Javier Hernández-Andrés, Raymond L. Lee, and Javier Romero, "Calculating correlated color temperatures across the entire gamut of daylight and skylight chromaticities," Appl. Opt. 38, 5703-5709 (1999)

Natural outdoor illumination daily undergoes large changes in its
correlated color temperature (CCT), yet existing equations for
calculating CCT from chromaticity coordinates span only part of this
range. To improve both the gamut and accuracy of these CCT
calculations, we use chromaticities calculated from our measurements of
nearly 7000 daylight and skylight spectra to test an equation that
accurately maps CIE 1931 chromaticities x and y
into CCT. We extend the work of McCamy [Color Res. Appl. 12, 285–287(1992)] by using a chromaticity epicenter
for CCT and the inverse slope of the line that connects it to
x and y. With two epicenters for different
CCT ranges, our simple equation is accurate across wide chromaticity
and CCT ranges (3000–10^{6} K) spanned by daylight and
skylight.

J. Schanda, M. Mészáros, G. Czibula, “Calculating correlated color temperature with a desktop programmable calculator,” Color Res. Appl. 3, 65–68 (1978).
[CrossRef]

J. Schanda, M. Dányi, “Correlated color temperature calculations in the CIE 1976 UCS diagram,” Color Res. Appl. 2, 161–163 (1977).
[CrossRef]

M. Krystek, “An algorithm to calculate correlated colour temperature,” Color Res. Appl. 10, 38–40 (1985).
[CrossRef]

Q. Xingzhong, “Formulas for computing correlated color temperature,” Color Res. Appl. 12, 285–287 (1987).
[CrossRef]

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates,” Color Res. Appl. 17, 142–144 (1992).
[CrossRef]

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates (Erratum),” Color Res. Appl. 18, 150 (1993).

LI-1800 spectroradiometer from Li-Cor, Inc., 4421 Superior St., Lincoln, Neb. 68504-1327.

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates (Erratum),” Color Res. Appl. 18, 150 (1993).

1992

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates,” Color Res. Appl. 17, 142–144 (1992).
[CrossRef]

1987

Q. Xingzhong, “Formulas for computing correlated color temperature,” Color Res. Appl. 12, 285–287 (1987).
[CrossRef]

1985

M. Krystek, “An algorithm to calculate correlated colour temperature,” Color Res. Appl. 10, 38–40 (1985).
[CrossRef]

1978

J. Schanda, M. Mészáros, G. Czibula, “Calculating correlated color temperature with a desktop programmable calculator,” Color Res. Appl. 3, 65–68 (1978).
[CrossRef]

J. Schanda, M. Mészáros, G. Czibula, “Calculating correlated color temperature with a desktop programmable calculator,” Color Res. Appl. 3, 65–68 (1978).
[CrossRef]

Dányi, M.

J. Schanda, M. Dányi, “Correlated color temperature calculations in the CIE 1976 UCS diagram,” Color Res. Appl. 2, 161–163 (1977).
[CrossRef]

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates (Erratum),” Color Res. Appl. 18, 150 (1993).

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates,” Color Res. Appl. 17, 142–144 (1992).
[CrossRef]

Mészáros, M.

J. Schanda, M. Mészáros, G. Czibula, “Calculating correlated color temperature with a desktop programmable calculator,” Color Res. Appl. 3, 65–68 (1978).
[CrossRef]

J. Schanda, M. Mészáros, G. Czibula, “Calculating correlated color temperature with a desktop programmable calculator,” Color Res. Appl. 3, 65–68 (1978).
[CrossRef]

J. Schanda, M. Dányi, “Correlated color temperature calculations in the CIE 1976 UCS diagram,” Color Res. Appl. 2, 161–163 (1977).
[CrossRef]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982), pp. 144–146, 306–310.

Tarrant, A. W. S.

A. W. S. Tarrant, “The spectral power distribution of daylight,” Trans. Illum. Eng. Soc. 33, 75–82 (1968).

J. Schanda, M. Mészáros, G. Czibula, “Calculating correlated color temperature with a desktop programmable calculator,” Color Res. Appl. 3, 65–68 (1978).
[CrossRef]

J. Schanda, M. Dányi, “Correlated color temperature calculations in the CIE 1976 UCS diagram,” Color Res. Appl. 2, 161–163 (1977).
[CrossRef]

M. Krystek, “An algorithm to calculate correlated colour temperature,” Color Res. Appl. 10, 38–40 (1985).
[CrossRef]

Q. Xingzhong, “Formulas for computing correlated color temperature,” Color Res. Appl. 12, 285–287 (1987).
[CrossRef]

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates,” Color Res. Appl. 17, 142–144 (1992).
[CrossRef]

C. S. McCamy, “Correlated color temperature as an explicit function of chromaticity coordinates (Erratum),” Color Res. Appl. 18, 150 (1993).

CIE 1931 x, y chromaticities of our Granada,
Spain, natural-light spectra (open circles) overlaid with the CIE
daylight locus (solid curve) and Planckian locus (curve with open
squares). The inset shows the entire CIE 1931 diagram and
Planckian locus.

Normalized spectral irradiances
E_{λ} measured for particular daylight (solid
curve) and calculated for a 5700-K blackbody (dashed
curve). Because the 5700-K spectrum yields the Planckian
chromaticity closest to the measured daylight chromaticity, this
particular daylight has a CCT of 5700 K.

Relative errors of the Robertson algorithm10
CCT’s calculated by our spectroradiometers compared with reference
CCT’s from a binary search algorithm that is given the same CIE 1931
x, y chromaticities.

CIE 1931 x, y chromaticities of our U.S.
natural-light spectra (open circles) overlaid with the CIE daylight
locus (plain curve) and Planckian locus (curve with open
squares).

Equation (3) CCT’s compared with reference CCT’s
calculated for the same chromaticities. If Eq. (3) were exact,
all points would lie exactly on the main diagonal (dashed line).

Relative CCT errors for Eq. (3) when it is given the
CIE 1931 x, y chromaticities of 138 Planckian radiators of
known temperatures. Note that the high-temperature Planckian
chromaticities lie far from the locus of natural-light chromaticities
used to develop Eq. (3) (see Figs. 1 and 4). Thus, when Eq.
(3) is properly limited to calculating CCT’s for natural-light
chromaticities, its errors will be much smaller (see Table 4) than
those shown here.

Table 1 Temporal and Spatial Details of Our Seven Spanish and U.S.
Observing Sites, Listed in Decreasing Order of the Number of Spectra
Measured at Each Site

Table 4 Maximum and Mean Percentage CCT Errors from Eq. (3)
and Its Maximum and Mean CIE 1931 Colorimetric Errors Compared with
Reference CCT’s for Our Measured Daylight and Skylight Spectra

Rural sites are denoted R and urban sites
U; all non-Granada sites are in the United States.

Table 2

CIE 1931 Best-Fit Colorimetric Epicenters
x_{
e
}, y_{
e
} and Constants
for Eq. (3)

Constants

Valid CCT Range (K)

3000–50,000

50,000–8 × 10^{5}

x_{
e
}

0.3366

0.3356

y_{
e
}

0.1735

0.1691

A_{
0
}

-949.86315

36284.48953

A_{
1
}

6253.80338

0.00228

t_{
1
}

0.92159

0.07861

A_{
2
}

28.70599

5.4535 × 10^{-36}

t_{
2
}

0.20039

0.01543

A_{
3
}

0.00004

t_{
3
}

0.07125

Note: Equation (3) has only two exponential
terms in the higher CCT range.

Table 3

Percentile Distribution of Eq. (3) CCT Errors Compared
with Reference CCT’s

Relative CCT Error (%)

Number of CCT’s

Percentile (%)

<0.1

5442

77.8

<0.2

6491

92.8

<0.5

6905

98.7

<1.0

6971

99.6

<2.0

6998

100

Note: A binary search algorithm calculates
reference CCT’s from our measured daylight and skylight chromaticities.

Table 4

Maximum and Mean Percentage CCT Errors from Eq. (3)
and Its Maximum and Mean CIE 1931 Colorimetric Errors Compared with
Reference CCT’s for Our Measured Daylight and Skylight Spectra

CCT Range (K)

Relative Error

CIE 1931 Colorimetric Error

CCT’s in Range

Maximum

Mean

Maximum

Mean

3000–5000

1.73

0.71

0.00317

0.00131

39

5000–9000

1.53

0.06

0.00268

0.00007

3726

9000–17,000

0.62

0.09

0.00041

0.00006

2632

17,000–50,000

1.02

0.14

0.00020

0.00004

520

50,000–10^{5}

1.37

0.56

0.00012

0.00005

46

10^{5}–8 × 10^{5}

1.40

0.58

0.00007

0.00002

35

Note: By comparison, MacAdam 1931 x, y
color matching ellipses on the Planckian locus from 3000 to 8 ×
10^{5} K have a mean semimajor axis of 0.0029294 (standard
deviation, 2.2424 × 10^{-4}) and a mean semiminor
axis of 9.3694 × 10^{-4} (standard deviation,
1.6345 × 10^{-4}).

Tables (4)

Table 1

Temporal and Spatial Details of Our Seven Spanish and U.S.
Observing Sites, Listed in Decreasing Order of the Number of Spectra
Measured at Each Site

Rural sites are denoted R and urban sites
U; all non-Granada sites are in the United States.

Table 2

CIE 1931 Best-Fit Colorimetric Epicenters
x_{
e
}, y_{
e
} and Constants
for Eq. (3)

Constants

Valid CCT Range (K)

3000–50,000

50,000–8 × 10^{5}

x_{
e
}

0.3366

0.3356

y_{
e
}

0.1735

0.1691

A_{
0
}

-949.86315

36284.48953

A_{
1
}

6253.80338

0.00228

t_{
1
}

0.92159

0.07861

A_{
2
}

28.70599

5.4535 × 10^{-36}

t_{
2
}

0.20039

0.01543

A_{
3
}

0.00004

t_{
3
}

0.07125

Note: Equation (3) has only two exponential
terms in the higher CCT range.

Table 3

Percentile Distribution of Eq. (3) CCT Errors Compared
with Reference CCT’s

Relative CCT Error (%)

Number of CCT’s

Percentile (%)

<0.1

5442

77.8

<0.2

6491

92.8

<0.5

6905

98.7

<1.0

6971

99.6

<2.0

6998

100

Note: A binary search algorithm calculates
reference CCT’s from our measured daylight and skylight chromaticities.

Table 4

Maximum and Mean Percentage CCT Errors from Eq. (3)
and Its Maximum and Mean CIE 1931 Colorimetric Errors Compared with
Reference CCT’s for Our Measured Daylight and Skylight Spectra

CCT Range (K)

Relative Error

CIE 1931 Colorimetric Error

CCT’s in Range

Maximum

Mean

Maximum

Mean

3000–5000

1.73

0.71

0.00317

0.00131

39

5000–9000

1.53

0.06

0.00268

0.00007

3726

9000–17,000

0.62

0.09

0.00041

0.00006

2632

17,000–50,000

1.02

0.14

0.00020

0.00004

520

50,000–10^{5}

1.37

0.56

0.00012

0.00005

46

10^{5}–8 × 10^{5}

1.40

0.58

0.00007

0.00002

35

Note: By comparison, MacAdam 1931 x, y
color matching ellipses on the Planckian locus from 3000 to 8 ×
10^{5} K have a mean semimajor axis of 0.0029294 (standard
deviation, 2.2424 × 10^{-4}) and a mean semiminor
axis of 9.3694 × 10^{-4} (standard deviation,
1.6345 × 10^{-4}).