Paul Tol's Notes

INTRODUCTION TO COLOUR SCHEMES

31 July 2026

Graphics with scientific data become clearer when the colours are chosen carefully. It is convenient to have good default schemes ready for each type of data, with colours that are:

distinct for all people, including colour-blind readers;
distinct from black and white;
distinct on screen and paper;
matching well together.

This site shows such schemes, developed with the help of mathematical descriptions of colour differences and the two main types of colour-blind vision. A colour scheme should reflect the type of data shown. There are three basic types of data:

Qualitative data – nominal or categorical data, where magnitude differences are not relevant. This includes lines in plots and text in presentations.
Diverging data – data ordered between two extremes where the midpoint is important, e.g. positive and negative deviations from zero or a mean.
Sequential data – data ordered from low to high.

My main schemes for each type of data are illustrated here. Click on any of them to go to the specification.

‘Bright’: 7 colours (plus black), qualitative, colour-blind safe, print-friendly but not good in grey

‘Banded 12 colours’ (plus black): here qualitative, all different in grey, colour-blind safe but less distinct than ‘Bright’, print-friendly

‘Nightfall’: diverging, colour-blind safe, print-friendly

‘Banded 20 colours’: here diverging, all different in grey, reasonably colour-blind safe, print-friendly

‘Iridescent’: sequential, optionally extended to white and black, good in grey, colour-blind safe, print-friendly

‘Smooth rainbow’: sequential, optionally extended to off-white and brown, reasonably colour-blind safe, print-friendly

‘Discrete rainbow’: sequential, 23 colours, reasonably colour-blind safe, print-friendly

Three following sections describe these and alternative colour schemes in detail for each type of data. Later topics:

Info on simulating approximately how any colour is seen if you are colour-blind.
The schemes that work well in a monochrome display or printout are highlighted, with tips on how to use them best.
A very specific colour scheme for the AVHRR global land cover classification.
A look at a traditional, bad rainbow scheme and how a scheme with fewer colours can be better.
Map projections for world maps that work well.
A way of splitting the data analysis and the production of maps.
A general LaTeX template for reports.

Schemes PDF version »

Schemes Python code »

Global CO column mixing ratio on 26 June 2023 (frame from movie of TROPOMI data)

The smooth rainbow sequential colour scheme is used for example in my test movies of atmospheric CO and methane as measured by TROPOMI since 2017, collected on a separate page.

MY WEBSITE WAS MOVED

7 July 2026

On 31 March 2025, my site www.sron.nl/~pault was moved to https://sronpersonalpages.nl/~pault.

QUALITATIVE COLOUR SCHEMES

16 August 2026

My default colour scheme for qualitative data is the bright scheme in Fig. 1. Colour coordinates (R,G,B) are given in the RGB colour system (red R, green G and blue B), decimal at the top and hexadecimal below. Click on this and other defining figures for the hexadecimal values as text. An alternative when fewer colours are enough is the high-contrast scheme in Fig. 2, which also works when converted to greyscale. A second alternative is the vibrant scheme in Fig. 3, designed for data visualization framework TensorBoard. A third alternative is the muted scheme in Fig. 4, which has more colours, but lacks a clear red or medium blue. A fourth alternative is the medium-contrast scheme in Fig. 5 with three colour pairs that can work in greyscale, but not as well as the high-contrast scheme.

If there is a requirement that the contrast ratio of the colours with white (but not necessarily with black or each other) is at least 4.5, the blue, green and purple from the bright scheme can be used, or any of the eleven colours from the fifth alternative, dark scheme in Fig. 6. This is useful to read text or see lines if vision is moderately low, but the colours in the dark scheme are less distinct from each other.

For more than nine distinct colours, a sequential colour scheme can be (mis)used: one of three banded colour schemes (Fig. 24), the discrete rainbow colour scheme (Fig. 25) skipping the lightest colours for lines on a white background, or the smooth rainbow colour scheme (Fig. 23) starting at purple, interpolated in large equal steps.

Bright scheme — Figure 1: *Bright* qualitative colour scheme that is colour-blind safe. The main scheme for lines and their labels.

High-contrast scheme — Figure 2: *High-contrast* qualitative colour scheme, an alternative to the bright scheme of Fig. 1 that is colour-blind safe and optimized for contrast. The samples underneath are shades of grey with the same luminance; this scheme also works well for people with monochrome vision and in a monochrome printout.

Vibrant scheme — Figure 3: *Vibrant* qualitative colour scheme, an alternative to the bright scheme of Fig. 1 that is equally colour-blind safe. It has been designed for data visualization framework TensorBoard, built around their signature orange FF7043. That colour has been replaced here to make it print-friendly.

Muted scheme — Figure 4: *Muted* qualitative colour scheme, an alternative to the bright scheme of Fig. 1 that is equally colour-blind safe with more colours, but lacking a clear red or medium blue. Pale grey is meant for bad data in maps.

Medium-contrast scheme — Figure 5: *Medium-contrast* qualitative colour scheme, an alternative to the high-contrast scheme of Fig. 2 that is colour-blind safe with more colours. It is also optimized for contrast to work in a monochrome printout, but the differences are inevitably smaller. It is designed for situations needing colour pairs, shown by the three rectangles, with the lower half in the greyscale equivalent.

Medium-contrast scheme application — Figure 5: *Medium-contrast* qualitative colour scheme, an alternative to the high-contrast scheme of Fig. 2 that is colour-blind safe with more colours. It is also optimized for contrast to work in a monochrome printout, but the differences are inevitably smaller. It is designed for situations needing colour pairs, shown by the three rectangles, with the lower half in the greyscale equivalent.

The bright, high-contrast, vibrant, muted, medium-contrast and dark schemes work well for plot lines and map regions, but the colours are too strong to use for backgrounds to mark (black) text, typically in a table. For that purpose, the pale scheme is designed (Fig. 7). The colours are inherently not very distinct from each other (pick pale purple last), but they are clear in a white area. This pale scheme can also be used in combination with the bright scheme, for example showing the full data range as a pale area and the mean as a bright line, or a pale fit line with the bright measured points on top.

There are situations where a scheme is needed between the bright and pale schemes, for example (Fig. 11) for backgrounds in a table where more colours are needed than available in the pale scheme and where the coloured areas are small. For this purpose, the light scheme of Fig. 8 is designed.

Light scheme — Figure 8: *Light* qualitative colour scheme that is reasonably distinct in both normal and colour-blind vision. It was designed to fill labelled cells with more and lighter colours than contained in the bright scheme of Fig. 1, using more distinct colours than that in the pale scheme of Fig. 7, but keeping black labels clearly readable (see Fig. 11). However, it can also be used for general qualitative maps.

Colour names have been added to the scheme definitions as mnemonics for the maker of a figure, not necessarily for use in text: a reader should not have to guess what olive looks like. Colours are identified uniquely by their names within the collective of the bright, pale and light schemes, whereas the high-contrast, vibrant, muted, medium-contrast and dark schemes reuse some names for different colours.

The colours within a qualitative scheme are given in order of changing hue (or luminance in the case of the high- and medium-contrast schemes), but the colours can be picked at random. Often, a data type suggests an appropriate choice or similar data types can be grouped by giving them similar colours. If the colours have to be picked in a fixed sequence, a good order for each scheme is as follows. Click on the defining figures above for the hexadecimal values in this order as text.

bright:: blue red green yellow cyan purple grey
high-contrast:: blue yellow red
vibrant:: orange blue cyan magenta red teal grey
muted:: rose indigo sand green cyan wine teal olive purple
medium-contrast:: light blue dark blue light yellow dark red dark yellow light red
dark:: blue red green sapphire maroon pine teal purple magenta brown grey
light:: light blue orange light yellow pink light cyan mint pear olive pale grey

Examples of the use of the qualitative schemes are given in Figs. 9 and 10 for lines of the Tokyo metro and in Fig. 11 for cell backgrounds and text blocks.

Tokyo metro in bright scheme — Figure 9: Metro lines in Tokyo coloured using the bright, high-contrast, vibrant, muted, medium-contrast, dark and (actually sequential) banded colour scheme.

Tokyo metro in high-contrast scheme — Figure 9: Metro lines in Tokyo coloured using the bright, high-contrast, vibrant, muted, medium-contrast, dark and (actually sequential) banded colour scheme.

Tokyo metro in light scheme — Figure 10: Metro lines in Tokyo coloured using the light scheme, showing that the light scheme is not meant for lines on a white background (see Fig. 11).

Cell backgrounds in pale scheme — Figure 11: Examples of filling labelled cells. The pale colour scheme can be used for most cases: large patterns with few colours. The highlighting works while the text remains easily readable. The light colour scheme is better when more colours are needed or the colours need to be more distinct, for example when the areas with one colour are small. Note that if the colours were darker, the small print would become difficult to read, while this scheme is much lighter than the schemes for lines (see Fig. 10). This caption is set in blue from the dark scheme, which is distinct from black but still easily readable on a white background.

Cell backgrounds in light scheme — Figure 11: Examples of filling labelled cells. The pale colour scheme can be used for most cases: large patterns with few colours. The highlighting works while the text remains easily readable. The light colour scheme is better when more colours are needed or the colours need to be more distinct, for example when the areas with one colour are small. Note that if the colours were darker, the small print would become difficult to read, while this scheme is much lighter than the schemes for lines (see Fig. 10). This caption is set in blue from the dark scheme, which is distinct from black but still easily readable on a white background.

The application to maps is shown in Fig. 12. It is a stylized variation of the diagnostic map used by the ColorBrewer website. In one area, all colours are shown in a random pattern. In other areas, basically one colour is shown, but with one small area of each other colour included. This indicates how well the colours within a scheme are identifiable when all of them are used.

Diagnostic map in bright scheme — Figure 12: From top to bottom, diagnostic maps for the bright, high-contrast, vibrant, muted, medium-contrast, dark and light qualitative schemes, given in Figs. 1, 2, 3, 4, 5, 6 and 8, respectively. For each scheme, the section at the left contains all colours, while each of the other sections contains one colour with one diamond of each other colour. This tests how well the colours are identifiable.

Diagnostic map in high-contrast scheme — Figure 12: From top to bottom, diagnostic maps for the bright, high-contrast, vibrant, muted, medium-contrast, dark and light qualitative schemes, given in Figs. 1, 2, 3, 4, 5, 6 and 8, respectively. For each scheme, the section at the left contains all colours, while each of the other sections contains one colour with one diamond of each other colour. This tests how well the colours are identifiable.

The design of the qualitative schemes involved four types of calculations:

for the distance between colours the CIEDE2000 colour difference ΔE₀₀ is used;
red-blind and green-blind vision is simulated by the method described later;
colours with the same product S × B of saturation S and brightness B in the HSB colour system (same vividness) match well together;
colours are called print-friendly if they are within the CMYK gamut provided by colour profile ISO Coated v2 300 % (discussed below).

To reduce the number of calculations without much loss of choice, only websmart colours were considered, meaning the hexadecimal RGB coordinates are only 00, 11, ..., FF. These colours can be specified using only 3 hexadecimal digits. In the diagram below, they are arranged using HSB coordinates, with hue H horizontally and vividness S × B vertically. Colour groups with the same hue and vividness are ordered by intensity, centred around the hue coordinate. The main values of hue and vividness are labelled with names designed by Bob Stein. Click on the diagram to get a version labelled with the 3-digit RGB coordinates.

All websmart colours

All colours on this site are defined in sRGB colour space, the default used by most software and displays. Printers work in a different colour space that also varies from model to model. When they conform to international standard ISO 12647-2 and the exact printing conditions are not known beforehand, it is recommended to assume the CMYK colour space provided by colour profile ISO Coated v2 300 %. All scheme colours are taken from the overlap between this and the sRGB colour spaces, shown in the diagram below (click on it for the detailed version). Individual printers may deviate, probably not so much that colours become unrecognizable, but enough to push some colours closer together. However, it is not possible to take individual printers into account.

All print-friendly websmart colours

The Netherlands Standardization Institute NEN has issued a code of practice which includes a recommended scheme with eight colours, three greys and white. The colours are bright, but differences between them in colour-blind vision are often much smaller than the smallest difference in the bright, vibrant or muted schemes, two colours are not print-friendly and they cannot be quoted without infringing copyright.

DIVERGING COLOUR SCHEMES

9 October 2022

Diverging schemes are for ordered data between two extremes where the midpoint is important. Such schemes could be constructed simply by scaling the colour coordinates linearly, e.g. from blue to white to red. However, by including subtle hue changes, the colours are more distinct and the schemes more attractive. Figures 13, 14, 15 and 16 show the sunset, nightfall, BuRd and PRGn schemes, which are tweaked versions of schemes on the ColorBrewer website. The darkest shades of the original versions have been removed, because they are too dark and similar to be used in practice. The circled colour is meant for bad data, without drawing attention away from good data with a large deviation from zero. The sunset and nightfall schemes were designed for situations where bad data have to be shown white, with more hues in the nightfall scheme. The four schemes look similar in colour-blind vision, so if more than one is used, do not reverse the direction in one of them. If more colours than shown are needed from a given scheme, use a continuous version of the scheme instead of the discrete colours, by linearly interpolating the colour coordinates. If fewer colours are needed, pick colours at equidistant points in the continuous version.

Sunset scheme — Figure 13: *Sunset* diverging colour scheme that also works in colour-blind vision. The colours can be used as given or linearly interpolated. The circled colour is meant for bad data. The scheme is related to the ColorBrewer RdYlBu scheme, but with darker central colours and made more symmetric.

Nightfall scheme — Figure 14: *Nightfall* diverging colour scheme that also works in colour-blind vision. The colours are linearly interpolated or every second colour is skipped when used as given. The circled colour is meant for bad data. The scheme is inspired by the ColorBrewer PuBuGn and YlOrRd schemes.

BuRd scheme — Figure 15: *BuRd* diverging colour scheme that also works in colour-blind vision. The colours can be used as given or linearly interpolated. The circled colour is meant for bad data. This is the reversed ColorBrewer RdBu scheme.

Figure 16: *PRGn* diverging colour scheme that also works in colour-blind vision. The colours can be used as given or linearly interpolated. The circled colour is meant for bad data. This is the ColorBrewer PRGn scheme, with green A6DBA0 shifted to ACD39E to make it print-friendly.

Examples of the use of the diverging schemes for maps are given in Figs. 17 and 18.

Map in discrete sunset scheme — Figure 17: Examples of maps using the diverging colour schemes of Figs. 13, Figs. 14, 15 and 16, each discrete and interpolated.

Map in discrete nightfall scheme — Figure 17: Examples of maps using the diverging colour schemes of Figs. 13, Figs. 14, 15 and 16, each discrete and interpolated.

Diagnostic map in sunset scheme — Figure 18: Diagnostic maps for the four diverging schemes given in Figs. 13, Figs. 14, 15 and 16. For each scheme, the section at the left contains all colours, while each of the other sections contains one colour with one diamond of each other colour. This tests how well the colours are identifiable when kept discrete.

Diagnostic map in nightfall scheme — Figure 18: Diagnostic maps for the four diverging schemes given in Figs. 13, Figs. 14, 15 and 16. For each scheme, the section at the left contains all colours, while each of the other sections contains one colour with one diamond of each other colour. This tests how well the colours are identifiable when kept discrete.

SEQUENTIAL COLOUR SCHEMES

7 July 2026

Sequential schemes are for ordered data from low to high. The YlOrBr scheme given in Fig. 19 is a tweaked version of the ColorBrewer YlOrBr scheme. The most distinct grey is also given, useful for data gaps; it is not meant for extreme values. If more colours than shown are needed from this scheme, use a continuous version of the scheme instead of the discrete colours, by linearly interpolating the colour coordinates. If fewer colours are needed, pick colours at equidistant points in the continuous version. An alternative continuous scale is provided by the iridescent scheme, which is the linear interpolation of the colours specified in Fig. 20. The luminance varies linearly, so this scheme also works well for people with monochrome vision and in a monochrome printout. The incandescent scheme in Fig. 21 also has a linearly varying luminance, useful on screen: it has brighter colours, but is not print-friendly, i.e. the colours may shift on paper.

Iridescent scheme — Figure 20: *Iridescent* sequential colour scheme with a linearly varying luminance that also works in colour-blind vision. The colours should be linearly interpolated, optionally extended towards white and black. The grey is meant for bad data.

Incandescent scheme — Figure 21: *Incandescent* sequential colour scheme with a linearly varying luminance that also works in colour-blind vision, but it is not print-friendly. The colours should be linearly interpolated, optionally extended towards white and black. The grey is meant for bad data. Take into account that pale cyan CEFFFF is almost white in red-blind vision.

There are many warnings that ordered data should not be shown with a rainbow scheme. The arguments are:

The spectral order of visible light carries no inherent magnitude message. However, a rainbow provides a scheme with many colours, showing subtle effects in the data clearer or making it easier to read the value in a map. There are also cases where a scheme is preferred that does not have pale colours at the low end or middle of the range.
Most rainbow schemes contain bands of almost constant hue with sharp transitions between them, which are perceived as jumps in the data. This can be avoided with careful design.
Colour-blind people have difficulty distinguishing some colours of the rainbow. This can be taken into account in the design.

Here are given a discrete rainbow scheme (Fig. 22) and a smooth rainbow scheme (Fig. 23) that are reasonably clear in colour-blind vision. To remain colour-blind safe, the discrete rainbow scheme should not be interpolated. The smooth rainbow scheme does not have to be used over the full range (see caption of Fig. 23).

Discrete rainbow scheme with 14 colours — Figure 22: *Discrete rainbow* colour scheme with 14 or 23 colours for maps. See Fig. 25 for the best subset if a different number of colours is needed. The colours have to be used as given: do not interpolate. For a smooth rainbow scheme, see Fig. 23. The grey is meant for bad data, but white can also be used, except in the case of exactly 23 colours. The colours have been numbered for easy referencing in Fig. 25.

Discrete rainbow scheme with 23 colours — Figure 22: *Discrete rainbow* colour scheme with 14 or 23 colours for maps. See Fig. 25 for the best subset if a different number of colours is needed. The colours have to be used as given: do not interpolate. For a smooth rainbow scheme, see Fig. 23. The grey is meant for bad data, but white can also be used, except in the case of exactly 23 colours. The colours have been numbered for easy referencing in Fig. 25.

Smooth rainbow scheme — Figure 23: *Smooth rainbow* colour scheme. The colours are meant to be linearly interpolated: for a discrete rainbow scheme, see Fig. 22. Often it is better to use only a limited range of these colours. Starting at purple, bad data can be shown white, whereas starting at off-white, the most distinct grey is given in the circle. If the lowest data value occurs often, start at off-white instead of purple. If the highest data value occurs often, end at red instead of brown. For colour-blind people, the light purples and light blues should not be mixed much.

A banded variation of the discrete rainbow limits the luminance range and picks samples from four colour bands with smoothly varying luminances, while optimizing the colour distances for colour-blind people and keeping the scheme print-friendly. Three versions are given in Fig. 24, with 12, 16 and 20 colours. The grey samples underneath indicate that the colours become increasingly similar, so use a scheme with fewer colours if possible. An example where the version with 12 colours is applied to lines is given in Fig. 9.

Banded scheme with 12 colours — Figure 24: Banded colour scheme, with 12, 16 or 20 colours. The samples underneath are the same colours in order of decreasing luminance and shades of grey with the same luminance. Bad data in maps can be indicated by a light colour such as white.

Banded scheme with 16 colours — Figure 24: Banded colour scheme, with 12, 16 or 20 colours. The samples underneath are the same colours in order of decreasing luminance and shades of grey with the same luminance. Bad data in maps can be indicated by a light colour such as white.

Examples of the use of all sequential schemes for maps are given in Figs. 26 and 27.

The discrete rainbow colour scheme is inspired by the temperature map of the weather forecast in newspaper de Volkskrant: unconnected curves in CIELAB colour space for purples, blues, greens and oranges, each sampled three times but the last one twice extra for yellow and red, in total 14 colours. The curves were straightened, shifted and sampled equidistantly to make the colours more distinct, reasonably colour-blind safe and print-friendly. Later, the lines were resampled with smaller distances and the scheme was extended towards white and black, to get 23 colours. Figure 25 shows how the two sets can be combined to make a scheme with any number of colours up to 23. If you have an application where one of the colours stands out too much (for example colour 18), use a set with one extra colour and skip the unwanted colour. Alternatively, consider one of the banded schemes above.

Discrete rainbow scheme with any number of colours up to 23 — Figure 25: Discrete rainbow colour scheme for maps, with 1 to 23 colours. The numbers refer to the colour definitions in Fig. 22.

Map in discrete YlOrBr scheme — Figure 26: Examples of maps using the sequential colour schemes given in Figs. 19, 20, 21, 22, 23 and 24.

Map in discrete rainbow scheme with 14 colours — Figure 26: Examples of maps using the sequential colour schemes given in Figs. 19, 20, 21, 22, 23 and 24.

Diagnostic map in YlOrBr scheme — Figure 27: Diagnostic maps for the discrete sequential schemes given in Figs. 19, 22 and 24. For each scheme, the section at the left contains all colours, while each of the other sections contains one colour with one diamond of each other colour. This tests how well the colours are identifiable.

Diagnostic map in discrete rainbow scheme with 14 colours — Figure 27: Diagnostic maps for the discrete sequential schemes given in Figs. 19, 22 and 24. For each scheme, the section at the left contains all colours, while each of the other sections contains one colour with one diamond of each other colour. This tests how well the colours are identifiable.

COLOUR BLINDNESS

16 August 2026

People usually find out at an early age whether they are colour-blind. However, there are subtle variants of colour-vision deficiency. The two main types are:

Green-blindness – the cone cells in the retina that are sensitive to medium wavelengths are absent or have their response shifted to the red (6 % of men, 0.4 % of women);
Red-blindness – the cone cells in the retina that are sensitive to long wavelengths are absent or have their response shifted to the green (2.5 % of men).

Effectively, one (and only one) of the three types of colour receptors is missing. In the much rarer monochrome vision, all working colour receptors are the same or missing and only brightness variations are seen. Figure 28 is a test on green-blindness and red-blindness. It works on a computer screen (when looking straight at it), projected with a beamer and sometimes even in unfaded print, although this will depend on the quality of the equipment.

Colour vision test — Figure 28: The readable text in this image is the colour-vision diagnosis of the reader. It is not a puzzle: there is no hidden message that requires much effort to see. The clarity of the text is not important, only whether it is readable at all; in normal vision one text is clearer than the other. This test only works for standard, red-blind and green-blind vision, not monochrome vision. Please do not make life-changing decisions based only on this test.

To simulate green-blindness, all RGB colours in an image are converted to R′G′B′ colours with
R ′ = (4211 + 0.677 G ^2.2 + 0.2802 R ^2.2)^1/2.2,
G ′ = (4211 + 0.677 G ^2.2 + 0.2802 R ^2.2)^1/2.2,
B ′ = (4211 + 0.95724 B ^2.2 + 0.02138 G ^2.2 − 0.02138 R ^2.2)^1/2.2,
with parameters R, G and B in the range 0–255 and the output values rounded. To simulate red-blindness, colours are shifted as follows:
R ′ = (782.7 + 0.8806 G ^2.2 + 0.1115 R ^2.2)^1/2.2,
G ′ = (782.7 + 0.8806 G ^2.2 + 0.1115 R ^2.2)^1/2.2,
B ′ = (782.7 + 0.992052 B ^2.2 − 0.003974 G ^2.2 + 0.003974 R ^2.2)^1/2.2.
These conversions should be applied in sRGB colour space, i.e. they work on a standard video display, but not necessarily on paper. The conversion can be performed with the free software suite ImageMagick. The following two commands make green-blind and red-blind versions of original image original.png, respectively:

convert original.png \( +clone -channel RG -fx "(0.02138+0.6770*G^2.2+0.2802*R^2.2)^(1/2.2)" \) +swap -channel B -fx "(0.02138(1+v.G^2.2-v.R^2.2)+0.9572*v.B^2.2)^(1/2.2)" greenblind.png

convert original.png \( +clone -channel RG -fx "(0.003974+0.8806*G^2.2+0.1115*R^2.2)^(1/2.2)" \) +swap -channel B -fx "(0.003974(1-v.G^2.2+v.R^2.2)+0.9921*v.B^2.2)^(1/2.2)" redblind.png

These are Unix-style commands, for Windows replace \( and \) by ( and ). Figure 29 shows the result when the conversions are applied to the grid of websmart colours.

Colour scale in normal vision — Figure 29: Websmart colours in normal, green-blind and red-blind vision. These conversions are only approximate and are designed for a computer screen.

Colour scale in green-blind vision — Figure 29: Websmart colours in normal, green-blind and red-blind vision. These conversions are only approximate and are designed for a computer screen.

Contrary to popular belief, pure red and green can be distinguished. Instead, yellow/green, blue/violet and (as in Fig. 28) cyan/pink/grey combinations can be problematic. Table 1 shows colour-blind vision simulations of the colour schemes given on this site.

Table 1: Colour schemes as given earlier and approximate simulations of green-blind and red-blind vision.

scheme	normal vision	green-blind vision	red-blind vision
bright
high-contrast
vibrant
muted
medium-contrast
dark
light
sunset
nightfall
BuRd
PRGn
YlOrBr
iridescent
incandescent
rainbow
banded 12
banded 16
banded 20

GREYSCALE CONVERSION

14 August 2026

According to the Web Content Accessibility Guidelines, a contrast ratio between colours of at least 3 is recommended by ISO-9241-3 for standard text and vision, but the Guidelines define a stronger criterion of at least 4.5 to make the colours useful for people with moderately low vision. This includes people with monochrome vision, who only see brightness variations. The criterion actually only applies to body text, but "charts, graphs, diagrams, and other non-text-based information [...] should also have good contrast to ensure that more users can access the information."

If the contrast ratio should be at least 4.5 between a colour and a white background, use the dark scheme (or blue, green or purple from the bright scheme), but if graphics with more than one colour should work after greyscale conversion, the contrast should also be high between the colours. Then even a contrast ratio of 3 cannot be reached using more than one print-friendly websmart colour plus white and black. The only blue shades are 4477BB and 5577AA, almost the same as the blue from the bright scheme. A websmart shade of grey is not available: only 757575 meets the criterion. The largest minimum contrast ratio in a set of two print-friendly websmart colours plus white and black is 2.8 and in a set with three such colours 2.1. One example with three colours is the high-contrast scheme. However, with some precautions it can still be applied to lines and symbols: use the colours in the order

blue yellow red from the high-contrast scheme

on a white background and use black only for annotation (e.g. text or a grid) that is not on top of blue. When the first two colours are used, the contrast between yellow and white is just 2.1, but that is probably less important than the contrast between blue and yellow, which is large enough: 4.52. Only in the case of all three colours does the contrast between them decrease to 2.1, but that is the highest achievable value. Use different types of lines and symbols for better clarity.

The largest minimum contrast ratio with six colours is 1.5. The medium-contrast scheme uses print-friendly websmart colours not darker than the blue in the high-contrast scheme (from light to dark):

light yellow light red light blue dark yellow dark red dark blue from the medium-contrast scheme

with the contrast ratio decreasing slightly to 1.4 between the lighter colours.

All other schemes fail the contrast-ratio criterion completely, as they contain too many colours and were designed for standard, red-blind and green-blind vision, relying not only on brightness differences, but also on hue differences. If one of the other qualitative schemes is used, the best subsets for greyscale conversion are (from light to dark):

yellow red blue from the bright scheme;
cyan teal red from the vibrant scheme;
cyan olive purple wine from the muted scheme.

These subsets are slightly clearer than the medium-contrast scheme (but with fewer colours) and less clear than the high-contrast scheme.

The YlOrBr, iridescent and incandescent sequential schemes work well (Fig. 30). The YlOrBr and iridescent schemes are print-friendly. The iridescent and incandescent schemes were designed to be useful in grey, with a linearly varying luminance. Python's default sequential scheme viridis has a similar property, but it is not print-friendly and seems to have fewer discernible colours. The banded schemes (Fig. 24) have a linearly varying luminance and are print-friendly, but with a smaller range and they should remain discrete. The rainbow schemes do not work. By definition, all diverging schemes do not work either after greyscale conversion.

YlOrBr scheme with grey version — Figure 30: The YlOrBr, iridescent and incandescent sequential colour schemes, with below the grey shades with the same luminance.

Iridescent scheme with grey version — Figure 30: The YlOrBr, iridescent and incandescent sequential colour schemes, with below the grey shades with the same luminance.

COLOUR SCHEME FOR GROUND COVER

31 May 2018

Some data sets need a very specific colour scheme. An example is the global land cover classification, as generated by the University of Maryland Department of Geography from AVHRR data acquired between 1981 and 1994, available at a resolution of 1 km. There is a recommended colour scheme, but the colours are not distinct, some not even in normal vision. Figure 31 gives a more subtle and logical scheme where all colours are distinct in all visions. Figure 32 shows the world with a reduced resolution of 20 km using this scheme. Figure 33 shows only North America at a resolution of 5 km, using almost all classes.

Map in ground-cover scheme — Figure 31: Colour-blind safe colour scheme for the AVHRR global land cover classification. The colours have been numbered with the data values.

Global land cover classification in North America — Figure 33: Land cover classification of North America at a resolution of 5 km. All possible classes except deciduous needleleaf forest are found.

GOOD AND BAD COLOUR SCHEMES COMPARED

22 November 2009

The following figures show the true physical figure of the Earth using two colour schemes: the smooth sunset scheme defined here and a traditional rainbow scheme as defined by many programs (e.g. IDL).

Most rainbow schemes contain bands of almost constant hue with sharp transitions between them, which are perceived as jumps in the data. In this example large areas are green without features, while for example the yellow line in northern Australia implies a sudden change that does not exist in reality. The oceanic trench at top right can be seen over the whole length with the sunset scheme, while it becomes difficult to see near Japan with the traditional scheme. This shows that it is more important that a scheme is smooth than that it contains many colours. In addition, colour-blind people have difficulty distinguishing some colours of the rainbow. With the traditional scheme, the yellow spot in the middle of the Sahara is not visible in red-blind vision, making the green areas effectively even larger than they are in normal vision. Last but not least, the sunset scheme emphasizes more clearly the average (light colours) and the extremes (dark colours of contrasting hues).

In this context the meaning of the figures is not really important, but if you're interested: they show the distance between the WGS84 ellipsoid and the geoid calculated with the EGM96 gravity model.

WORLD MAPS

5 September 2026

There are many websites on map projections, so I try to keep it brief. A flat map has to make a compromise between distortion of area, (small-feature) shapes and things like scale and direction. If you are not navigating the globe, it is best to use an equal-area (area-preserving) world map. Uninformed news reports suggest that must mean the Equal Earth projection from 2018, but there are many equal-area projections since the 16th century. I will concentrate here on reasonably well-known projections with straight parallels (to see latitude dependencies of data clearly), poles shown as lines instead of points (to avoid crowded polar areas), no splits except an east-west boundary and the smallest average scale and shape errors over the entire world (not just the continents).

If you want pointed corners, go for Equal Earth. If like me you don't mind a round planet, Eckert IV from 1906 has a bit less distortion. The shape distortion is even smaller if you squeeze the vertical dimension by 10 %, making the aspect ratio 0.45 instead of 0.5. Most noticeable, Africa is then less elongated. If you need the calculation to be simple, use Nell-Hammer (modified by Wagner in 1949: x = (g/2)λ(1+cos φ), y = (2/g)(φ−tan φ/2), g = √(2−4/π) with latitude φ, longitude λ), an approximation of Eckert IV. You get extra points if the world is centred around 11.5° E. The east-west boundary is then between Siberia and Alaska instead of through Siberia. In addition, Australia and New Zealand get less distorted and move away from the world edge.

Equal Earth

Eckert IV

Eckert IV with aspect ratio 0.45

World map in Eckert IV projection with aspect ratio 0.45

Nell-Hammer

Nell-Hammer with aspect ratio 0.45

World map in Nell-Hammer projection with aspect ratio 0.45

The differences may seem small, but that is due to the selection criteria. Dropping the requirement that the projection should be reasonably well-known, I like a Hufnagel projection with parameters A = 1/18, B = −1/9, ψ_max = π/2 and α = 2.2 that I call Hufnagel Z. The equations are: x ≈ 0.2157λ(cos ψ)√(18 + cos 2ψ − 2cos 4ψ), y ≈ 0.3279(sin ψ)√(18 + cos 2ψ − 2cos 4ψ), 72ψ + 40sin 2ψ − sin 4ψ − 2sin 6ψ = 36πsin φ. It has on average less distortion than Eckert IV.

Hufnagel Z

An equal-area projection where the polar areas are shown much better is Lambert Azimuthal Equal-Area from 1772, although it is normally used for regions instead of the whole world. Note that the shape of the central area, in this case Africa, is accurate.

Lambert Azimuthal Equal-Area

MAP MAKING

31 December 2012

Currently I produce maps in two steps. First, I perform the analysis in whatever program and export the data as a particular type of ASCII table. Then, I apply a colour look-up table to the data and export the result as a PNG file. This way, I don't have to redo the analysis if I want a different colour scale, e.g. to emphasize a different value range. The second step is performed on the command line with the command "convert" of ImageMagick, a free, cross-platform and open-source program suite for image manipulation.

The data are exported to a plain portable graymap (PGM) file, which is an ASCII file starting with "P2 w h 65535 ", where w is the width and h the height of the map, followed by a list of integers in the range 0 to (in my case) 65535 with a space or newline between the values. Because the width and height are given, the spaces and newlines can be put wherever you like. The "P2" identifies the file as a PGM file. I scale the data so all values are in the range 0–65532 (but you don't have to fill this whole range). The other possible values are reserved: 65533 for bad data, 65534 for text and lines and 65535 for no data (e.g. outside the projection of the world).

The colour look-up table (clut) is exported to a plain portable pixmap (PPM) file, which is an ASCII file starting with "P3 1 n 255 ", where n is the number of colours, followed by a list of RGB coordinates of the colours as integers in the range 0–255 with a space or newline between the values. I don't use perfectly white, because that may become transparent later on.

Now the image can be produced. The data is stored in input.pgm, the clut in clut.ppm and the image should be output.png. If none of the 3 reserved values are used, then the command is:

convert input.pgm -level 0,65532 clut.ppm -interpolate integer -clut -depth 8 output.png

To spread all available colours over the data range [b,w], change the level option to "-level b,w" (values outside this range will have the first or last colour). To interpolate the colours (for a smooth gradient instead of discrete steps), remove option "-interpolate integer". If any of the reserved values are used, then the command is:

convert input.pgm \( +clone -level 0,65532 clut.ppm -interpolate integer -clut \) \( -clone 0 -black-threshold 65533 -transparent black -fill \#000 -opaque \#FFFEFFFEFFFE -fill \#777 -opaque \#FFFDFFFDFFFD \) -flatten -transparent white -depth 8 output.png

Here I have assumed that text should be black (#000), bad data grey (#777) and areas without data transparent. If you want a white instead of a transparent background, remove option "-transparent white". To treat bad data as background, set it to #FFF. The command above is given in Linux syntax; in Windows, remove all backslashes. If the background is transparent, for clarity a white outline can be added around the objects:

convert output.png -bordercolor none -border 1 \( +clone -channel A -morphology edgeout square +channel +level-colors white \) -compose dstover -composite output_outlined.png

An example: given a map of the albedo at 750 nm including the legend (input.pgm) and a list of colours (clut.ppm), the output of the last two commands above (with "-level 0,9999" and without "-interpolate integer") is this image.

LATEX TEMPLATE FOR REPORTS

20 August 2026

General reports, technotes, etc. can be written in LaTeX and pdfLaTeX with two files:

plain.cls, the class file;
template_latex.tex, an example and starting point for new documents.

Running (pdf)LaTeX on template_latex.tex should give a result that looks like this. Under Debian, you may need to execute first (once) the command "sudo apt-get install texlive-fonts-extra".