reflectance 

factor of a uniform sample. 

The signals are assumed to be directly proportional to the radiant fluxes reflected 

by the sample and reference standard. However, non-linearity of the detector results in 

the ratio of the signals not being equal to the ratio of the reflected fluxes. Non-linearity 

arises both within the ranges of the current-to-voltage and lock-in amplifiers and between 

ranges of the lock-in amplifier. The relative standard uncertainty in the signal from the 

sample due to non-linearity is 

002.0

),(

)),((1 = σλ

λ σ

ix

ix

S

Su (5.1) 

 Stray-light is radiant flux that reaches the detector at wavelengths outside the 

nominal spectral bandwidth of the instrument, which contributes to the signal. This 

contribution depends upon the spectral distribution of the illuminator, the spectral 

reflectance of both the sample and reference standard, and the responsivity of the 

receiver. The stray-light rejection of the monochromator is less than 10-6 at wavelengths 

10 nm and larger from the set wavelength, and 10-3 at wavelengths 5 nm from the set 

wavelength. Therefore, the estimated error in the signals due to stray-light is given by 

( ) = ∑ [ ++−+⋅ ] −−

'

36 ),5(),5(1010),'(),(

λ

s i s σλσλε s i s i σλσλ (5.2) 

where s is the signal from either the sample or reference standard and the summation is 

over all wavelengths from 350 nm to 1100 nm for the source used at wavelength λi, 

excluding those at wavelengths of λi – 5 nm, λi, and λi + 5 nm. 

 The wavelength calibration of the monochromator was discussed in Section 4.2 

and resulted in a standard uncertainty in wavelength of u(λi) = 0.05 nm at all 

wavelengths. 

The bandpass of the monochromator can cause errors when the spectral 

reflectance factor curve is not linear within the full-bandwidth of the bandpass, and can 

cause significant errors where spectral reflectance factor curves change rapidly, e.g., for 
19

saturated color samples. The measured spectral reflectance factor is a convolution of the 

true reflectance factor and the normalized bandpass function. At each wavelength point, 

measured reflectance factor Rm (λ0 ) is calculated as a product of the true spectral 

reflectance factor 

R(λ) and the normalized bandpass function z0(λ), thus, 

 

Rm (λ0 ) = R(λ0 − λ)

λ

∫ ⋅ z(λ0 − λ) dλ with z(λ) =1

λ

∫ . (5.3) 

An example of the bandpass function z0(λ) is shown in Fig. 5.1. 

0.0

0.5

1.0

1.5

-10 -5 0 5 10 l (nm)

Normalized Bandpass Function

Fig. 5.1 An example of a bandpass function z0(λ). 

To estimate this error without knowing the true reflectance factor R(λ), the measured 

data is used as 

R(λ) for an approximation. The spectral reflectance factor data is first 

interpolated to 1 nm interval, and the z(λ) function (normally a triangular function) also 

prepared at 1 nm intervals, normalized so that the sum of all z(λ) values is equal to 1, 

then a sum-product calculation is performed as given in Eq. (5.3). The relative error due 

to the bandpass can be evaluated by looking at the difference between Rm (λ) and 

R(λ). 

 

ε B,rel(λ) = Rm (λ) − R(λ)

R(λ) (5.4) 

In substitution measurements, this error tends to be cancelled out if the sample and 

reference standard have similar spectral reflectance factor curves. In this case, the relative 

error for the sample ) ( ε samplerel,B, λ and for the reference standard 

εB,rel,std (λ) can be 

calculated, respectively, and the substitution error εB,sub(λ) in the measured test sample is 

given by 

)( )()( ε subB, λ = ε samplerel,B, λ − ε stdrel,B, λ (5.5) 

An example of the calculation of bandpass error for a yellow BCRA tile for a triangular 

bandpass of 5 nm (FWHM) is shown in Fig. 5.2. 
20

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

350 400 450 500 550 600 650 700 750

Wavelength (nm)

Reflectance factor

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

Error (%) in Reflectance Factor

BCRA yellow

 (%)Error

Fig. 5.2 An example of the error due to a 5 nm (FWHM) triangular bandpass of the 

monochromator for the measurement of a yellow BCRA tile. The 

corresponding error in color difference is ΔEab

* = 0.09. 

The error in a color quantity Γ (such as L*, a*, b*) is given by calculating the color 

quantity value from Rm (λ) and 

R(λ), 

))(())(()( ε subB, Rm λ Γ−Γ=Γ R λ (5.6) 

If this error is not corrected, this estimated error is converted to a standard uncertainty by 

3/)()( u2 ε subB, Γ=Γ (5.7) 

The calculation presented above is only for estimation of error. For correction of 

bandpass errors, other methods [22, 23] are available. 

 The relative standard uncertainties due to random and systematic effects during 

the calibration of the reference standards with STARR are 

0002.0

),(

)),((

s

s3 = σλ

λ σ

i

i

R

Ru and (5.8) 

002.0

),(

)),((

s

s4 = σλ

λ σ

i

i

R

Ru , (5.9) 
21

respectively, and are both evaluated using Type B methods for the uncertainty analysis of 

the 0:45 reflectometer. 

5.2 Uncertainty and Correlation of R(λ) 

 Determining the standard uncertainty u(R(λi)) for each source of uncertainty 

caused by systematic effects uses Eqs. (2.1) and (2.2), which relate the reflectance factor 

of the sample to the signals and the reflectance factor of the reference standard, and the 

reflectance factor for unpolarized incident radiant flux to the reflectance factors when the 

incident radiant flux is polarized. Beginning with polarized incident radiant flux, from 

Eq. 2.1, the standard uncertainty due to uncertainties in the signals is given by 

( ) ( ) ( )

( ) ( ) ( )),(),,( ),(

),(

),(

),( 2 

),(

),(

),(

),(

),(

),(

x

x

x

2

s

s

2

2

x

x

2

2

2

σλσλ

σλ

σλ

σλ

σλ

σλ

σλ

σλ

σλ

σλ

σλ

isi

is

is

i

i

i

i

i

i

i

i

ssr

s

su

s

su

s

su

s

su

R

Ru

−

= +

 (5.10) 

For non-linearity of the signal from the sample, u(ss(λi, σ) = 0 and Eq. 5.10 reduces to 

( ) ( )

),(

),(

),(

),( 5 4

σλ

λ σ

σλ

λ σ

ix

ix

i

i

s

su

R

Ru = . (5.11) 

For stray-light effects on both signals, the correlation coefficient r(sx(λi, σ), ss(λi, σ)) = 1 

and Eq. 5.10 reduces to 

( ) ( ) ( )

⎥

⎦

⎤ ⎢

⎣

⎡ = − ),(

),(

),(

),(

),(

),( 6 5 s5

σλ

σλ

σλ

σλ

σλ

σλ

is

i

ix

ix

i

i

s

su

s

su

R

Ru . (5.12) 

The uncertainty in R due to the wavelength uncertainty is also fully correlated between 

the signals from the sample and the reference standard, and the relative standard 

uncertainty is given by 

( ) )( ),(

),(

),(

),( 7 λ

λ

σλ

σλ

σλ

σλ

u

S

S

R

Ru is

ix

is

i ⋅ ∂

⎟

⎠

⎞ ⎜

⎝

⎛ ∂

= . (5.13) 

 Proceeding to unpolarized incident radiant flux, from Eq. 2.2, the reference 

standard uncertainty is given by 
22

( ) ( ) ( )

( )( ) ( ))90,(),0,()90,()0,( 2

1 

)90,( 4

1 )0,( 41 )( 2 2 2

+ ° °°⋅°

= +° °

i i i i

i i i

RRrRuRu

RuRu Ru

λλ λλ

λ λ λ

 (5.14) 

Assuming the sources of uncertainty are completely correlated between polarizations, so 

r(R(λi,0°), R(λi,90°)) = 1 and Eq. 5.14 reduces to 

( ) [ ] ( )( ))90,()0,( 2

1 )( = +° ° Ru λi Ru λi Ru λi . (5.15) 

 From Eq. 5.10, the correlation coefficients r(R(λi), R(λj)) between reflectance 

factors at wavelengths λi and λj are needed for the calculation of uncertainties in L*, a*, 

and b*. For uncertainties caused by non-linearity of the signal, the correlation coefficient 

is one, while for uncertainties caused by stray-light the correlation coefficient is 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⋅ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = − )(

1

)(

1

sgn )(

1

)(

1 sgn))(),((

ix s i jx s j

ji ss ss

RRr

λλ λλ

λλ . (5.16)

However, since ss > sx at all wavelengths for the reference standard used, the correlation 

coefficient is one. For uncertainties caused by wavelength, which are treated as constant 

in magnitude and sign at all wavelengths, the correlation coefficient is 

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

∂

⎟

⎠

⎞ ⎜

⎝

⎛

∂

⋅

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

∂

⎟

⎠

⎞ ⎜

⎝

⎛ ∂

= λ

λ

λ

λ

λ

λ

λλ )(

)(

sgn

)(

)(

sgn))(),(( s

x

s

x

j

j

i

i

ji

s

s

s

s

RRr , (5.17) 

while for uncertainties caused by the reflectance factor of the reference standard, the 

correlation coefficient is zero for those caused by random effects and is assumed to be 

one for those caused by systematic effects. 

5.3 Uncertainties in L*, a*, and b* 

The uncertainties in L*, a*, and b* are obtained by propagating the uncertainties in R(λi) 

using 

))(())(())(),(( )()(2 

))(( )( )(

1

1 1

2

2

1

2

ji i j

j

m

i

m

ij i

i

m

i i

RuRuRRr

RR

Ru

R

u

λλλλ

λλ

λ

λ

⋅ ⋅⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

Γ∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

Γ∂

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

Γ∂ =Γ

∑ ∑

∑

−

= + =

=

(5.18) 
23

where Γ represents any of the color space values, u(R(λi)) is the standard uncertainty of 

the reflectance factor at wavelength λi, and r(R(λi), R(λj)) is the correlation coefficient 

between reflectance factors at different wavelengths. The sensitivity coefficients are 

given by 

 

)( 1

3

116

)(

* 3

1

n

2 iy

i

W

R YY

L λ

λ ⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

, (5.19) 

 

)( 1

3

500 )( 1

3

500

)(

* 3

1

n

2

3

1

n

2 ix iy

i

W

YY

W

R XX

a λ λ

λ ⋅

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −⋅ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

 , and (5.20) 

)( 1

3

200 )( 1

3

200

)(

* 3

1

n

2

3

1

n

2 iy iz

i

W

ZZ

W

R YY

b λ λ

λ ⋅

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −⋅ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

 . (5.21) 

 

 The uncertainties in L*, a*, and b* for a set of color tiles available from NIST as 

calibrated standards were calculated for each source of uncertainty for CIE Standard 

Illuminant D65 and the 1931 CIE 2° Standard Observer. The uncertainties caused by 

random effects were calculated from the standard deviations of L*, a*, and b* from 

repeat measurements of each color tile. Therefore, only systematic effects are considered 

for other sources of uncertainty. The results are given in Table 5.1 for a representative 

set of color tiles. The expanded uncertainty U is the root-sum-square of the standard 

uncertainties from all the sources, multiplied by a coverage factor k = 2. The 

uncertainties for the CIE 1964 Standard Observer are similar to those in Table 5.1. 

Except for the uncertainties caused by repeatability, which change slightly with each set 

of measurements of a color tile set, the other uncertainties are those listed in the 

calibration report that accompanies a set of calibrated color tiles. 

 Color difference and combined color uncertainty are two distinct yet related 

concepts in color space. The following discussion is meant to provide a simple approach 

on the relationship between the two; a full analysis is beyond the scope of this paper. 

Color difference ΔE*ab in the CIELAB color space applies to comparisons between the 

color space values of two different specimens, or to a single specimen measured by two 

different instruments. The color difference is given by 

2 2 2

ab

* Δ+Δ+Δ=Δ baLE *)(*)(*)( , (5.22) 

where ΔL*, Δa*, and Δb* are the differences in color space values. Combined color 

uncertainty ΔE applies to a single measurement on a single specimen, and is given by 

*)(*)(*)( 2 2 2 ++=Δ buauLuE . (5.23) 
24

 The combined color uncertainty assists with interpreting a color difference. Let 

ΔE1 and ΔE2 be the combined color uncertainties of two different specimens, or a single 

specimen measured by two different instruments. If the relationship 

ΔE*ab < 2

2

2

1 Δ+Δ EE )()( is valid, then the color space values agree within the limits 

imposed by the uncertainties. If the relationship is not valid, then the color space values 

disagree. Values of ΔE for all the color tiles are given in Table 5.1. Note that the 

expanded, combined color uncertainties are all less than 0.5, indicating that the design 

goal for the NIST 0:45 reflectometer was met for all the standard color tiles. 

 Since tables of numbers are often difficult to easily interpret, the uncertainties for 

selected color tiles (those used in Ref. [19]) are presented in Fig. 5.3. The two most 

significant differences between the previous analysis presented in ref. [19] and the one 

presented here are the inclusion of the polarization of the incident radiant flux and 

evaluating the effects of signal noise from repeat measurements rather than assuming a 

constant noise of 0.1 % for all signals. Other differences are an uncertainty from an nonlinearity of the signal of 0.2 % rather than 0.1 %, a more sophisticated calculation of the 

contribution to the signals from stray-light rejection level by including signals from both 

sources over the entire wavelength range of responsivity of the silicon detector, and an 

improved wavelength uncertainty of 0.05 nm rather than 0.1 nm. The standard 

uncertainties of the reference standard remained unchanged. 

 

 Several generalities and conclusions can be drawn from Table 5.1 and Figure 5.3. 

For the neutral color tiles (black, deep gray, mid gray, diff gray, pale gray, and white), 

there is negligible uncertainty due to stray-light and wavelength since the spectral shapes 

of the reflectance factors of these samples are similar to that of the reference standard. 

The uncertainties due to signal non-linearity affect only the uncertainty in L* and are 

generally directly proportional to L*, whereas uncertainties due to repeatability affect the 

uncertainties in L*, a*, and b* and are generally inversely proportional to L*. 

 

 For the colored color tiles (deep pink, red, orange, yellow, green, diff green, cyan, 

and deep blue), uncertainties from all the sources contribute to the combined 

uncertainties in L*, a*, and b*. Uncertainties due to repeatability affect all the color 

space values, and are roughly directly proportional to L*, as are uncertainties due to 

signal non-linearity. Uncertainties due to signal stray-light depend on the spectral shape 

of the reflectance factor, and are largest for those colors with rapid changes from low to 

high reflectance factor (red, orange, and yellow). Overall, those color tiles whose 

reflectance factors have a spectral shape similar to that of the reference standard have the 

lowest uncertainties in color space values. 
25

Table 5.1 Standard, u, and expanded (k = 2), U, uncertainties of the CIELAB values for 

each source of uncertainty for CIE Illuminant D65 and the CIE 1931 Standard 

Observer and the indicated color tiles. The standard uncertainties are u1 – 

repeatability, u2 – bandpass of the monochromator, u3 – random effects from 

the calibration of the reference standard, u4 – systematic effects from the 

calibration of the reference standard, u5 – non-linearity of the signals due to 

non-linearity of the detectors, u6 – stray light effects, and u7 – wavelength of 

the monochromator. 

Color 

Tile Value Standard Uncertainty uc U 

 u1 u2 u3 u4 u5 u6 u7 

Black 

L* 5.63 0.025 0.000 0.000 0.014 0.014 0.005 0.000 0.03 0.06

a* -1.36 0.043 0.002 0.001 0.001 0.001 0.005 0.001 0.04 0.09

b* 0.05 0.060 -0.001 0.001 0.000 0.000 0.021 0.000 0.06 0.13

ΔE 0.078 0.002 0.001 0.014 0.014 0.022 0.001 0.08 0.17

Deep 

Gray 

L* 26.97 0.005 0.000 0.001 0.029 0.029 0.004 0.000 0.04 0.08

a* -0.14 0.007 0.003 0.002 0.000 0.000 0.003 0.000 0.01 0.02

b* 1.33 0.032 -0.003 0.001 0.001 0.001 0.017 0.001 0.04 0.07

ΔE 0.033 0.004 0.002 0.029 0.029 0.018 0.001 0.06 0.11

Mid 

Gray 

L* 57.63 0.004 0.000 0.001 0.049 0.049 0.000 0.000 0.07 0.14

a* -0.15 0.014 0.000 0.003 0.000 0.000 0.000 0.000 0.01 0.03

b* 0.55 0.018 -0.003 0.003 0.000 0.000 0.001 0.000 0.02 0.04

ΔE 0.023 0.003 0.004 0.049 0.049 0.001 0.001 0.07 0.15

Diff 

Gray 

L* 58.21 0.003 0.000 0.001 0.049 0.049 0.000 0.000 0.07 0.14

a* -2.50 0.007 0.002 0.003 0.002 0.002 0.000 0.001 0.01 0.02

b* 2.41 0.011 -0.004 0.003 0.002 0.002 0.001 0.004 0.01 0.03

ΔE 0.013 0.004 0.004 0.050 0.050 0.001 0.005 0.07 0.14

Pale 

Gray 

L* 83.43 0.002 0.000 0.001 0.066 0.066 0.000 0.000 0.09 0.19

a* -0.41 0.008 0.000 0.004 0.000 0.000 0.000 0.000 0.01 0.02

b* 0.37 0.011 -0.005 0.003 0.000 0.000 0.001 0.001 0.01 0.02

ΔE 0.014 0.005 0.005 0.066 0.066 0.001 0.001 0.10 0.19
26

Color 

Tile Value Standard Uncertainty uc U 

 u1 u2 u3 u4 u5 u6 u7 

White 

L* 95.99 0.010 0.000 0.001 0.075 0.075 0.000 0.000 0.11 0.21

a* -0.41 0.007 0.001 0.004 0.000 0.000 0.000 0.001 0.01 0.02

b* 1.13 0.010 -0.006 0.004 0.001 0.001 0.001 0.001 0.01 0.02

ΔE 0.015 0.006 0.006 0.075 0.075 0.001 0.001 0.11 0.21

Deep 

Pink 

L* 40.47 0.008 0.006 0.001 0.038 0.038 0.005 0.011 0.06 0.11

a* 30.12 0.022 -0.008 0.002 0.020 0.020 0.006 0.006 0.04 0.08

b* 5.32 0.037 0.004 0.002 0.004 0.004 0.017 0.032 0.05 0.11

ΔE 0.044 0.011 0.003 0.043 0.043 0.018 0.035 0.09 0.17

Red 

L* 38.00 0.011 0.013 0.001 0.036 0.036 0.011 0.023 0.06 0.12

a* 52.65 0.015 -0.019 0.002 0.035 0.035 0.015 0.032 0.06 0.12

b* 36.56 0.032 0.018 0.002 0.025 0.025 0.140 0.037 0.15 0.31

ΔE 0.037 0.029 0.002 0.056 0.056 0.141 0.049 0.18 0.35

Orange 

L* 65.89 0.010 0.002 0.001 0.055 0.055 0.005 0.022 0.08 0.16

a* 40.35 0.025 -0.030 0.002 0.027 0.027 0.005 0.036 0.07 0.13

b* 60.88 0.086 -0.001 0.002 0.041 0.041 0.085 0.035 0.14 0.28

ΔE 0.090 0.030 0.004 0.073 0.073 0.086 0.055 0.17 0.35

Yellow 

L* 83.63 0.008 -0.004 0.001 0.066 0.066 0.002 0.010 0.09 0.19

a* -4.49 0.027 0.013 0.004 0.003 0.003 0.006 0.027 0.04 0.08

b* 84.87 0.039 -0.050 0.002 0.057 0.057 0.087 0.026 0.14 0.27

ΔE 0.048 0.052 0.005 0.087 0.087 0.087 0.039 0.17 0.34

Green 

L* 53.44 0.012 -0.002 0.001 0.046 0.046 0.001 0.008 0.07 0.13

a* -35.79 0.044 0.036 0.003 0.024 0.024 0.005 0.012 0.07 0.13

b* 15.80 0.036 -0.023 0.002 0.011 0.011 0.010 0.043 0.06 0.13

ΔE 0.058 0.043 0.004 0.053 0.053 0.011 0.045 0.11 0.23

Diff 

Green 

L* 53.66 0.013 -0.002 0.001 0.046 0.046 0.001 0.007 0.07 0.13

a* -35.77 0.034 0.037 0.003 0.024 0.024 0.005 0.009 0.06 0.12

b* 19.81 0.026 -0.025 0.002 0.013 0.013 0.013 0.042 0.06 0.12

ΔE 0.044 0.044 0.004 0.054 0.054 0.014 0.044 0.11 0.22

Cyan 

L* 50.88 0.012 0.004 0.001 0.045 0.045 0.002 0.015 0.07 0.13

a* -13.64 0.048 0.000 0.003 0.009 0.009 0.003 0.043 0.07 0.13

b* -33.00 0.030 0.011 0.003 0.022 0.022 0.005 0.032 0.06 0.11

ΔE 0.058 0.012 0.004 0.051 0.051 0.006 0.055 0.11 0.22

Deep 

Blue 

L* 9.95 0.005 0.003 0.000 0.017 0.017 0.014 0.004 0.03 0.06

a* 22.65 0.119 0.000 0.002 0.015 0.015 0.023 0.025 0.13 0.25

b* -32.64 0.108 -0.005 0.001 0.022 0.022 0.008 0.029 0.12 0.23

ΔE 0.161 0.006 0.002 0.032 0.032 0.029 0.038 0.17 0.35
27

Figure 5.3 Standard uncertainties in L*, a*, and b* caused by the indicated sources of 

uncertainty for selected samples for CIE Standard Illuminant D65 and the CIE 

1964 Standard Observer. The standard uncertainties are u1 – repeatability, u2

– bandpass of the monochromator, u3 – random effects from the calibration of 

the reference standard, u4 – systematic effects from the calibration of the 

reference standard, u5 – non-linearity of the signals due to non-linearity of the 

detectors, u6 – stray light effects, and u7 – wavelength of the monochromator. 

Red

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

u1 u2 u3 u4 u5 u6 u7

Source of Uncertainty

Standard Uncertainty

L*

a*

b*

Mid Gray

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Source of Uncertainty

Standard Uncertainty

L*

a*

b*

u1 u2 u3 u4 u5 u6 u7

Red

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

u1 u2 u3 u4 u5 u6 u7

Source of Uncertainty

Standard Uncertainty

L*

a*

b*

Mid Gray

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

Source of Uncertainty

Standard Uncertainty

L*

a*

b*

u1 u2 u3 u4 u5 u6 u7
28

Acknowledgments 

 

The success of the measurement service for 0:45 Color measurements was aided 

by the efforts of Dr. Gerald Fraser and Dr. Yoshihiro Ohno. 
29

References 

1. D. B. Judd and G. Wyszecki, Color in Business, Science, and Industry (3rd ed.), John 

Wiley & Sons, Inc., New York, 1975. 

2. D. B. Judd, “Contributions to Color Science”, NBS Special Publication 545, U. S. 

Dept of Commerce (1979). 

3. NIST Calibration Program: Calibration Services Users Guide, SP 250 Appendix Fee 

Schedule 2007. 

4. ISO/IEC Guide 17025:2005, “General Requirements for the Competence of Testing 

and Calibration Laboratories.” 

5. M. E. Nadal and E. A. Early, “0:45 NIST Reference Reflectometer,” accepted for 

publication in Color Res. Appl 

6. R.W.G. Hunt, “Measuring Color”, 3rd ed”, Fountain Press, England, 1998. 

7. G. Wyszecki and W.S. Styles “Color Science: Concepts and Methods, Quantitative 

Data and Formulae” Wiley-Interscience, 2000. 

8. Standard Terminology of Appearance, E284. West Conshohocken, Pennsylvania: 

American Society for Testing and Materials; 1999.ASTM “Standards on Color and 

Appearance Measurements, 6th edition”, American Society for Testing and Materials, 

2000. 

9. CIE Publication no. 17.4, CIE International Lighting Vocabulary (1987). 

10. Standard Practice for Computing the Colors of Objects by Using the CIE System, 

E308. West Conshohocken, Pennsylvania: American Society for Testing and 

Materials; 2000. 

11. CIE Publication No. 15:2004, Colorimetry, Vienna, Austria. 

12. CIE S014-1/E:2006: Colorimetry – Part I: CIE Standard Colorimetric Observer. 

13. CIE S014-2/E:2006: Colorimetry – Part II: CIE Standard Illuminants. 

14. P. Y. Barnes, E. A. Early, and A. C. Parr, “NIST Measurement Services: Spectral 

Reflectance,” NIST Spec. Pub. 250-48, U. S. Dept. of Commerce (1998). 

15. ISO, International Vocabulary of Basic and General Terms in Metrology, second 

edition (International Organization for Standardization, Geneva, Switzerland, 1993), 

definition 6.10. 

16. Guide to the Expression of Uncertainty in measurement. Geneva: International 

Organization for Standardization; 1993. 

17. J. L. Gardner and R. B. Frenkel, “Correlation coefficients for Tristimulus Response 

Value Uncertainties,” Metrologia, 36, 477-480 (1999). 

18. J. L. Gardner, “Uncertainty Estimation in Colour Measurement,” Color Res. Appl., 

25, 349-355 (2000). 

19. E. A. Early and M. E. Nadal, “Uncertainty Analysis for Reflectance Colorimetry,” 

Color Res. Appl., 29, 205-216 (2004). 

20. E. A. Early and M. E. Nadal, “Uncertainty Analysis for the 0:45 Reference 

Reflectometer,” accepted for publication in Color Res. Appl. 

21. B. N. Taylor and C. E. Kuyatt, “Guidelines for Evaluating and Expressing the 

Uncertainty of NIST Measurement Results,” Natl. Inst. Stand. Technol. Tech. Note 

1297 (1994). 
30

22. E. I. Stearns and R. E. Stearns, “An Example of a Method for Correcting Radiance 

Data for Bandpass Error,” Color Res. Appl. 13-4, 257-259 (1988). 

23. Y. Ohno, “A Flexible Bandpass Correction Method for Spectrometers,” Proc. AIC’05 

Granada, Spain, May 9-13, 2005, pp. 697-700 (2005). 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

1

Appendix A 

REPORT OF CALIBRATION 

38091S 0:45 Surface Color 

for 

Color Tile Set 

Submitted by: 

Any Company, Inc. 

Attn.: Ms. Jane Doe 

123 Calibration Street 

Measurement City, MD 20800-1234 

(See your Purchase Order No. 12345, dated January 1, 2006) 

1. Description of Calibration Items 

One color tile set consisting of fourteen BCRA glossy color tiles manufactured by 

CERAM Research, 10 cm square, set number NISTCS-XX, supplied by NIST. The set is 

divided into a wooden box containing twelve colored tiles and a twin box containing a 

black and a white tile. Each tile in the set is identified by its color designation. 

2. Description of Calibration 

The surface color of an item depends on the spectral power distribution of the illuminant, 

the spectral responsivity of the observer, the geometrical conditions of illumination and 

observation, and the optical properties of the item. Only the last factor depends on the 

item, and is quantified by the spectral reflectance factor. The spectral reflectance factor R

is the ratio of the radiant flux reflected by the item to the radiant flux that would be 

reflected by a perfect Lambertian diffuser, under identical measurement conditions, as a 

function of wavelength. The designation 0:45 refers to the bi-directional geometry of the 

measurement, where the illumination angle is 0o and the viewing angle is 45o. 

The calibration items were measured using the NIST 0:45 Reference Reflectometer. A 

spherical mirror focuses radiant flux from a source through an order-sorting filter onto 

the entrance slit of a 1⁄2 m double-grating monochromator. The beam emerging from the 

exit slit of the monochromator is incident on a circular aperture and focused by an 

elliptical mirror in a telecentric arrangement, passes through a Glan-Taylor polarizer, and 

is incident upon the item. Radiant flux reflected from the item is collected by a receiver 

consisting of a lens and circular aperture in a telecentric arrangement and a Si 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

2

photodiode. A sample wheel with twenty positions contains the measured items. At each 

wavelength and polarization, signals are measured from each position of the sample 

wheel containing an item. These signals are proportional to the reflected radiant flux. 

Since the receiver uses a lock-in amplifier to measure the signal from the detector, no 

dark signals are measured. 

Measuring the 0:45 spectral reflectance factor of the calibration items is a relative 

measurement and therefore requires comparison to a transfer standard with a calibrated 

0:45 spectral reflectance factor. The transfer standard is Spectralon, a sintered 

polytetrafluoroethylene (PTFE) manufactured by Labsphere, Inc., whose 0:45 spectral 

reflectance factor was determined using the NIST Spectral Tri-function Automated 

Reference Reflectometer (STARR) [1]. 

Three Spectralon standards and the calibration items were cleaned with an air bulb and 

mounted in the sample wheel with the front surfaces aligned normal to the illumination 

beam and at the same distance from the receiver aperture. 

The sampling aperture, defined by the illumination beam, was 10 mm by 12 mm and was 

located at the center of the reference standards and calibration items. The angle of 

illumination was 0o from the normal of the item, and the maximum deviation of any ray 

within the illumination beam from the angle of illumination was 0.92o. The angle of 

viewing was 45o from the normal of the item, the solid angle of collection was 

0.00176 sr, and the maximum deviation of any ray within the receiver beam from the 

angle of viewing was 1.36o. This geometry is designated as 0:45. 

The reference standards and calibration items were measured at wavelengths from 

380 nm to 780 nm every 5 nm. The spectral bandwidth of the illumination beam was 

5 nm. The source was a Xe arc lamp for wavelengths of 460 nm and shorter and a 

quartz-tungsten-halogen incandescent lamp for longer wavelengths. Each reference 

standard and calibration item was measured three times for polarizations of the 

illumination beam both parallel and perpendicular to the plane of illumination 

 

3. Results of Calibration 

The reflectance factor R at each wavelength λ and polarization σ is given by 

 

),( ),(

),( ),( σλ

σλ

λ σ σλ s

s

R

S

S R ⋅= , 3.1 

where S and Ss are the measured signals from the item and reference standard, 

respectively, and Rs is the reflectance factor of the reference standard. 

The final 0:45 spectral reflectance factor was obtained by averaging the values from 

multiple scans. The certified 0:45 spectral reflectance factor of each calibration item, 

measured according to the details given above, is given in Tables 1 to 14. 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

3

The tristimulus values X, Y, and Z of each item were calculated for CIE standard 

illuminant D65 and CIE 1931 standard observer [2]. The chromaticity coordinates x and 

y were derived from the tristimulus values [3]. 

The tristimulus values, for reflectance factors regularly spaced in wavelength, are given 

by 

∑=

⋅=

m

i

RWX ii

1

x, , 3.2

∑=

⋅=

m

i

RWY ii

1

y, , and 3.3 

∑=

⋅=

m

i

RWZ ii

1

z, , 3.4 

where i indexes the wavelengths, Wx,i, Wy,i, and Wz,i are the normalized tristimulus 

weighting factors for a particular illuminant and observer combination, and Ri is the 

spectral reflectance factor. The chromaticity coordinates are given by 

ZYX

X

x

++ = and 3.5

 

ZYX

Y

y

++ = . 3.6 

The tristimulus value Y and chromaticity coordinates x and y for CIE standard illuminant 

D65 and CIE 1931 standard observer of each of the calibrated items is given in Table 15. 

Uncertainties were calculated according to the procedures outlined in Reference 4 and 5]. 

From the measurement equation for the spectral reflectance factor, Eq. (3.1), 

uncertainties in S, Ss, Rs, and λ all contribute to an uncertainty in R. An important 

consideration for determining the uncertainty in R is correlations between signals S and Ss

at the same wavelength. The uncertainties in Y, x, and y were calculated by propagating 

the uncertainties in the spectral reflectance factor thorough the calculations, including 

correlations between the uncertainties in R at different wavelengths. 

The sources of uncertainty are signal noise, offset in the signal, and uncertainties in the 

reference standard and the wavelength. Signal noise is the result of random processes in 

both the source and the detector. Offsets in the signal include non-linearity and stray 

light. Non-linearity is a deviation of the signal from proportionality with the radiant flux 

as the signal changes value. Stray light is radiant flux that reaches the detector at 

wavelengths outside the nominal spectral bandwidth of the instrument. The reference 

standard has an uncertainty from its calibration, and there is an uncertainty associated 

with the wavelength setting of the measuring instrument. 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

4

The uncertainties from signal noise include source stability and detector noise. The 

uncertainty contributions caused by these effects were evaluated by multiple 

measurements of the items. The uncertainties from non-linearity, stray light, wavelength, 

and reference standard were evaluated as part of the characterization of the 0:45 

Reference Reflectometer and their values are given in Table 16. 

The expanded uncertainty was obtained from the root-sum-square of the uncertainty 

contributions multiplied by a coverage factor k = 2. The expanded uncertainty in 0:45 

spectral reflectance factor of each calibration item is given in Tables 1 to 14. The 

expanded uncertainties in tristimulus value and chromaticity coordinates of each 

calibration item are given in Table 15. 

4. General Information 

1) The values in Tables 1 to 15 apply only to the central 10 mm by 12 mm area of the 

calibration item. 

2) This calibration report may not be reproduced except in full without the written 

consent of this Laboratory. 

Prepared by: Approved by: 

Maria E. Nadal Gerald T. Fraser 

Optical Technology Division For the Director, 

Physics Laboratory National Institute of 

(301) 975-4632 Standards and Technology 

(301) 975-3797 

References: 

[1] P. Y. Barnes, E. A. Early, and A. C. Parr, “NIST Measurement Services: Spectral 

Reflectance,” NIST Special Publication 250-48 (1998). 

[2] Commission International de l’Eclairage: International Lighting Vocabulary, 

Colorimetry, 2nd ed., Publ. No. 15.2 (1986). 

[3] ASTM E308, “Standard Practice for Computing the Colors of Objects by Using the 

CIE System Gloss,” in ASTM standards on Color and Appearance Measurement, 

6th ed., (ASTM, Philadelphia, PA, 2000). 

[4] B. N. Taylor and C. E. Kuyatt, “Guidelines for Evaluating and Expressing the 

Uncertainty of NIST Measurement Results,” NIST Technical Note 1297 (1994). 

[5] E. A. Early and M. E. Nadal, “Uncertainty Analysis for Reflectance Colorimetry,” 

Color Res. Appl. 29:205-216 (2004). 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

5

Table 1. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Black color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.0051 0.0007 515 0.0067 0.0003 650 0.0058 0.0001 

385 0.0055 0.0012 520 0.0067 0.0001 655 0.0059 0.0001 

390 0.0056 0.0011 525 0.0067 0.0001 660 0.0060 0.0001 

395 0.0056 0.0008 530 0.0068 0.0000 665 0.0062 0.0001 

400 0.0059 0.0008 535 0.0070 0.0001 670 0.0066 0.0001 

405 0.0059 0.0007 540 0.0069 0.0001 675 0.0070 0.0001 

410 0.0062 0.0003 545 0.0070 0.0002 680 0.0078 0.0001 

415 0.0061 0.0002 550 0.0068 0.0002 685 0.0088 0.0001 

420 0.0060 0.0003 555 0.0067 0.0001 690 0.0100 0.0001 

425 0.0061 0.0004 560 0.0065 0.0001 695 0.0115 0.0001 

430 0.0061 0.0002 565 0.0062 0.0001 700 0.0133 0.0001 

435 0.0061 0.0000 570 0.0060 0.0001 705 0.0151 0.0001 

440 0.0062 0.0001 575 0.0058 0.0002 710 0.0171 0.0001 

445 0.0062 0.0001 580 0.0056 0.0002 715 0.0190 0.0002 

450 0.0062 0.0001 585 0.0056 0.0001 720 0.0209 0.0002 

455 0.0063 0.0001 590 0.0055 0.0001 725 0.0229 0.0002 

460 0.0062 0.0004 595 0.0054 0.0000 730 0.0247 0.0002 

465 0.0063 0.0005 600 0.0053 0.0001 735 0.0265 0.0002 

470 0.0062 0.0002 605 0.0052 0.0001 740 0.0282 0.0003 

475 0.0064 0.0007 610 0.0051 0.0001 745 0.0298 0.0002 

480 0.0062 0.0004 615 0.0052 0.0001 750 0.0313 0.0002 

485 0.0061 0.0002 620 0.0053 0.0001 755 0.0329 0.0002 

490 0.0062 0.0004 625 0.0055 0.0001 760 0.0344 0.0002 

495 0.0062 0.0005 630 0.0057 0.0001 765 0.0359 0.0002 

500 0.0063 0.0004 635 0.0057 0.0001 770 0.0373 0.0002 

505 0.0063 0.0002 640 0.0058 0.0001 775 0.0388 0.0003 

510 0.0065 0.0001 645 0.0059 0.0001 780 0.0404 0.0003 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

6

Table 2. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Cyan color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.1665 0.0020 515 0.3246 0.0019 650 0.0712 0.0004 

385 0.1911 0.0017 520 0.3007 0.0019 655 0.0722 0.0004 

390 0.2164 0.0018 525 0.2769 0.0017 660 0.0735 0.0004 

395 0.2404 0.0030 530 0.2540 0.0017 665 0.0753 0.0004 

400 0.2627 0.0024 535 0.2322 0.0014 670 0.0776 0.0005 

405 0.2846 0.0020 540 0.2121 0.0013 675 0.0804 0.0005 

410 0.3050 0.0021 545 0.1935 0.0012 680 0.0838 0.0005 

415 0.3235 0.0023 550 0.1767 0.0011 685 0.0877 0.0005 

420 0.3413 0.0021 555 0.1618 0.0010 690 0.0922 0.0006 

425 0.3570 0.0027 560 0.1483 0.0009 695 0.0974 0.0006 

430 0.3719 0.0022 565 0.1365 0.0008 700 0.1032 0.0006 

435 0.3857 0.0023 570 0.1260 0.0008 705 0.1095 0.0006 

440 0.3984 0.0026 575 0.1168 0.0007 710 0.1159 0.0007 

445 0.4095 0.0024 580 0.1088 0.0007 715 0.1218 0.0007 

450 0.4200 0.0025 585 0.1018 0.0006 720 0.1269 0.0007 

455 0.4287 0.0025 590 0.0958 0.0006 725 0.1311 0.0007 

460 0.4370 0.0026 595 0.0906 0.0005 730 0.1342 0.0008 

465 0.4416 0.0027 600 0.0862 0.0005 735 0.1364 0.0008 

470 0.4432 0.0026 605 0.0824 0.0005 740 0.1376 0.0008 

475 0.4428 0.0026 610 0.0792 0.0005 745 0.1381 0.0008 

480 0.4387 0.0028 615 0.0766 0.0005 750 0.1380 0.0008 

485 0.4316 0.0025 620 0.0746 0.0004 755 0.1374 0.0008 

490 0.4208 0.0026 625 0.0729 0.0004 760 0.1365 0.0008 

495 0.4069 0.0024 630 0.0717 0.0004 765 0.1352 0.0008 

500 0.3900 0.0023 635 0.0710 0.0004 770 0.1340 0.0008 

505 0.3700 0.0022 640 0.0707 0.0004 775 0.1326 0.0008 

510 0.3480 0.0022 645 0.0708 0.0004 780 0.1314 0.0008 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

7

Table 3. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Deep Blue color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.1169 0.0038 515 0.0116 0.0003 650 0.0078 0.0002 

385 0.1253 0.0023 520 0.0114 0.0001 655 0.0080 0.0001 

390 0.1322 0.0013 525 0.0113 0.0003 660 0.0085 0.0001 

395 0.1379 0.0016 530 0.0114 0.0001 665 0.0092 0.0001 

400 0.1407 0.0010 535 0.0116 0.0002 670 0.0101 0.0001 

405 0.1410 0.0012 540 0.0120 0.0001 675 0.0117 0.0001 

410 0.1388 0.0010 545 0.0121 0.0001 680 0.0142 0.0001 

415 0.1336 0.0015 550 0.0121 0.0001 685 0.0182 0.0002 

420 0.1256 0.0010 555 0.0115 0.0001 690 0.0246 0.0003 

425 0.1160 0.0012 560 0.0106 0.0002 695 0.0342 0.0030 

430 0.1050 0.0011 565 0.0096 0.0001 700 0.0508 0.0005 

435 0.0933 0.0010 570 0.0088 0.0001 705 0.0737 0.0007 

440 0.0816 0.0009 575 0.0080 0.0001 710 0.1049 0.0010 

445 0.0703 0.0009 580 0.0075 0.0001 715 0.1444 0.0012 

450 0.0593 0.0008 585 0.0072 0.0001 720 0.1913 0.0015 

455 0.0500 0.0007 590 0.0070 0.0001 725 0.2436 0.0019 

460 0.0399 0.0008 595 0.0070 0.0001 730 0.2981 0.0021 

465 0.0329 0.0008 600 0.0072 0.0000 735 0.3519 0.0024 

470 0.0273 0.0007 605 0.0073 0.0001 740 0.4012 0.0026 

475 0.0232 0.0004 610 0.0074 0.0001 745 0.4440 0.0027 

480 0.0197 0.0018 615 0.0075 0.0001 750 0.4785 0.0028 

485 0.0178 0.0003 620 0.0075 0.0001 755 0.5042 0.0029 

490 0.0162 0.0003 625 0.0076 0.0001 760 0.5221 0.0030 

495 0.0148 0.0005 630 0.0075 0.0000 765 0.5331 0.0034 

500 0.0137 0.0003 635 0.0075 0.0000 770 0.5379 0.0031 

505 0.0124 0.0011 640 0.0076 0.0002 775 0.5392 0.0031 

510 0.0121 0.0002 645 0.0074 0.0007 780 0.5368 0.0031 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

8

Table 4. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Deep Gray color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.0452 0.0028 515 0.0491 0.0003 650 0.0540 0.0003 

385 0.0459 0.0019 520 0.0496 0.0003 655 0.0543 0.0003 

390 0.0466 0.0011 525 0.0501 0.0003 660 0.0551 0.0003 

395 0.0470 0.0011 530 0.0507 0.0003 665 0.0564 0.0003 

400 0.0477 0.0008 535 0.0513 0.0004 670 0.0585 0.0003 

405 0.0477 0.0006 540 0.0520 0.0003 675 0.0615 0.0004 

410 0.0479 0.0004 545 0.0525 0.0003 680 0.0654 0.0004 

415 0.0480 0.0006 550 0.0527 0.0003 685 0.0702 0.0004 

420 0.0479 0.0005 555 0.0526 0.0003 690 0.0758 0.0005 

425 0.0479 0.0003 560 0.0522 0.0003 695 0.0819 0.0005 

430 0.0478 0.0004 565 0.0516 0.0003 700 0.0884 0.0005 

435 0.0478 0.0004 570 0.0510 0.0003 705 0.0954 0.0006 

440 0.0479 0.0003 575 0.0507 0.0003 710 0.1028 0.0006 

445 0.0479 0.0003 580 0.0505 0.0003 715 0.1104 0.0007 

450 0.0479 0.0003 585 0.0503 0.0003 720 0.1182 0.0007 

455 0.0482 0.0003 590 0.0503 0.0003 725 0.1262 0.0007 

460 0.0482 0.0008 595 0.0501 0.0003 730 0.1343 0.0008 

465 0.0485 0.0010 600 0.0498 0.0003 735 0.1426 0.0009 

470 0.0483 0.0009 605 0.0495 0.0003 740 0.1507 0.0009 

475 0.0484 0.0006 610 0.0495 0.0003 745 0.1585 0.0009 

480 0.0484 0.0006 615 0.0499 0.0003 750 0.1657 0.0010 

485 0.0484 0.0003 620 0.0507 0.0003 755 0.1721 0.0010 

490 0.0484 0.0004 625 0.0515 0.0003 760 0.1779 0.0010 

495 0.0483 0.0004 630 0.0523 0.0003 765 0.1827 0.0011 

500 0.0484 0.0003 635 0.0529 0.0003 770 0.1870 0.0011 

505 0.0487 0.0004 640 0.0534 0.0003 775 0.1909 0.0011 

510 0.0488 0.0004 645 0.0537 0.0003 780 0.1946 0.0012 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

9

Table 5. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Deep Pink color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.1483 0.0034 515 0.0648 0.0005 650 0.3584 0.0021 

385 0.1487 0.0018 520 0.0650 0.0004 655 0.3765 0.0022 

390 0.1489 0.0017 525 0.0658 0.0004 660 0.3944 0.0023 

395 0.1471 0.0013 530 0.0672 0.0005 665 0.4114 0.0024 

400 0.1459 0.0012 535 0.0694 0.0005 670 0.4276 0.0026 

405 0.1435 0.0011 540 0.0723 0.0004 675 0.4428 0.0026 

410 0.1400 0.0015 545 0.0758 0.0004 680 0.4567 0.0026 

415 0.1358 0.0011 550 0.0800 0.0005 685 0.4690 0.0028 

420 0.1311 0.0013 555 0.0849 0.0005 690 0.4797 0.0028 

425 0.1260 0.0011 560 0.0905 0.0005 695 0.4899 0.0028 

430 0.1207 0.0008 565 0.0970 0.0006 700 0.4990 0.0028 

435 0.1154 0.0009 570 0.1046 0.0008 705 0.5064 0.0033 

440 0.1102 0.0008 575 0.1129 0.0007 710 0.5138 0.0031 

445 0.1052 0.0006 580 0.1226 0.0007 715 0.5191 0.0036 

450 0.1002 0.0007 585 0.1335 0.0008 720 0.5236 0.0030 

455 0.0957 0.0008 590 0.1458 0.0009 725 0.5271 0.0030 

460 0.0901 0.0008 595 0.1593 0.0010 730 0.5290 0.0031 

465 0.0861 0.0007 600 0.1742 0.0012 735 0.5308 0.0032 

470 0.0824 0.0007 605 0.1902 0.0012 740 0.5310 0.0030 

475 0.0789 0.0008 610 0.2073 0.0013 745 0.5305 0.0031 

480 0.0759 0.0008 615 0.2252 0.0014 750 0.5296 0.0031 

485 0.0732 0.0006 620 0.2438 0.0015 755 0.5282 0.0030 

490 0.0696 0.0063 625 0.2630 0.0015 760 0.5264 0.0030 

495 0.0688 0.0004 630 0.2822 0.0017 765 0.5241 0.0031 

500 0.0672 0.0005 635 0.3016 0.0018 770 0.5217 0.0030 

505 0.0646 0.0055 640 0.3209 0.0019 775 0.5187 0.0030 

510 0.0651 0.0005 645 0.3399 0.0020 780 0.5158 0.0029 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

10

Table 6. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Diff Gray color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.2002 0.0026 515 0.2663 0.0014 650 0.2538 0.0015 

385 0.2063 0.0024 520 0.2668 0.0015 655 0.2534 0.0015 

390 0.2100 0.0013 525 0.2670 0.0015 660 0.2538 0.0015 

395 0.2137 0.0014 530 0.2672 0.0015 665 0.2542 0.0015 

400 0.2173 0.0018 535 0.2671 0.0015 670 0.2549 0.0014 

405 0.2203 0.0016 540 0.2672 0.0015 675 0.2557 0.0015 

410 0.2236 0.0014 545 0.2668 0.0015 680 0.2567 0.0015 

415 0.2264 0.0015 550 0.2661 0.0015 685 0.2577 0.0015 

420 0.2292 0.0014 555 0.2651 0.0015 690 0.2586 0.0015 

425 0.2322 0.0015 560 0.2637 0.0015 695 0.2596 0.0015 

430 0.2348 0.0015 565 0.2622 0.0015 700 0.2605 0.0015 

435 0.2377 0.0015 570 0.2606 0.0015 705 0.2610 0.0015 

440 0.2405 0.0015 575 0.2590 0.0015 710 0.2612 0.0015 

445 0.2433 0.0014 580 0.2577 0.0014 715 0.2612 0.0015 

450 0.2459 0.0015 585 0.2566 0.0015 720 0.2611 0.0015 

455 0.2483 0.0015 590 0.2562 0.0015 725 0.2607 0.0015 

460 0.2508 0.0015 595 0.2561 0.0015 730 0.2602 0.0015 

465 0.2530 0.0015 600 0.2561 0.0016 735 0.2595 0.0014 

470 0.2547 0.0015 605 0.2561 0.0015 740 0.2585 0.0015 

475 0.2567 0.0015 610 0.2562 0.0015 745 0.2579 0.0015 

480 0.2588 0.0015 615 0.2561 0.0015 750 0.2569 0.0014 

485 0.2603 0.0014 620 0.2560 0.0015 755 0.2557 0.0014 

490 0.2620 0.0015 625 0.2557 0.0015 760 0.2546 0.0014 

495 0.2634 0.0015 630 0.2554 0.0015 765 0.2534 0.0015 

500 0.2645 0.0015 635 0.2548 0.0015 770 0.2521 0.0014 

505 0.2655 0.0015 640 0.2544 0.0015 775 0.2509 0.0014 

510 0.2661 0.0015 645 0.2543 0.0015 780 0.2496 0.0014 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

11

Table 7. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Diff Green color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.0511 0.0027 515 0.3366 0.0020 650 0.1127 0.0006 

385 0.0544 0.0020 520 0.3328 0.0020 655 0.1139 0.0006 

390 0.0565 0.0020 525 0.3225 0.0019 660 0.1157 0.0007 

395 0.0584 0.0009 530 0.3079 0.0018 665 0.1182 0.0006 

400 0.0604 0.0009 535 0.2908 0.0018 670 0.1212 0.0007 

405 0.0624 0.0008 540 0.2731 0.0016 675 0.1249 0.0007 

410 0.0641 0.0008 545 0.2549 0.0015 680 0.1293 0.0007 

415 0.0661 0.0007 550 0.2377 0.0014 685 0.1345 0.0008 

420 0.0685 0.0006 555 0.2215 0.0013 690 0.1402 0.0008 

425 0.0712 0.0006 560 0.2066 0.0012 695 0.1468 0.0008 

430 0.0746 0.0006 565 0.1931 0.0011 700 0.1542 0.0009 

435 0.0787 0.0006 570 0.1808 0.0010 705 0.1620 0.0009 

440 0.0835 0.0007 575 0.1696 0.0011 710 0.1697 0.0010 

445 0.0893 0.0007 580 0.1599 0.0009 715 0.1769 0.0010 

450 0.0966 0.0008 585 0.1513 0.0009 720 0.1830 0.0010 

455 0.1051 0.0010 590 0.1438 0.0008 725 0.1880 0.0011 

460 0.1182 0.0009 595 0.1373 0.0008 730 0.1914 0.0011 

465 0.1315 0.0012 600 0.1318 0.0008 735 0.1939 0.0011 

470 0.1479 0.0012 605 0.1269 0.0007 740 0.1953 0.0011 

475 0.1675 0.0012 610 0.1229 0.0007 745 0.1958 0.0011 

480 0.1901 0.0015 615 0.1195 0.0007 750 0.1956 0.0011 

485 0.2159 0.0015 620 0.1168 0.0007 755 0.1949 0.0011 

490 0.2440 0.0016 625 0.1147 0.0006 760 0.1938 0.0011 

495 0.2730 0.0018 630 0.1133 0.0006 765 0.1923 0.0011 

500 0.2991 0.0020 635 0.1123 0.0006 770 0.1907 0.0011 

505 0.3199 0.0019 640 0.1119 0.0006 775 0.1891 0.0010 

510 0.3326 0.0020 645 0.1121 0.0006 780 0.1875 0.0011 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

12

Table 8. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Green color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.0584 0.0041 515 0.3401 0.0020 650 0.1075 0.0006 

385 0.0616 0.0018 520 0.3335 0.0020 655 0.1086 0.0007 

390 0.0653 0.0017 525 0.3212 0.0020 660 0.1104 0.0006 

395 0.0675 0.0012 530 0.3052 0.0019 665 0.1129 0.0007 

400 0.0698 0.0010 535 0.2871 0.0018 670 0.1160 0.0007 

405 0.0721 0.0007 540 0.2686 0.0017 675 0.1196 0.0007 

410 0.0744 0.0006 545 0.2502 0.0015 680 0.1240 0.0007 

415 0.0768 0.0007 550 0.2326 0.0014 685 0.1291 0.0008 

420 0.0796 0.0006 555 0.2163 0.0013 690 0.1350 0.0008 

425 0.0828 0.0006 560 0.2012 0.0012 695 0.1416 0.0008 

430 0.0863 0.0007 565 0.1875 0.0011 700 0.1489 0.0009 

435 0.0909 0.0007 570 0.1752 0.0010 705 0.1567 0.0009 

440 0.0962 0.0007 575 0.1641 0.0010 710 0.1645 0.0010 

445 0.1026 0.0008 580 0.1543 0.0009 715 0.1717 0.0010 

450 0.1106 0.0010 585 0.1457 0.0009 720 0.1779 0.0010 

455 0.1197 0.0010 590 0.1383 0.0008 725 0.1828 0.0011 

460 0.1339 0.0010 595 0.1318 0.0008 730 0.1865 0.0011 

465 0.1481 0.0012 600 0.1264 0.0008 735 0.1889 0.0011 

470 0.1653 0.0012 605 0.1215 0.0007 740 0.1903 0.0011 

475 0.1857 0.0013 610 0.1175 0.0007 745 0.1908 0.0011 

480 0.2088 0.0014 615 0.1142 0.0007 750 0.1907 0.0011 

485 0.2347 0.0015 620 0.1115 0.0006 755 0.1899 0.0011 

490 0.2624 0.0017 625 0.1095 0.0006 760 0.1888 0.0011 

495 0.2897 0.0018 630 0.1080 0.0006 765 0.1873 0.0011 

500 0.3130 0.0020 635 0.1071 0.0006 770 0.1857 0.0011 

505 0.3302 0.0020 640 0.1067 0.0006 775 0.1841 0.0010 

510 0.3397 0.0020 645 0.1068 0.0006 780 0.1826 0.0011 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

13

Table 9. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Mid Gray color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.2162 0.0023 515 0.2533 0.0015 650 0.2548 0.0015 

385 0.2217 0.0026 520 0.2540 0.0015 655 0.2544 0.0015 

390 0.2255 0.0015 525 0.2547 0.0015 660 0.2543 0.0015 

395 0.2298 0.0016 530 0.2555 0.0015 665 0.2545 0.0015 

400 0.2337 0.0014 535 0.2563 0.0015 670 0.2547 0.0015 

405 0.2368 0.0014 540 0.2572 0.0016 675 0.2552 0.0015 

410 0.2399 0.0017 545 0.2579 0.0015 680 0.2556 0.0015 

415 0.2424 0.0014 550 0.2581 0.0015 685 0.2562 0.0015 

420 0.2451 0.0015 555 0.2582 0.0015 690 0.2566 0.0015 

425 0.2476 0.0015 560 0.2577 0.0015 695 0.2569 0.0015 

430 0.2496 0.0016 565 0.2573 0.0015 700 0.2570 0.0015 

435 0.2514 0.0016 570 0.2565 0.0015 705 0.2568 0.0015 

440 0.2525 0.0015 575 0.2556 0.0015 710 0.2565 0.0015 

445 0.2539 0.0015 580 0.2551 0.0015 715 0.2558 0.0015 

450 0.2544 0.0016 585 0.2548 0.0015 720 0.2551 0.0015 

455 0.2548 0.0015 590 0.2548 0.0014 725 0.2542 0.0014 

460 0.2547 0.0015 595 0.2552 0.0015 730 0.2534 0.0014 

465 0.2544 0.0018 600 0.2558 0.0015 735 0.2523 0.0014 

470 0.2536 0.0016 605 0.2563 0.0015 740 0.2514 0.0014 

475 0.2532 0.0015 610 0.2566 0.0015 745 0.2503 0.0014 

480 0.2528 0.0016 615 0.2568 0.0015 750 0.2493 0.0014 

485 0.2525 0.0015 620 0.2570 0.0015 755 0.2481 0.0014 

490 0.2524 0.0014 625 0.2569 0.0015 760 0.2471 0.0014 

495 0.2524 0.0015 630 0.2567 0.0015 765 0.2458 0.0014 

500 0.2525 0.0015 635 0.2562 0.0015 770 0.2445 0.0014 

505 0.2526 0.0015 640 0.2558 0.0015 775 0.2431 0.0014 

510 0.2530 0.0015 645 0.2555 0.0015 780 0.2418 0.0014 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

14

Table 10. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Orange color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.0520 0.0066 515 0.0806 0.0006 650 0.7699 0.0044 

385 0.0529 0.0042 520 0.0838 0.0005 655 0.7742 0.0044 

390 0.0545 0.0028 525 0.0876 0.0006 660 0.7789 0.0045 

395 0.0555 0.0021 530 0.0937 0.0006 665 0.7834 0.0045 

400 0.0566 0.0014 535 0.1051 0.0007 670 0.7871 0.0045 

405 0.0573 0.0013 540 0.1260 0.0010 675 0.7913 0.0046 

410 0.0581 0.0011 545 0.1608 0.0014 680 0.7947 0.0045 

415 0.0591 0.0011 550 0.2124 0.0018 685 0.7983 0.0047 

420 0.0601 0.0010 555 0.2791 0.0023 690 0.8015 0.0046 

425 0.0607 0.0007 560 0.3539 0.0026 695 0.8047 0.0046 

430 0.0616 0.0009 565 0.4280 0.0029 700 0.8081 0.0046 

435 0.0625 0.0008 570 0.4934 0.0031 705 0.8113 0.0046 

440 0.0623 0.0059 575 0.5475 0.0034 710 0.8150 0.0047 

445 0.0643 0.0008 580 0.5903 0.0035 715 0.8180 0.0047 

450 0.0656 0.0009 585 0.6240 0.0036 720 0.8211 0.0047 

455 0.0666 0.0009 590 0.6512 0.0038 725 0.8231 0.0047 

460 0.0617 0.0010 595 0.6726 0.0039 730 0.8244 0.0047 

465 0.0625 0.0010 600 0.6902 0.0040 735 0.8255 0.0048 

470 0.0634 0.0008 605 0.7041 0.0040 740 0.8263 0.0047 

475 0.0641 0.0009 610 0.7155 0.0041 745 0.8273 0.0047 

480 0.0649 0.0006 615 0.7257 0.0042 750 0.8281 0.0048 

485 0.0657 0.0006 620 0.7336 0.0046 755 0.8293 0.0047 

490 0.0670 0.0005 625 0.7415 0.0042 760 0.8305 0.0047 

495 0.0681 0.0006 630 0.7486 0.0043 765 0.8308 0.0047 

500 0.0700 0.0006 635 0.7543 0.0043 770 0.8318 0.0048 

505 0.0731 0.0006 640 0.7601 0.0043 775 0.8323 0.0047 

510 0.0769 0.0006 645 0.7651 0.0044 780 0.8329 0.0048 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

15

Table 11. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Pale Gray color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.5670 0.0044 515 0.6306 0.0036 650 0.6273 0.0036 

385 0.5783 0.0039 520 0.6310 0.0036 655 0.6269 0.0036 

390 0.5864 0.0047 525 0.6312 0.0036 660 0.6270 0.0036 

395 0.5935 0.0042 530 0.6318 0.0036 665 0.6271 0.0036 

400 0.5994 0.0034 535 0.6322 0.0036 670 0.6271 0.0036 

405 0.6043 0.0035 540 0.6324 0.0037 675 0.6278 0.0036 

410 0.6081 0.0041 545 0.6325 0.0037 680 0.6283 0.0036 

415 0.6113 0.0043 550 0.6320 0.0036 685 0.6286 0.0036 

420 0.6148 0.0039 555 0.6316 0.0036 690 0.6291 0.0037 

425 0.6175 0.0037 560 0.6307 0.0036 695 0.6289 0.0038 

430 0.6203 0.0040 565 0.6297 0.0036 700 0.6292 0.0036 

435 0.6229 0.0040 570 0.6287 0.0036 705 0.6286 0.0036 

440 0.6236 0.0036 575 0.6276 0.0036 710 0.6284 0.0036 

445 0.6261 0.0037 580 0.6272 0.0037 715 0.6278 0.0036 

450 0.6269 0.0041 585 0.6266 0.0037 720 0.6273 0.0036 

455 0.6282 0.0039 590 0.6270 0.0036 725 0.6263 0.0036 

460 0.6285 0.0036 595 0.6272 0.0036 730 0.6252 0.0036 

465 0.6286 0.0036 600 0.6280 0.0038 735 0.6242 0.0036 

470 0.6288 0.0036 605 0.6280 0.0036 740 0.6234 0.0036 

475 0.6288 0.0036 610 0.6285 0.0036 745 0.6224 0.0036 

480 0.6286 0.0036 615 0.6287 0.0036 750 0.6212 0.0036 

485 0.6288 0.0037 620 0.6290 0.0036 755 0.6201 0.0035 

490 0.6289 0.0036 625 0.6289 0.0036 760 0.6187 0.0038 

495 0.6292 0.0036 630 0.6286 0.0036 765 0.6176 0.0035 

500 0.6292 0.0039 635 0.6282 0.0036 770 0.6164 0.0036 

505 0.6300 0.0036 640 0.6280 0.0036 775 0.6150 0.0035 

510 0.6302 0.0036 645 0.6277 0.0036 780 0.6139 0.0035 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

16

Table 12. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Red color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.0156 0.0058 515 0.0269 0.0004 650 0.6471 0.0037 

385 0.0165 0.0036 520 0.0280 0.0003 655 0.6619 0.0038 

390 0.0170 0.0024 525 0.0294 0.0003 660 0.6764 0.0039 

395 0.0181 0.0018 530 0.0309 0.0003 665 0.6891 0.0039 

400 0.0187 0.0014 535 0.0324 0.0003 670 0.7009 0.0040 

405 0.0191 0.0012 540 0.0337 0.0003 675 0.7111 0.0041 

410 0.0196 0.0010 545 0.0350 0.0003 680 0.7207 0.0041 

415 0.0202 0.0007 550 0.0364 0.0003 685 0.7297 0.0043 

420 0.0206 0.0007 555 0.0373 0.0035 690 0.7373 0.0042 

425 0.0208 0.0005 560 0.0405 0.0003 695 0.7460 0.0043 

430 0.0212 0.0005 565 0.0442 0.0003 700 0.7544 0.0044 

435 0.0214 0.0004 570 0.0500 0.0003 705 0.7623 0.0044 

440 0.0218 0.0004 575 0.0584 0.0004 710 0.7703 0.0044 

445 0.0217 0.0020 580 0.0699 0.0005 715 0.7782 0.0045 

450 0.0224 0.0003 585 0.0851 0.0006 720 0.7845 0.0045 

455 0.0228 0.0003 590 0.1049 0.0008 725 0.7891 0.0045 

460 0.0223 0.0010 595 0.1314 0.0010 730 0.7914 0.0045 

465 0.0228 0.0009 600 0.1674 0.0014 735 0.7925 0.0047 

470 0.0231 0.0006 605 0.2159 0.0019 740 0.7932 0.0048 

475 0.0233 0.0005 610 0.2773 0.0022 745 0.7944 0.0045 

480 0.0236 0.0005 615 0.3480 0.0026 750 0.7957 0.0046 

485 0.0235 0.0022 620 0.4193 0.0029 755 0.7967 0.0048 

490 0.0248 0.0004 625 0.4834 0.0031 760 0.7978 0.0045 

495 0.0251 0.0003 630 0.5355 0.0032 765 0.7986 0.0046 

500 0.0252 0.0004 635 0.5753 0.0033 770 0.7991 0.0046 

505 0.0255 0.0004 640 0.6057 0.0035 775 0.7996 0.0046 

510 0.0262 0.0003 645 0.6292 0.0037 780 0.8002 0.0046 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

17

Table 13. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the White color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.7564 0.0050 515 0.8988 0.0051 650 0.9028 0.0051 

385 0.7791 0.0053 520 0.8992 0.0052 655 0.9028 0.0051 

390 0.8014 0.0053 525 0.8998 0.0052 660 0.9035 0.0052 

395 0.8199 0.0048 530 0.9003 0.0051 665 0.9047 0.0052 

400 0.8356 0.0050 535 0.9008 0.0052 670 0.9052 0.0052 

405 0.8482 0.0055 540 0.9008 0.0052 675 0.9057 0.0052 

410 0.8577 0.0053 545 0.9009 0.0052 680 0.9064 0.0052 

415 0.8631 0.0050 550 0.9010 0.0052 685 0.9067 0.0052 

420 0.8687 0.0060 555 0.9010 0.0053 690 0.9069 0.0052 

425 0.8736 0.0052 560 0.9010 0.0052 695 0.9069 0.0053 

430 0.8764 0.0051 565 0.9010 0.0051 700 0.9075 0.0052 

435 0.8775 0.0052 570 0.9002 0.0051 705 0.9076 0.0052 

440 0.8798 0.0052 575 0.8996 0.0051 710 0.9076 0.0054 

445 0.8811 0.0052 580 0.8994 0.0051 715 0.9084 0.0052 

450 0.8824 0.0054 585 0.8994 0.0052 720 0.9082 0.0052 

455 0.8860 0.0058 590 0.9006 0.0052 725 0.9084 0.0052 

460 0.8869 0.0051 595 0.9010 0.0051 730 0.9084 0.0052 

465 0.8896 0.0051 600 0.9021 0.0054 735 0.9085 0.0052 

470 0.8910 0.0052 605 0.9017 0.0052 740 0.9082 0.0052 

475 0.8924 0.0051 610 0.9019 0.0052 745 0.9081 0.0052 

480 0.8935 0.0051 615 0.9020 0.0052 750 0.9084 0.0052 

485 0.8942 0.0053 620 0.9020 0.0052 755 0.9084 0.0052 

490 0.8952 0.0051 625 0.9026 0.0051 760 0.9084 0.0052 

495 0.8962 0.0052 630 0.9022 0.0052 765 0.9079 0.0052 

500 0.8971 0.0052 635 0.9021 0.0052 770 0.9082 0.0052 

505 0.8980 0.0051 640 0.9026 0.0052 775 0.9079 0.0052 

510 0.8986 0.0051 645 0.9032 0.0051 780 0.9078 0.0052 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

18

Table 14. 0:45 spectral reflectance factor R and expanded uncertainty U (k = 2) as a 

function of wavelength λ of the Yellow color tile, set number NISTCS-XX 

λ [nm] R U λ [nm] R U λ [nm] R U 

380 0.0240 0.0075 515 0.4630 0.0030 650 0.7904 0.0045 

385 0.0249 0.0051 520 0.5179 0.0033 655 0.7916 0.0047 

390 0.0253 0.0034 525 0.5656 0.0034 660 0.7943 0.0046 

395 0.0252 0.0023 530 0.6050 0.0036 665 0.7966 0.0046 

400 0.0255 0.0017 535 0.6364 0.0037 670 0.7989 0.0047 

405 0.0262 0.0015 540 0.6611 0.0040 675 0.8011 0.0046 

410 0.0270 0.0013 545 0.6802 0.0040 680 0.8028 0.0046 

415 0.0277 0.0010 550 0.6951 0.0040 685 0.8047 0.0046 

420 0.0289 0.0009 555 0.7072 0.0041 690 0.8060 0.0047 

425 0.0304 0.0008 560 0.7165 0.0041 695 0.8077 0.0046 

430 0.0322 0.0007 565 0.7249 0.0042 700 0.8095 0.0046 

435 0.0345 0.0006 570 0.7309 0.0042 705 0.8107 0.0046 

440 0.0375 0.0005 575 0.7326 0.0042 710 0.8125 0.0046 

445 0.0413 0.0006 580 0.7361 0.0042 715 0.8137 0.0047 

450 0.0463 0.0005 585 0.7404 0.0042 720 0.8149 0.0046 

455 0.0522 0.0007 590 0.7455 0.0043 725 0.8159 0.0047 

460 0.0619 0.0010 595 0.7520 0.0043 730 0.8168 0.0047 

465 0.0728 0.0011 600 0.7591 0.0045 735 0.8163 0.0048 

470 0.0863 0.0009 605 0.7644 0.0044 740 0.8144 0.0047 

475 0.1040 0.0009 610 0.7694 0.0045 745 0.8167 0.0047 

480 0.1265 0.0011 615 0.7735 0.0044 750 0.8158 0.0047 

485 0.1550 0.0012 620 0.7771 0.0044 755 0.8184 0.0047 

490 0.1911 0.0015 625 0.7796 0.0047 760 0.8204 0.0047 

495 0.2348 0.0017 630 0.7824 0.0045 765 0.8228 0.0048 

500 0.2857 0.0022 635 0.7846 0.0045 770 0.8245 0.0047 

505 0.3430 0.0024 640 0.7869 0.0045 775 0.8246 0.0056 

510 0.4036 0.0028 645 0.7893 0.0045 780 0.8266 0.0048 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

19

Table 15. Tristimulus values Y and chromaticity coordinates x and y, with expanded 

uncertainties U (k=2) for CIE standard illuminant D65 and CIE 1931 standard 

observer for each color tile, set number NISTCS-XX 

Tile Y U x U y U 

Black 0.624 0.006 0.3010 0.0011 0.3355 0.0018 

Cyan 19.161 0.113 0.1986 0.0000 0.2416 0.0003 

Deep Blue 1.119 0.009 0.1994 0.0003 0.1206 0.0008 

Deep Gray 5.084 0.029 0.3182 0.0003 0.3356 0.0003 

Deep Pink 11.531 0.068 0.4127 0.0004 0.3059 0.0004 

Diff Gray 26.181 0.148 0.3140 0.0000 0.3382 0.0000 

Diff Green 21.651 0.126 0.2845 0.0002 0.4379 0.0004 

Green 21.455 0.125 0.2748 0.0002 0.4236 0.0005 

Mid Gray 25.577 0.145 0.3139 0.0000 0.3307 0.0001 

Orange 35.184 0.209 0.5228 0.0005 0.3976 0.0005 

Pale Gray 62.975 0.357 0.3128 0.0001 0.3301 0.0000 

Red 10.086 0.067 0.5850 0.0011 0.3337 0.0005 

White 89.994 0.511 0.3141 0.0001 0.3314 0.0001 

Yellow 63.350 0.362 0.4467 0.0003 0.4850 0.0002 
REPORT OF CALIBRATION 

38091S 0:45 Surface Color Color Tile Set 

Any Company, Inc. Serial No.: NISTCS-XX 

20

Table 16. Sources of uncertainty and their standard uncertainty or value for a color tile 

set, set number NISTCS-XX 

Source of Uncertainty Standard Uncertainty 

or Value 

Signal 

 Noise 0.01 % - 0.2 % 

 Non-linearity 0.2 % 

 Stray Light 10-6 

Wavelength 0.05 nm 

Standard 

 Random 0.02 % 

 Systematic 0.2 % 

2 

Appendix B 

Appeared in Color Research Applications, Vol. 29, June 2004 

Uncertainty Analysis for Reflectance Colorimetry 

E. A. Early and M. E. Nadal 

Optical Technology Division 

National Institute of Standards and Technology 

100 Bureau Drive, Stop 8441 

Gaithersburg, MD 20899-8441 USA 

Abstract 

The uncertainty associated with the quantitative description of the color of an object is 

often necessary for determining the acceptability of that object for its intended 

application. Uncertainties are also required for establishing the traceability of a 

measurement to a national metrology institute. A systematic, analytical approach to 

uncertainty analysis, which conforms to currently accepted practice, is presented for 

reflectance colorimetry. Two important concepts are stressed – the measurement 

equation describing the relation between signals and spectral reflectance factor, and 

correlations both between signals at the same wavelength and between reflectance factors 

at different wavelengths. The approach is illustrated by considering representative object 

colors and instrument parameters. 

Keywords: colorimetry, correlation, uncertainty 
3 

I. Introduction 

Determining the uncertainty in the measured color of a specimen has been a topic 

of interest for many years.1-9 It has received additional attention with the publication of 

the ISO Guide to the Expression of Uncertainty in Measurement,

10 which describes 

methods for estimating uncertainties in a consistent manner. Recent publications have 

applied the principles of this Guide to uncertainty analysis for colorimetry, emphasizing 

the treatment of correlation.11,12 

The development of a new reference instrument for reflectance colorimetry at the 

National Institute of Standards and Technology (NIST) has prompted a detailed analysis 

of the uncertainties associated with this measurement. From a review of the literature, 

three important aspects of previous uncertainty analyses for colorimetry became 

apparent. First, while sources of uncertainty can be caused by either random or 

systematic effects, most of the previous analyses were restricted to considering only 

uncertainties caused by random effects.1,5,6,8,9 Where uncertainties caused by systematic 

effects were considered, modeling was performed to calculate2,4 or minimize7

 the effects 

of these uncertainties. Second, the importance of correlations due to spectral overlap of 

the color-matching functions was recognized from the beginning,1,4,5 and has received 

renewed attention.9,11,12 Third, the analyses started with a given uncertainty in the 

spectral reflectance factor,1,2,4-9 without considering the underlying measurement used to 

calculate it. 

The analysis detailed here presents a systematic, analytical method for calculating 

uncertainties in reflectance colorimetry and emphasizes two key points. First, the 

spectral properties of the measurement instrument are important because ratios of signals 

are used to calculate the spectral reflectance factor of a specimen. Therefore, the 

measurement equation is used to determine the uncertainty in the spectral reflectance 

factor. Second, since correlations exist both between signals at the same wavelength and 

between reflectance factors at different wavelengths, the effects of these correlations are 

included in the analysis. These concepts are illustrated by considering common sources 

of uncertainty for a double-grating, single-detector instrument and applying the analysis 

to representative object colors. 

II. Equations for Reflectance Colorimetry 

The surface color of a specimen depends upon several factors – the spectral power 

distribution of the illuminant, the spectral sensitivity of the observer, the geometrical 

conditions of illumination and observation, and the optical properties of the specimen. 

Only the last factor depends upon the specimen, and is quantified by the spectral 

reflectance factor under the given geometrical conditions. Once the spectral reflectance 

factor is known, the tristimulus values are calculated for a given illuminant and observer, 

and from these values other quantities are derived, such as chromaticity coordinates or 

color space values. 

The measurement of the spectral reflectance factor of a specimen is performed by 

comparing signals from the specimen and a standard, under the same measurement 

conditions. This process is described by the measurement equation 
4 

)( )(

)( )( s

s

i

i

i

i R

S

S R λ

λ

λ λ ⋅= , (2.1) 

where λi is the wavelength setting of the instrument, S and Ss are the measured signals 

from the specimen and standard, respectively, and R and Rs are the reflectance factors for 

the specimen and standard, respectively. For ease of notation, in the following the 

wavelength dependence of the signals and reflectance factors will be indicated by 

subscripts i and j. Therefore, Eq. (2.1) becomes 

i

i

i

i R

S

S R ,s

,s

⋅= . (2.2) 

The tristimulus values X, Y, and Z, for reflectance factors regularly spaced in wavelength, 

are given by 

∑=

⋅=

m

i

RWX ii

1

x, , (2.3) 

∑=

⋅=

m

i

RWY ii

1

y, , and (2.4) 

∑=

⋅=

m

i

RWZ ii

1

z, , (2.5) 

where Wx,i, Wy,i, and Wz,i are the normalized tristimulus weighting factors for a particular 

illuminant and observer combination and spectral bandwidth and interval.13 These 

tristimulus values are then used to calculate chromaticity coordinates, such as x and y or 

u’ and v’, or color space values, such as L*, a*, and b* or L*, u*, and v*, as given in Ref. 

[13]. 

III. Basics of Uncertainty Analysis 

 The approach to the uncertainty analysis presented here follows the Guide.10 This 

section details the basic definitions and equations used in the uncertainty analysis. 

 In general, the purpose of a measurement is to determine the value of a measurand 

y, which is obtained from n other quantities xk through the functional relationship f, given 

by 

),,,,,( 21 k n = KK xxxxfy . (3.1) 

An example of such a functional relationship, which is used extensively in this paper, is 

Eq. (2.2), where Ri = f(Si, Ss,i, Rs,i, λ). The standard uncertainty of an input quantity xk is 

the estimated standard deviation associated with this quantity and is denoted by u(xk). 

The estimated standard uncertainty in the measurand y is the combined standard 

uncertainty uc(y), given by the law of propagation of uncertainty as 

)( 2)( ),(

1

1 1

2

2

1

2

c lk

n

k

n

kl lk

k

n

k k

xxu

x

f

x

f xu

x

f yu ∑ ∑ ∑

−

= = + =

⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ = . (3.2) 

Here, ∂f/∂xk is the sensitivity coefficient with respect to xk and u(xk, xl) is the covariance 

of xk and xl. The covariance is related to the correlation coefficient r(xk, xl) by 

)()(

),( ),(

k l

lk

lk

xuxu

xxu

xxr ⋅ = , (3.3) 
5 

where the correlation coefficient is a pure number between –1 and +1, inclusive. For two 

input quantities that are uncorrelated, r(xk, xl) = 0, while two quantities that are fully 

correlated have r(xk, xl) = ±1. In general, sources of uncertainty arising from random and 

independent effects are uncorrelated, while those arising from systematic effects are 

correlated. In terms of the correlation coefficients, Eq. (3.2) becomes 

)( 2)( .)()(),(

1

1 1

2

2

1

2

c lklk

n

k

n

kl lk

k

n

k k

xuxuxxr

x

f

x

f xu

x

f yu ⋅⋅⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ = ∑ ∑ ∑

−

= = + =

 (3.4) 

 The covariance, and hence the correlation coefficient, can be calculated using two 

different techniques, depending on the knowledge of the input quantities xk and xl. If the 

quantities are obtained from coincident experimental data with H values for each 

quantity, then the covariance is given by 

∑=

= −−

H

h

lk lhlkhk xxxx

H

xxu

1

, , ))(( 1 ( , (3.5) ),

where k x and l x are the averages of xk and xl, respectively. The standard deviations of 

the quantities are 

∑=

= −

H

h

k khk xx

H

xu

1

2

, )( 1 )( and (3.6) 

∑=

= −

H

h

l lhl xx

H

xu

1

2

, )( 1 )( . (3.7) 

Substituting Eqs. (3.5) to (3.7) into Eq. (3.3) yields the correlation coefficient 

∑ ∑

∑

= =

=

− −

−−

= H

h

H

l

khk lhl

H

h

ljlkhk

lk

xxxx

xxxx

xxr

1 1

2

,

2

,

1

, ,

)()(

))((

( . (3.8) ),

 In the other technique, analytical expressions for the input quantities are known, 

so that the quantities are functions of additional variables, given by 

),,,,,( kk 21 h H = KK qqqqgx . (3.9) 

Then, if u(qh) is the standard uncertainty associated with the quantity qh, 

2 and)()(),(

)( )(

1

1 1

2

2

1

2

ph h p

p

k

H

h

H

hp h

k

h

H

h h

k

k

ququqqr

q

g

q

g

qu

q

g

xu

⋅⋅⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ =

∑ ∑

∑

−

= + =

=

 (3.10) 

.)()(),(

),( )(

1

1 1

1

2

∑ ∑

∑

−

= + =

=

⋅⋅⋅

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

∂

∂ =

H

h

H

hp

ph h p

h

l

p

k

p

l

h

k

H

h

h

h

l

h

k

lk

ququqqr

q

g

q

g

q

g

q

g

qu

q

g

q

g xxu

(3.11) 
6 

Substitution of Eqs. (3.10) and (3.11) into Eq. (3.3) yields the correlation coefficient 

r(xk, xl). For this technique to work, the correlation coefficients for the additional 

variables r(qh, qp) must be known. 

 As an example of the second technique, consider a common situation in which the 

input quantities xk and xl are given by 

 ) k α =+= k agax ,( α and (3.12) 

 ) l β =+= l bgbx ,( β , (3.13) 

where the variables are a, b, α, and β, and only α and β are possibly correlated. The 

standard uncertainties, using Eq. (3.10) and the sensitivity coefficients from Eqs. (3.12) 

and (3.13), are 

 ) ()()(0)(1)(0)(1)( 2 2 2 2 2 2 2

k α β +=⋅+⋅+⋅+⋅= uauuubuauxu α and (3.14) 

)()()(1)(0)(1)(0)( 2 2 2 2 2 2 2

l βα +=⋅+⋅+⋅+⋅= ubuuubuauxu β . (3.15) 

The covariance, from Eq. (3.11), is 

 ) xxu lk ⋅⋅+⋅= α β ⋅ α ⋅uur ()(),()0011(),( β . (3.16) 

Substituting Eqs. (3.14) to (3.16) into Eq. (3.3) yields the correlation coefficient 

)()()()(

)()(),( ),( 2 2 2 2 α β

α β α β

ubuuau

uur

xxr lk

+ +

⋅ ⋅ = . (3.17) 

If the only source of uncertainty in Eqs. (3.12) and (3.13) is α and β, so that 

u(a) = u(b) = 0, then r(xk, xl) = r(α, β). To continue this example further, let the variables 

α and β be given by 

α ⋅= sA and (3.18) 

β ⋅= sB , (3.19) 

where the variables A and B are uncorrelated and the variable s is a common factor. 

Using Eqs. (3.10) and (3.11), assuming no uncertainty in A and B, and the sensitivity 

coefficients derived from Eqs. (3.18) and (3.19), yields 

 ) ()( 2 22 α ⋅= suAu , (3.20) 

 ) ()( 2 22 β ⋅= suBu , and (3.21) 

 ) ),( (

2 βα ⋅⋅= suBAu . (3.22) 

Therefore, 

1

)()(

)( ),( 22 22

2

=

⋅⋅

⋅⋅ =

suBsuA

suBA

r βα . (3.23) 

IV. Application to Reflectance Colorimetry 

A. Overview 

As detailed in Section II, the progression from measurement to surface color is 

spectral reflectance factor, tristimulus values, and chromaticity coordinates or color space 

values. Therefore, to arrive at the uncertainties in the final values describing the surface 

color of a specimen, the uncertainties in spectral reflectance factor must be propagated 

through the calculations. From the measurement equation for the spectral reflectance 
7 

factor Ri, given by Eq. (2.2), uncertainties in Si, Ss,i, Rs,i and λ all contribute to an 

uncertainty in Ri. 

Uncertainties in the final color values can be calculated using two different 

approaches. In the first, the uncertainties in Ri are propagated through to uncertainties in 

the tristimulus values, which are then used to calculate the uncertainties in the 

chromaticity coordinates or color space values. In the second approach, Eqs. (2.3) to 

(2.5) are substituted into the definitions of the final values to obtain equations in terms of 

Ri. The uncertainties in the final color values given by these equations are calculated 

directly from the uncertainties in Ri. In both approaches the correlations between the 

uncertainties in Ri at different wavelengths must be included in the calculations. In 

addition, if the first approach is used, the correlations between the tristimulus values are 

also required. 

These concepts of propagating uncertainties and taking correlations into account 

are developed below by considering common sources of uncertainty in Ri

14,15 – signal 

noise, non-linearity in the signal, and uncertainties in the standard and the wavelength. 

Signal noise is the result of random processes in both the optical source and detector. 

Non-linearitys in the signal can arise from several sources; those considered here are an 

improper zero, mis-alignment of the specimen, and stray-light. An improper zero adds a 

constant value to all signals, while mis-alignment of the specimen is the result of a 

difference in location, with respect to the reference plane, between the specimen and the 

standard. For example, if the specimen and standard are physically different, it may not 

be possible to align one of the front surfaces with the reference plane. Stray-light is 

radiant flux that reaches the detector at wavelengths outside the nominal spectral 

bandwidth of the instrument, which adds a value to each signal. This value depends upon 

the spectral shape of the illumination, the spectral reflectance of both the specimen and 

standard, and the responsivity of the receiver. The standard has an uncertainty from its 

calibration, and there is an uncertainty associated with the wavelength setting of the 

measuring instrument. 

Uncertainties in the normalized tristimulus weighting factors Wx,i, Wy,i, and Wz,i

can be an additional source of uncertainty for reflectance colorimetry. For example, the 

weighting factors chosen for the calculation may not be appropriate for the spectral 

bandwidth and interval of the measuring instrument. However, the effect of this source 

of uncertainty is beyond the scope of this paper. Also, this paper does not address the 

method for obtaining the uncertainties in the signal, standard, and wavelength, other than 

to recommend applying the techniques found in the Guide.10

In the following, each source of uncertainty – signal noise, signal non-linearity, 

standard, and wavelength – is considered separately, and the sensitivity coefficients of Ri

for each source are obtained using the measurement equation. The correlations between 

signals at the same wavelength and between reflectance factors at different wavelengths 

for each of these sources are considered in the analysis. 

B. Uncertainty in Spectral Reflectance Factor 

The sensitivity coefficients of Ri for the signals Si and Ss,i are 

i

i

i

i

i

i

S

R

S

R

S

R == ∂∂

,s

,s and (4.1) 
8 

i

i

i

ii

i

i

S

R

S

SR

S

R

,s

2

,s

,s

,s

− = − = ∂∂ . (4.2) 

Substituting these into Eq. (3.4) and dividing by Ri

2

 yields the component of standard 

uncertainty in Ri due to uncertainty in the signals, 

),( )( )( 2 )()( )( ,s

,s

,s

2

,s

,s

2

2

2

2

2

c

ii

i

i

i

i

i

i

i

i

i

i SSr

S

Su

S

Su

S

Su

S

Su

R

Ru

−+= . (4.3) 

 The correlation coefficient r(Si, Ss,i) between the two signals depends upon the 

source of uncertainty. Since signal noise is a random effect, the standard uncertainties 

are uncorrelated and r(Si, Ss,i) = 0. From Eq. (4.3), the resulting component of standard 

uncertainty in Ri due to signal noise is 

2

,s

,s

2

2

2

2

2

c )()( )(

i

i

i

i

i

i

S

Su

S

Su

R

Ru

+= . (4.4) 

 For an non-linearity in the signals, the example given by Eqs. (3.12) and (3.13) 

and following applies by replacing xk and xl by Si and Ss,i, respectively. If the nonlinearity affects only one of the signals, for example a mis-alignment of the specimen 

relative to the standard, then β = 0 and there is no uncertainty in Ss,i, so Eq. (4.3) becomes 

2

2

2

2

c )()(

i

i

i

i

S

Su

R

Ru = . (4.5) 

If the non-linearity affects both signals, then the correlation coefficient r(Si, Ss,i) is 

needed. For an improper zero, α = β since it affects both signals with the same value, so 

r(α, β) = 1 and therefore r(Si, Ss,i) = 1. For an non-linearity caused by stray-light, the 

magnitudes of the non-linearity will be different for the two signals because of the 

different spectral reflectance factors of the specimen and standard. However, the nonlinearity can be modeled as Eqs. (3.18) and (3.19) (see Eqs. (5.2) and (5.3)), so r(α, β) = 

1 and hence r(Si, Ss,i) = 1. For the non-linearity affecting both signals considered here, 

where r(Si, Ss,i) = 1, Eq. (4.3) becomes 

2

,s

,s

2

2

c )()( )(

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ −=

i

i

i

i

i

i

S

Su

S

Su

R

Ru . (4.6) 

Note that for the case of an non-linearity that is the same for both signals, Eq. (4.6) 

implies that if the signals from the specimen and standard are the same, meaning they are 

similar materials, then the standard uncertainty in Ri is zero. 

 The sensitivity coefficient of Ri for the spectral reflectance factor of the standard 

Rs,i is 

i

i

i

i

i

i

R

R

S

S

R

R

,s,s,s

== ∂

∂ , (4.7) 

and Eq. (3.4) becomes 

2

,s

,s

2

2

2

c )( )(

i

i

i

i

R

Ru

R

Ru = . (4.8) 

 Finally, the sensitivity coefficient of Ri for wavelength λ is 
9 

i

i

i

i R

S

S

R ,s

,s

∂λλ

⎟

⎠

⎞ ⎜

⎝

⎛

∂

= ∂

∂ (4.9) 

and Eq. (3.4) becomes 

)( )( 2

2

,s

2

2

,s

2

2

c λ

λ

u

S

S

R

R

R

Ru i

i

i

i

i

i

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

∂

⎟

⎠

⎞ ⎜

⎝

⎛

∂

= . (4.10) 

Any uncertainty in Rs,i due to wavelength is included in Eq. (4.8). 

C. Correlations between Wavelengths 

 The correlation coefficients r(Ri, Rj) for spectral reflectance factors at different 

wavelengths i and j are calculated for the sources of uncertainty considered here. These 

correlation coefficients are used in succeeding sections to determine the uncertainty in 

tristimulus values and chromaticity coordinates and color space values. For all 

calculations in the following, the signals and spectral reflectance factors are assumed to 

be positive. 

 The sources of uncertainty associated with the signals are noise and an nonlinearity. Since signal noise is a random effect, there is no correlation between 

wavelengths, and hence r(Ri, Rj) = 0. For an non-linearity, an extension of the approach 

leading to Eq. (4.6) is used, but now with four signals (from the specimen and the 

standard at the two wavelengths). If the non-linearity affects only one pair of signals, for 

example those from the specimen, then r(Ri, Rj) = 1. If the non-linearity affects all four 

signals, as for an improper zero or stray-light, then the correlation coefficient is given by 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −=

ii jj

ji SSSS

RRr

,s ,s

11

sgn

11 sgn),( , (4.11) 

where sgn(x) = +1 if x > 0, –1 if x < 0, and 0 if x = 0. Equation (4.11) is a result of taking 

the absolute values of the sensitivity coefficients for the denominator of Eq. (3.3). For 

the common situation where the standard has a greater reflectance factor than the 

specimen, Ss,i > Si for all i and r(Ri, Rj) = +1 for all i and j. On the other hand, if the 

standard and specimen have similar spectral reflectance factors, then r(Ri, Rj) can be –1, 

0, or +1 depending upon the four signals. 

 The spectral reflectance factor of the standard, Rs,i, has uncertainties caused by 

both random and systematic effects. For the random effects, r(Ri, Rj) = 0, while 

additional analysis is needed for the systematic effects. The spectral reflectance factor of 

the standard is assigned based upon absolute measurements, which involve ratios of 

signals and other factors. Therefore, let Rs,i be a function of these variables, so that 

),,( ,s i = SSfR ,2,1 ii ξ , (4.12) 

where ξ accounts for the other factors that are common to all wavelengths. The 

sensitivity coefficients of the spectral reflectance factor of the specimen with respect to 

the variable ξ are 
10 

ξ ξ ∂ξ

∂ = ∂∂

∂

∂ = ∂∂ i

i

i i

i

ii R

R

R R

R

RR ,s

,s

,s

,s

 and (4.13) 

ξ ξ ∂ξ

∂ = ∂

∂

∂

∂ = ∂

∂ j

j

jj

j

j j R

R

RR

R

RR ,s

,s

,s

,s

 . (4.14) 

Therefore, the correlation coefficient is given by 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ ⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ = ξ ξ

i j

ji

R R

RRr ,s ,s sgn),( sgn . (4.15) 

In nearly all cases, this reduces to r(Ri, Rj) = +1. For example, for the common 0:45 

geometry (illumination angle of 0°, viewing angle of 45°), the spectral reflectance factor 

of the standard is obtained from signals proportional to the illuminating and reflected 

radiant flux, SI and SR, and from the projected solid angle Ω from the center of the 

standard to the entrance pupil of the receiver, and is given by 

Ω = 1

,I

,R

,s

i

i

i S

S

R . (4.16) 

The common factor is Ω, and applying Eq. (4.15) yields r(Ri, Rj) = +1. 

 Finally, uncertainty in the wavelength of the instrument can be correlated between 

wavelengths. If the uncertainty arises from random effects, then r(Ri, Rj) = 0, as usual. 

However, the uncertainty can also arise from systematic effects. For example, an nonlinearity in wavelength may be constant or vary in a systematic manner across the 

wavelength range of the instrument. The sensitivity coefficients are given by Eq. (4.9), 

and the correlation coefficient is 

 

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

∂

⎟

⎠

⎞ ⎜

⎝

⎛

∂

⋅

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

∂

⎟

⎠

⎞ ⎜

⎝

⎛

∂

= λ λ

j

j

i

i

ji

S

S

S

S

RRr ,s ,s sgn),( sgn . (4.17) 

Therefore, the correlation coefficient for an uncertainty caused by wavelength depends 

upon the slopes of the signal ratios at the two wavelengths, and can have values of –1, 0, 

or +1. 

 The correlation coefficients, both between signals at the same wavelength and 

between spectral reflectance factors at different wavelengths, are summarized in Table 1 

for the common sources of uncertainty considered here. 

D. Calculations Based on Tristimulus Values 

 The uncertainties in spectral reflectance factor, u(Ri) as given in Section IV.B, and 

their correlations between wavelengths, r(Ri, Rj) as given in Section IV.C, are used to 

calculate the uncertainty in the tristimulus values. The uncertainties in the tristimulus 

values are then used to calculate the uncertainty in the chromaticity coordinates or the 

color space values, taking correlations between tristimulus values into account. 

 The uncertainty in the tristimulus value X is presented here, and similar 

considerations apply for Y and Z. For a given source of uncertainty, the uncertainty in X

is, from Eq. (3.4), given by 
11 

)( 2)( .)()(),(

1

1 1

2

2

1

2

c iji j

j

m

i

m

ij i

i

m

i i

RuRuRRr

R

X

R

X Ru

R

X Xu ⋅⋅⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂ = ∑ ∑ ∑

−

= = + =

(4.18) 

From Eq. (2.3), the sensitivity coefficients are 

i

i

W

R

X = ,x ∂∂ , (4.19) 

which, when substituted into Eq. (4.18) yield 

)( 2)( )()(),(

1

1 1

,x,x

2

1

2

,x

2

c iji j

m

i

m

ij

i ji

m

i

∑ i RuWXu +⋅= ∑ ∑ ⋅⋅⋅⋅ RuRuRRrWW

−

= = + =

 . (4.20) 

If r(Ri, Rj) = 0 for all wavelengths, Eq. (4.20) reduces to 

∑=

⋅=

m

i

RuWXu ii

1

22

,x

2

c )( )( , (4.21) 

while if r(Ri, Rj) = +1 for all wavelengths it becomes 

2

1

,x

2

c )( )( ⎥

⎦

⎤ ⎢

⎣

⎡

∑ ⋅= =

m

i

RuWXu ii . (4.22) 

However, if r(Ri, Rj) = –1 for all wavelengths or depends upon the wavelengths, then 

Eq. (4.20) does not reduce to a simple form. 

 The correlations between tristimulus values X and Y are presented here, and as 

before similar considerations apply for correlations between X and Z and between Y and 

Z. Substituting Eq. (4.19) and its equivalent for Y into Eq. (3.11) yields 

[ ] .)()(),(

)(

)()(),(

),( )(

1

1 1

,y,x,y,x

2

1

,y,x

1

1 1

2

1

iji j

m

i

m

ij

ijji

i

m

i

ii

iji j

m

i

m

ij ijji

i

i

m

i i

RuRuRRrWWWW

RuWW

RuRuRRr

R

Y

R

X

R

Y

R

X

Ru

R

Y

R

X YXu

+ ⋅⋅⋅⋅+⋅

⋅⋅=

⋅⋅⋅ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

∂

∂

∂

∂

+

∂

∂

∂

∂

+

∂

∂

∂

∂ =

∑ ∑

∑

∑ ∑

∑

−

= + =

=

−

= + =

=

 (4.23) 

Substituting Eq. (4.23) and the square roots of Eq. (4.20) and its equivalent for Y into 

Eq. (3.3) yields the correlation coefficient between X and Y, 

)()(

),( ),( YuXu

YXu YXr ⋅ = . (4.24) 

Note that r(X, Y) can have values between –1 and +1, depending on the values of 

r(Ri, Rj), u(Ri), and u(Rj), and hence is a function of the source of uncertainty. 

 The uncertainties in chromaticity coordinates and color space values are obtained 

from their dependence on the tristimulus values. Let Γ represent either a chromaticity 

coordinate or a color space value. Using Eq. (3.4) with xk = X, Y, or Z, the combined 

uncertainty from each source is given by 
12 

2 .)()(),(

2 )()(),(

2 )()(),(

)( )( )( )(2

2

2

2

2

2

2

ZuYuZYr

ZY

ZuXuZXr

ZX

YuXuYXr

YX

Zu

Z

Yu

Y

Xu

X

uc

⎟ ⋅⋅⋅ ⎠

⎞ ⎜

⎝

⎛

∂

Γ∂ ⎟

⎠

⎞ ⎜

⎝

⎛

∂

Γ∂

+

⎟ ⋅⋅⋅ ⎠

⎞ ⎜

⎝

⎛

∂

Γ∂ ⎟

⎠

⎞ ⎜

⎝

⎛

∂

Γ∂

+

⎟ ⋅⋅⋅ ⎠

⎞ ⎜

⎝

⎛

∂

Γ∂ ⎟

⎠

⎞ ⎜

⎝

⎛

∂

Γ∂

+

⎟

⎠

⎞ ⎜

⎝

⎛

∂

Γ∂ ⎟ + ⎠

⎞ ⎜

⎝

⎛

∂

Γ∂ ⎟ + ⎠

⎞ ⎜

⎝

⎛

∂

Γ∂ =Γ

 (4.25) 

 The sensitivity coefficients for common chromaticity coordinates and color space 

values are given in Appendix A, while the standard uncertainties u(X), u(Y), and u(Z) are

given by Eq. (4.18) and its equivalents, and the correlation coefficients r(X, Y), r(X, Z) 

and r(Y, Z) are given by Eq. (4.24) and its equivalents. An interesting result for the 

chromaticity coordinates x and y is that, if the standard uncertainty is a constant 

proportion of the reflectance factor at all wavelengths, and the correlation coefficient 

between wavelengths is +1, then the uncertainty in x and y is zero. This is usually the 

case for an uncertainty in the standard caused by systematic effects. 

 The uncertainties in chroma C and hue h can be derived is a similar fashion. 

Using the L*, a*, b* color space as an example, the chroma Cab* is given by 

2 2

ab += baC *)(*)(* (4.26) 

 and the combined uncertainty in chroma is given by 

.*)(*)(*)*,( *

*

*

*

2

*)( *

*

*)( *

*

*)(

ab ab

2

2

2 ab

2

ab

ab

2

c

buaubar

b

C

a

C

bu

b

C

au

a

C Cu

⎟ ⋅⋅⋅ ⎠

⎞ ⎜

⎝

⎛

∂

∂ ⎟

⎠

⎞ ⎜

⎝

⎛

∂

∂

+

⎟

⎠

⎞ ⎜

⎝

⎛

∂

∂ ⎟ + ⎠

⎞ ⎜

⎝

⎛

∂

∂ =

 (4.27) 

A similar equation applies for hab*. The correlation coefficient r(a*, b*) is given by 

*)(*)(

*)*,( *)*,( buaubau bar ⋅ = , (4.28) 

where u(a*) and u(b*) are given by Eq. (4.25) and u(a*, b*), from Eq. (3.11), is given by 

.)()(),(

****

)()(),(

 

)()(),(

** **

)(

**

*)*,( 2

ZuYuZYr

Y

b

Z

a

Z

b

Y

a

ZuXuZXr

X

b

Z

a

Z

b

X

a

YuXuYXr

X

b

Y

a

Y

b

X

a

Yu

Y

b

Y

a bau

⋅⋅⋅ ⎥

⎦

⎤ ⎢

⎣

⎡

∂

∂

∂

∂

+

∂

∂

∂

∂

+

⋅⋅⋅ ⎥

⎦

⎤ ⎢

⎣

⎡

∂

∂

∂

∂

+

∂

∂

∂

∂

+

⋅⋅⋅ ⎥

⎦

⎤ ⎢

⎣

⎡

∂

∂

∂

∂

+

∂

∂

∂

∂

+

∂

∂

∂

∂ =

 (4.29) 

Note that the correlation coefficient r(a*, b*) applies to both Cab* and hab*. 
13 

E. Calculations Based on Spectral Reflectance Factors 

 The chromaticity coordinates and color space values can also be calculated 

directly from the spectral reflectance factor by substituting Eqs. (2.3) to (2.5) into their 

definitions. The uncertainties in the chromaticity coordinates and color space values are 

then calculated from the uncertainties in the spectral reflectance factor using Eq. (3.4). 

The sensitivity coefficients in terms of the spectral reflectance factor are obtained by 

applying the chain rule, so again letting Γ represent either a chromaticity coordinate or a 

color space value 

i i i Ri

Z

ZR

Y

YR

X

XR ∂

∂

∂

∂Γ

+

∂

∂

∂

Γ∂

+

∂

∂

∂

Γ∂ = ∂Γ∂ . (4.30) 

The derivatives with respect to the tristimulus values are given in Appendix A. Using 

Eq. (4.30) with Eq. (3.4) yields the combined uncertainty in Γ for each source of 

uncertainty, 

)( 2)( )()(),(

1

1 1

2

2

1

2

c iji j

j

m

i

m

ij i

i

m

i i

RuRuRRr

RR

Ru

R

u ⋅⋅⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

Γ∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

Γ∂

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

Γ∂ =Γ ∑ ∑ ∑

−

= = + =

. (4.31) 

The uncertainties in chroma and hue in terms of uncertainties in the spectral reflectance 

factor are also obtained by applying the chain rule. Using the example of Cab*, 

i i Ri

b

b

C

R

a

a

C

R

C

∂

∂

∂

∂

+

∂

∂

∂

∂ = ∂∂ *

*

**

*

ab ab

**

ab , (4.32) 

where ∂a*/∂Ri and ∂b*/∂Ri are given by Eq. (4.30). 

F. Summary 

 A step-by-step summary of the procedure for analyzing uncertainties in 

reflectance colorimetry is presented below. The relevant equations and sections for each 

step are given in brackets. The variable Γ below represents the final value, either a 

chromaticity coordinate such as u’ or v’ or a color space value such as L*, a*, or b*. 

1. Identify all independent sources of uncertainty. 

2. Assign standard uncertainties to these sources, as described in the Guide. 

3. Calculate the standard uncertainty in spectral reflectance factor, u(Ri), for each source 

of uncertainty, using the measurement equation and taking any correlations between 

signals at the same wavelength into account with the correlation coefficient r(Si, Ss,i). 

[Eq. (2.2) and Section IV.B] 

4. Calculate the correlation coefficient between spectral reflectance factors at different 

wavelengths, r(Ri, Rj), for each source of uncertainty. [Section IV.C] 

5. If using the tristimulus values to calculate Γ

 a. Calculate the uncertainties in the tristimulus values based on u(Ri) and r(Ri, Rj) 

from steps 3 and 4 for each source of uncertainty. [Eq. (4.18)] 

 b. Calculate the correlation coefficients between the tristimulus values for each 

source of uncertainty. [Eq. (4.24)] 

 c. Calculate the sensitivity coefficients for Γ with respect to the tristimulus values, 

∂Γ/∂X, ∂Γ/∂Y, and ∂Γ/∂Z. [Appendix A] 
14 

 d. Calculate the uncertainty in Γ using the results from steps a, b, and c for each 

source of uncertainty. [Eq. (4.25)] 

6. If using the spectral reflectance factor to calculate Γ

 a. Calculate the sensitivity coefficient for Γ with respect to Ri. [Eq. (4.30) and 

Appendix A] 

 b. Calculate the uncertainty in Γ using the results from steps 3 and 4 for each source 

of uncertainty. [Eq. (4.31)] 

7. Calculate the final uncertainty in Γ using the root-sum-square of the uncertainties 

from all the sources obtained in step 5.d or 6.b. 

V. Example Calculations 

 The general concepts of uncertainty analysis for reflectance colorimetry detailed 

above are applied to an example. The uncertainties in the L*, a*, b* color space values 

from the common sources of uncertainty considered above for four representative 

specimens are calculated for the Commission Internationale de l’Eclairage (CIE) D65 

Standard Illuminant and the 1964 Standard Colorimetric Observer.16 The signals used in 

the calculations are taken from measurements on a reference instrument. The instrument 

consists of a quartz-tungsten-halogen lamp, a monochromator illuminator, and a silicon 

photodiode detector, the geometry of measurement is 0:45, and the wavelength increment 

and spectral bandwidth were both 5 nm. While the uncertainties calculated below are 

specific to the specimens and instrument, the purpose of this example is to illustrate the 

uncertainty analysis so that it can be more easily applied to other specimens and 

instruments. 

 The spectral reflectance factors of the four representative specimens and the 

standard are shown in Fig. 1, along with designations based upon their color. These 

designations will be used for the remainder of this section. The standard was 

Spectralon*

, which has a nearly constant reflectance factor over the wavelength range. 

The gray specimen is spectrally neutral, while the red specimen has a reflectance factor 

that is small at short wavelengths and then increases rapidly to become large at long 

wavelengths. The green and deep blue specimens have maxima of their reflectance 

factors at different wavelengths. The wavelength range in Fig. 1 extends past the limits 

for color calculations, 380 nm to 780 nm, because stray-light contributes to the signal 

from all wavelengths at which the detector is sensitive, in this case as long as 1100 nm. 

 The weighting factors for the D65 Illuminant and the 1964 Observer are shown in 

Fig. 2. The signals from the instrument when measuring the Spectralon standard are 

shown in Fig. 3 as a function of wavelength. The features in the signal at wavelengths of 

460 nm and 750 nm are due to a change of the order-sorting filter. From the 

measurement equation, Eq. (2.2), since the reflectance factor of the standard is nearly 

constant, the signals from the specimen are essentially the reflectance factors multiplied 

by the signal shown in Fig. 3. 

 

*

 Certain commercial equipment, instruments, or materials are identified in this paper to foster 

understanding. Such identification does not imply recommendation or endorsement by the National 

Institute of Standards and Technology, nor does it imply that the materials or equipment identified are 

necessarily the best available for the purpose. 
15 

 Following the first two steps in Section IV.F, the sources of uncertainty are the 

same as those considered above and listed in Table 1. The standard uncertainties for each 

source are chosen to be representative of a high-quality instrument. The uncertainty due 

to noise in the signals is a constant 0.1 % for both the specimen and the standard, so 

001.0 )( )(

,s

,s ==

i

i

i

i

S

Su

S

Su . (5.1) 

The uncertainty due to an non-linearity of the specimen affects only the signal from the 

specimen, and is also a constant 0.1 %. The uncertainty due to stray-light affects both the 

specimen and standard signals. The uncertainty is modeled by 

∑ − ⋅= i

N ( i SSu i 10) and (5.2) 

∑ − ⋅= i

N

,s i)( SSu ,s i 10 , (5.3) 

where N is the stray-light rejection, taken to be N = 6 for this example. The uncertainty 

due to wavelength is u(λ) = 0.1 nm across the entire wavelength range. The uncertainty 

in the reflectance factor of the standard due to random effects is 0.02 %, while the 

uncertainty due to systematic effects is 0.2 %. These uncertainties for the standard are 

typical of those from a national metrology institute.17

 The third step in Section IV.F is to calculate the standard uncertainty in the 

spectral reflectance factor for each source of uncertainty. Using Eq. (5.1) for the 

uncertainty in the signals due to noise, and Eq. (4.4), the standard uncertainty in the 

reflectance factor is 

0014.0)001.0()001.0( )( c 2 2 = + =

i

i

R

Ru . (5.4) 

For an uncertainty due to an non-linearity affecting only the signal from the specimen, 

Eq. (4.5) is applicable, and so 

 001.0 )( c =

i

i

R

Ru . (5.5) 

However, for an uncertainty due to stray-light, which affects both signals, Eq. (4.6) is 

applicable, so combining this with Eqs. (5.2) and (5.3) and taking the positive square root 

yields 

i

i

i

i

i

i

S

S

S

S

R

Ru

,s

c 6 ,s 10 )( ∑∑ = − − . (5.6) 

 For an uncertainty due to wavelength, Eq. (4.10) applies, and so 

nm1.0 )( ,s ,s c ⋅ ∂

⎟

⎠

⎞ ⎜

⎝

⎛

∂

= λ

i

i

i

i

i

i

S

S

R

R

R

Ru . (5.7) 

For the uncertainties in the standard due to both random and systematic effects, Eq. (4.8) 

applies, so 

0002.0 )( c =

i

i

R

Ru (5.8) 
16 

for the random effects and 

002.0 )( c =

i

i

R

Ru (5.9) 

for the systematic effects. 

 The fourth step in Section IV.F is to calculate the correlation coefficient between 

spectral reflectance factors at different wavelengths for each source of uncertainty. These 

are given in Table 1. The fifth and sixth steps are to calculate the uncertainties in L*, a*, 

and b* from either the tristimulus values or directly from the spectral reflectance factors. 

Both approaches are presented here. 

 Given the uncertainties in spectral reflectance factor and their correlations for 

each source of uncertainty, and the weighting factors for the Illuminant and Observer, the 

uncertainties in the tristimulus values are given by Eq. (4.20) for X and its equivalent for 

Y and Z. The correlation coefficients between tristimulus values are given by Eqs. (4.23) 

and (4.24) for X, with similar equations for Y and Z. For uncertainties due to systematic 

effects (those with r(Ri, Rj) = –1 or +1), the correlation coefficients are identically one. 

The correlation coefficients for uncertainties due to random effects are given in Table 2 

for each specimen. Referring to Figs. 1 and 2, the correlation coefficient for X and Y is 

largest for all the colors in Table 2 because these weighting functions have the most 

overlap with each other. This behavior is enhanced for the red specimen because the 

reflectance factor is small for wavelengths shorter than 600 nm. 

 The uncertainties in L*, a*, and b* are calculated using Eq. (4.25). The 

sensitivity coefficients ∂L*/∂Y, ∂a*/∂X, ∂a*/∂Y, ∂b*/∂Y, and ∂b*/∂Z are given by 

Eqs. (A.13) to (A.17), the uncertainties in the tristimulus values u(X), u(Y), and u(Z) are 

given by Eq. (4.20) and its equivalents, and the correlation coefficients between the 

tristimulus values r(X, Y), r(X, Z) and r(Y, Z) are either +1 for uncertainties caused by 

systematic effects or are given in Table 2 for uncertainties caused by random effects. 

For example, L* is a function only of Y, so the uncertainty in L* is given by 

)( 1

3

116 *)( 2

2

3

1

2

2

c Yu

YY

Lu

n ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = (5.10) 

while a* is a function of X and Y so the uncertainty in a* is given by 

.)()(),( 11

3

500

3

500 2

)( 1

3

500 )( 1

3

500 *)(

3

1

n

2

3

1

n

2

2

2

3

1

n

2

2

2

3

1

n

2

2

c

YuXuYXr

YYXX

Yu

YY

Xu

XX

au

⋅⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⋅⋅−

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

− ⎛

+ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

 (5.11) 

 The uncertainties in Cab* and hab* are calculated using Eq. (4.27) and its 

equivalent for hab* and the uncertainties u(a*) and u(b*) given by the procedure detailed 

in the previous paragraph. The correlation coefficient r(a*, b*) = –1 or +1 for 

uncertainties caused by systematic effects, and is given in Table 2 for uncertainties 

caused by random effects. The sensitivity coefficients ∂Cab*/∂a*, ∂Cab*/∂b*, ∂hab*/∂a*, 

and ∂hab*/∂b* are given by Eqs. (A.18) to (A.21). For example, the uncertainty in Cab* is 

given by 
17 

.*)(*)(*)*,( *

*

*

*

2

*)( *

*

*)( *

*

*)(

ab ab

2

2

ab

2

2

ab

ab

2

c

buaubar

C

b

C

a

bu

C

b

au

C

a Cu

⋅⋅⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⋅+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

 (5.12) 

 To calculate the uncertainties is L*, a*, and b* directly from the spectral 

reflectance factors, Eq. (4.31) is used. The sensitivity coefficients are given by Eq. (4.30) 

using Eqs. (A.13) to (A.17) and (4.19). For example, the sensitivity coefficients for L* 

are given by 

i

i

W

R YY

L

,y

3

1

n

2

1

3

116* ⋅

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂ , (5.13) 

for a* by 

i i

i

W

YY

W

R XX

a

,y

3

1

n

,x 2

3

1

n

2

1

3

1 500

3

500* ⋅

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

− ⎛

+⋅ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂ , (5.14) 

and for Cab* by 

. 1

3

1 200

3

200

*

*

1

3

1 500

3

500

*

* *

,z

3

1

n

,y 2

3

1

n

2

2

ab

,y

3

1

n

,x 2

3

1

n

2

ab

ab

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⋅

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

− ⎛

+⋅ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⋅

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

− ⎛

+⋅ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

i i

i i

i

W

ZZ

W

C YY

b

W

YY

W

C XX

a

R

C

 (5.15) 

The uncertainties in the values are given by substituting the appropriate equation from 

Eqs. (5.13) to (5.15) into Eq. (4.31). Using L* as an example, this results in 

 

,)()(),( 1

3

116 2

)( 1

3

116 *)(

,y,y

2

1

1 1

3

1

n

2

1

2

2

,y

3

1

n

2

2

c

jijiji

m

i

m

ij

m

i

i i

RuRuRRrWW

YY

RuW

YY

Lu

⋅⋅⋅⋅⋅ ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⋅ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ =

∑ ∑

∑

−

= + =

=

 (5.16) 

where u(Ri), u(Rj), and r(Ri, Rj) are the uncertainties and correlation coefficients 

appropriate to the source of uncertainty. 

 The absolute uncertainties (k = 1) in the color scale values L*, a*, and b* from 

each source of uncertainty considered here are shown in Fig. 4 for the four representative 

specimens. Note that the vertical scale for the red specimen is three times that of the 

other specimens. Several conclusions can be drawn from the results presented in Fig. 4. 

Signal noise contributes uncertainty for all the specimens, with the uncertainties in a* and 

b* larger than that of L*. The effect of correlation is also apparent. Even though the 

uncertainties due to noise and non-linearity are both 0.1 % of the signal, the former is 

uncorrelated between wavelengths, while the later is correlated. This results in the 

uncertainty in L* being larger for the correlated uncertainty and the uncertainties in a* 
18 

and b* being smaller. The uncertainties caused by random effects in the standard are 

simply reduced in magnitude from those caused by signal noise. 

 The uncertainties caused by systematic effects in the standard are significant for 

all the specimens, and can be the largest source of uncertainty for those specimens with 

nearly constant reflectance factor over the visible wavelength region. The uncertainties 

caused by wavelength and stray-light also depend on the spectral shape of the reflectance 

factor. The largest uncertainties due to wavelength are for the red specimen, whose 

reflectance factor increases rapidly with wavelength. The reflectance factors of the deep 

blue and green specimens also depend upon wavelength, but the slope of the reflectance 

factor, and hence the sensitivity coefficients, changes sign several times over the 

wavelength range. In addition, the reflectance factor of the deep blue specimen is nearly 

constant over much of the visible wavelength range. The uncertainty due to stray-light is 

largest for the red specimen because of the combination of the spectral reflectance factor 

of this color, shown in Fig. 1, and the radiant flux illuminating the specimen, which is 

proportional to the signal in Fig. 3. Since the reflectance factor is small over the 

wavelength range where the flux is also low, while both are large at longer wavelengths, 

there is a significant stray-light contribution to the signals at short wavelengths and hence 

a contribution to the uncertainty in L*, a*, and b*. Finally, the uncertainties are smallest 

for those specimens with a spectral reflectance factor similar to that of the standard, 

namely gray, while the uncertainties are largest for those specimens that are most 

dissimilar. 

 The results shown in Fig. 4 are not sensitive to the Standard Illuminant or 

Observer chosen for the analysis, provided an illuminant with sharp spectral features at 

only one or a few wavelengths is not used. Also, the uncertainties are proportional to the 

magnitudes of uncertainties of the sources. If the uncertainty for one of the sources is 

increased by a factor of ten, then the corresponding uncertainty shown in Fig. 4 is also 

increased by a factor of ten. The combined uncertainty in L*, a*, or b* is the root-sumsquare of the uncertainties due to all the sources. For example, for the red specimen, the 

combined uncertainty in b* is uc(b*) = [0.0102

 + 0.0122 + 0.2842 + 0.0712 + 0.0012 + 

0.0242

]

1/2 = 0.294. The expanded uncertainty of b*, with a coverage factor k = 2, is 

0.588. 

VI. Summary 

 Uncertainties in reflectance colorimetry are important, and have received attention 

over the years. Application of the ISO Guide to the Expression of Uncertainty in 

Measurement allows the uncertainties to be calculated using a systematic, analytical 

approach. This requires consideration of the measurement equation, relating signals to 

spectral reflectance factor, and correlations both between signals at the same wavelength 

and between reflectance factors at different wavelengths. This approach was detailed for 

common sources of uncertainty and applied to representative object colors and instrument 

parameters. 

Acknowledgements 

James Gardner is thanked for several useful discussions. 
19 

References of Appendix B 

1. Nimeroff I. Propagation of errors in spectrophotometric colorimetry. J Opt Soc Am 

1953; 43: 531-533. 

2. Shipley T, Walker GL. Chromatic significance of spectrophotometric errors. J Opt 

Soc Am 1956; 46: 1052-1060. 

3. Nimeroff I. Propagation of errors in tristimulus colorimetry. J Opt Soc Am 1957; 

47: 697-702. 

4. Robertson AR,. Colorimetric significance of spectrophotometric errors. J Opt Soc 

Am 1967; 57: 691-698. 

5. Kishner SJ. Effect of spectrophotometric errors on color difference. J Opt Soc Am 

1977; 67: 772-778. 

6. Wyszecki G, Stiles WS, Color Science: Concepts and Methods, Quantitative Data 

and Formulae, 2nd Edition. New York: John Wiley and Sons; 1982, p. 328-330. 

7. Berns RS, Petersen KH. Empirical modeling of systematic spectrophotometric 

errors. Color Res Appl 1988; 13: 243-256. 

8. Fairchild MD, Reniff L. Propagation of random errors in spectrophotometric 

colorimetry. Color Res Appl 1991; 16: 360-367. 

9. Burns PD, Berns RS. Error propagation analysis in color measurement and imaging. 

Color Res Appl 1997; 22: 280-289. 

10. Guide to the Expression of Uncertainty in Measurement. Geneva: International 

Organization for Standardization; 1993. 

11. Gardner JL, Frenkel RB. Correlation coefficients for tristimulus response value 

uncertainties. Metrologia 1999; 36: 477-480. 

12. Gardner JL. Uncertainty estimation in colour measurement. Color Res Appl 2000; 

25: 349-355. 

13. Standard Practice for Computing the Colors of Objects by Using the CIE System, 

E 308. West Conshohocken, Pennsylvania: American Society for Testing and 

Materials; 1999. 

14. Zwinkels JC. Errors in colorimetry caused by the measuring instrument. Text Chem 

Color 1989; 21: 23-29. 

15. Good Practice Guide to Surface Colour Measurement. European Commission, EUR 

19552; 2001. 

16. CIE Publication No. 15.2, Colorimetry, 2nd Edition. Vienna, Austria: Commission 

Internationale de l’Eclairage; 1986. 

17. Barnes PY, Early EA, Parr AC. Spectral reflectance, NIST Special Publication 250-

48. U. S. Dept. of Commerce; 1998. 
20 

Appendix A. Sensitivity coefficients for common chromaticity coordinates and color 

space values 

1. Chromaticity coordinates 

 a. x, y 

 2 ZYX )(

ZY

X

x

++

+ = ∂∂ (A.1) 

 2 ZYX )(

X

Y

x

++

− = ∂∂ (A.2) 

 2 ZYX )(

X

Z

x

++

− = ∂∂ (A.3) 

 2 ZYX )(

Y

X

y

++

− = ∂∂ (A.4) 

 2 ZYX )(

ZX

Y

y

++

+ = ∂∂ (A.5) 

 2 ZYX )(

Y

Z

y

++

− = ∂∂ (A.6) 

 b. u’, v’ 

 2

)315(

1260

ZYX

ZY

X

u

++

+ = ∂∂ ′ (A.7) 

 2

)315(

60

ZYX

X

Y

u

++

− = ∂∂ ′ (A.8) 

 2

)315(

12

ZYX

X

Z

u

++

− = ∂∂ ′ (A.9) 

 2

)315(

9

ZYX

Y

X

v

++

− = ∂∂ ′ (A.10) 

 2

)315(

279

ZYX

ZX

Y

v

++

+ = ∂∂ ′ (A.11) 

 2

)315(

27

ZYX

Y

Z

v

++

− = ∂∂ ′ (A.12) 

2. Color space values 

 a. L*, a*, b* 

 

3

1

n

2

1

3

116*

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

Y YY

L (A.13) 
21 

 

3

1

n

2

1

3

500*

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

X XX

a (A.14) 

 

3

1

n

2

1

3

500*

⎟

⎟

⎠

⎞

⎜

⎜

⎝

− ⎛ = ∂∂

Y YY

a (A.15) 

 

3

1

n

2

1

3

200*

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

YYY

b (A.16) 

 

3

1

n

2

1

3

200*

⎟

⎟

⎠

⎞

⎜

⎜

⎝

− ⎛ = ∂∂

Z ZZ

b (A.17) 

 *

*

*

*

ab

ab

C

a

a

C = ∂∂ (A.18) 

 *

*

*

*

ab

ab

C

b

b

C = ∂∂ (A.19) 

 ( )2

ab

ab

*

*

*

*

C

b

a

h − = ∂∂ (A.20) 

 ( )2

ab

ab

*

*

*

*

C

a

b

h = ∂∂ (A.21) 

 b. L*, u*, v* 

 

3

1

n

2

1

3

116*

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂

Y YY

L (A.22) 

 2

)315(

1260 *13*

ZYX

ZY L

X

u

++

+ ⋅= ∂

∂ (A.23) 

 n 2

3

1

n

2 )315(

60 *13)( 1

3

116 13*

ZYX

X Luu

Y YY

u

++

− ′ − ′ ⋅+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂ (A.24) 

 2

)315(

12 *13*

ZYX

X L

Z

u

++

− ⋅= ∂∂ (A.25) 

 2

)315(

9 *13*

ZYX

Y L

X

v

++

− ⋅= ∂∂ (A.26) 

 n 2

3

1

n

2 )315(

279 *13)( 1

3

116 13*

ZYX

ZX Lvv

Y YY

v

++

+ ′ − ′ ⋅+ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∂

∂ (A.27) 

 2

)315(

27 *13*

ZYX

Y L

Z

v

++

− ⋅= ∂∂ (A.28) 

 *

*

*

*

uv

uv

C

u

u

C = ∂∂ (A.29) 
22 

 *

*

*

*

uv

uv

C

v

v

C = ∂∂ (A.30) 

 ( )2 *

*

*

*

uv

uv

C

v

u

h − = ∂∂ (A.31) 

 ( )2 *

*

*

*

uv

uv

C

u

v

h = ∂∂ (A.32) 
23 

Tables 

Table 1. Correlation coefficients for common sources of uncertainty 

 Correlation Coefficient 

Source of Uncertainty r(Si, Ss,i) r(Ri, Rj) 

Signal 

 Noise 0 0 

 Non-linearity 

 One signal -- +1 

 Two signals +1 +1 a [Eq. (4.11)] 

Standard 

 Random effects -- 0 

 Systematic effects -- +1 b [Eq. (4.15)] 

Wavelength -- –1, 0, or +1 [Eq. (4.17)] 

Notes: 

a

 r(Ri, Rj) = +1 if Ss,i > Si or Ss,i < Si for all wavelengths i. Otherwise, r(Ri, Rj) = –1, 0, or 

+1 depending on the magnitudes of the signals. 

b

 +1 in nearly all cases 

Table 2. Correlation coefficients between tristimulus values and a* and b* due to 

uncertainties caused by random effects 

Correlation 

Coefficient 

Specimen Color 

Gray Deep Blue Green Red 

r(X, Y) 0.7746 0.7289 0.7299 0.9648 

r(X, Z) 0.2957 0.3070 0.2299 0.0281 

r(Y, Z) 0.1381 0.1549 0.1850 0.0330 

r(a*, b*) -0.4215 -0.3653 -0.4344 0.5539 
24 

Figure Captions 

Figure 1. Reflectance factor as a function of wavelength for the example specimens and 

the standard. The colors of the specimens are indicated. 

Figure 2. Weighting factors as a function of wavelength for the D65 Illuminant and 1964 

Observer. 

Figure 3. Signal as a function of wavelength when measuring the standard. 

Figure 4. Absolute uncertainties (k = 1) in the L*, a*, b* color scale values caused by the 

indicated sources of uncertainty for the example specimens. The sources of 

uncertainty are noise in the signals, an non-linearity of the specimen signals, 

stray-light contribution to the signals, the wavelength of the instrument (WL), 

and the reflectance factor of the standard (Std) caused by random (R) and 

systematic (S) effects. 
25 

300 400 500 600 700 800 900 1000 1100

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Green

Deep Blue

Red

Gray

Standard

Reflectance Factor

Wavelength [nm]

Figure 1. 

300 400 500 600 700 800

0

1

2

3

4

5

6

7

8

9

10

11

Wz

Wy

Wx

Weighting Factor

Wavelength [nm]

Figure 2. 
26 

300 400 500 600 700 800 900 1000 1100

0

10

20

30

40

50

Signal [mV]

Wavelength [nm]

Figure 3. 
27 

Noise Offset Stray WL Std (R)Std (S)

0.00

0.03

0.06

0.09

0.12 (c) Green

Uncertainty

Source of Uncertainty

 L*

 a*

 b*

Noise Offset Stray WL Std (R)Std (S)

0.00

0.09

0.18

0.27

0.36 (d) Red

Uncertainty

Source of Uncertainty

 L*

 a*

 b*

Noise Offset Stray WL Std (R)Std (S)

0.00

0.03

0.06

0.09

0.12

(b) Deep Blue

Uncertainty

Source of Uncertainty

 L*

 a*

 b*

Noise Offset Stray WL Std (R)Std (S)

0.00

0.03

0.06

0.09

0.12 (a) Gray

Uncertainty

Source of Uncertainty

 L*

 a*

 b*

Figure 4.