Colour Science & Perception
what the eye settles, and what it cannot
What a scene does

A shadow has its own illuminant

Outdoors there are two lights, not one. The direct beam is a radiator reddened by the atmosphere; the sky is precisely the power that reddening removed. A shadow is lit by the second alone, so shadowed ground and sunlit ground sit under illuminants 21 units of ΔE apart — before any surface, any eye or any opinion is involved.

Assumes A bounce is a multiplication and The sun is not one illuminant.

17 min read 9 figures Computed, not quotedSay which colour

Photographs of snow have blue shadows. This is usually explained as a defect of film, a quirk of white balance, or an artistic convention borrowed from the Impressionists — and it is none of those. The shadows are blue because they are lit by a different light, and the difference is large enough that an instrument reports it without needing an eye to agree.

The two components of daylight, computed from one radiator and one scattering lawA 5800 K radiator through the atmosphere at air mass 1. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 21.4 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.powerdirect sunskyopen groundopen groundin shadowΔE00 21.4400450500550600650700wavelength / nmair mass 1CIE 1931 2° observer
Fig. 1 A 5800 K radiator through the atmosphere at air mass 1. The direct beam is the radiator times e−τme^{-\tau m}; the sky is precisely the power that extinction removed, so the two are complements of one calculation rather than two separate models. Open ground sees their sum; shadowed ground sees the second alone. The handle runs the air mass, which changes the proportion of sun to sky and therefore what a shadow is lit by.

The claim

An outdoor scene has two illuminants with different spectra, and every surface in it is lit by a different mixture of them depending only on what it can see of the sky.

The two are not independently modelled here. Both come out of the same line of arithmetic, which is what makes the result a derivation rather than a pair of assumptions that happen to fit.

One extinction, two lights

Start with a radiator at the top of the atmosphere and take Beer’s law on the Rayleigh optical depth:

τ(λ)=τ550(550λ)4\tau(\lambda) = \tau_{550}\left(\frac{550}{\lambda}\right)^{4}

The λ−4\lambda^{-4} is Rayleigh’s, from scattering by particles much smaller than the wavelength. Only the scale is quoted — τ550=0.0973\tau_{550} = 0.0973 at sea level — and the shape follows from the exponent. At air mass mm the direct beam that survives is S(λ) e−τ(λ)mS(\lambda)\,e^{-\tau(\lambda)m}.

The steepness of that is the whole story. Computed on this site’s grid, τ\tau is 0.3478 at 400 nm and 0.0371 at 700 nm: the optical depth in the violet is 9.38 times the optical depth in the deep red.

Now the step that makes this one calculation instead of two. The power removed from the beam has to go somewhere, and to first order it goes into the diffuse hemisphere. So the sky is

S(λ)(1−e−τ(λ)m)S(\lambda)\left(1 - e^{-\tau(\lambda)m}\right)

and needs no separate model, no fitted blue, and no appeal to what the sky looks like. The sky is blue here because τ∝λ−4\tau \propto \lambda^{-4} and for no other reason. If the exponent were wrong the figure would be wrong, and nothing else in the file would have to change to make it so.

At air mass 1 the two land at chromaticities of (0.3384, 0.3485) for the direct beam and (0.2403, 0.2440) for the sky, under the CIE 1931 2° observer. Those are not close. The sky sits far below the Planckian locus in the blue; the direct beam sits above D65 towards the yellow.

What a shadow is

A shadow, in this model, is a place that cannot see the sun but can still see the sky.

That is a geometric statement, and it is the only thing distinguishing the two illuminants at a point. Open ground receives sun plus sky. Shadowed ground receives sky alone. The two illuminants differ by

ΔE00=21.4\Delta E_{00} = 21.4

between their normalised whites — a difference several times larger than the tolerance any coating would be certified against, arising from nothing but whether a patch of ground has a wall next to it.

Sunlight, skylight, and the two whites an outdoor scene has. The direct beam, the scattered sky, their sum in the open, and the sky alone in a shadow — all computed from one radiator at 5800 K and the Rayleigh λ⁻⁴ law at air mass 1. The shadow's white and the open ground's white differ by ΔE00 = 21.4, which is a fact about the light and involves no surface and no observer's opinion. Nothing here is tabulated: the sky is blue because τ ∝ λ⁻⁴ and for no other reason.
Fig. 2 The four whites of an outdoor scene on the chromaticity diagram: the direct beam, the sky alone, their sum in the open, and the shadow that gets only the second. D65 is marked for scale. The open-ground white sits between the two components, at a position set entirely by the diffuse fraction.

The exact position of “open ground” depends on how much of the illuminance is diffuse, which depends on cloud, on time of day, and on how much sky the surface can see. That fraction is a parameter here rather than a derived quantity, and it is stated in every caption: the figures above use 25%, which is a clear day with the sun reasonably high.

A shadow is not a switch

Open ground and full shadow are the two ends of something continuous, and the continuum is far steeper at the open end than the two endpoints suggest.

Let g be the fraction of the sun’s disc a point can see — 1 in the open, 0 in full shadow, anything between in a penumbra or under a partly obstructed view. The illuminant is then g times the beam plus all of the sky, and its white moves away from open ground’s by 1.32 ΔE00 at g = 0.9, 3.52 at 0.75, 7.93 at a half, 13.65 at a quarter and 21.44 at zero.

Solving for the crossings: a point that can still see 92 per cent of the sun is already one unit from open ground; 85 per cent is two units; 66 per cent is five. The illuminant’s colour begins to move as soon as anything begins to occlude, and it has passed a coating tolerance while the shadow is barely visible as a shadow at all.

Read as a colour temperature the same ramp runs 7,001 K in the open, 7,186 with a tenth of the sun hidden, 7,550 with a quarter, 8,601 with a half, 11,435 with three quarters — and off the top of the model’s scale in full shadow, where the sky alone sits at or beyond 25,000 K with a Duv of −0.0084.

That last pair is the reason a shadow cannot be described by a colour temperature at all. The sky’s chromaticity is far enough below the Planckian locus that the nearest point on it barely means anything: the correlated temperature saturates at whatever the search’s upper bound happens to be, and eight thousandths of Duv is several times what a lamp specification would accept before refusing to quote a temperature.

So shade is about 7,500 kelvin is a statement about a partly shaded point rather than about shade. At three quarters of the sun visible the model gives 7,550; in full shadow it gives a number with nothing behind it, and the only honest description of the light in a shadow is its spectrum or its chromaticity.

Why this is a fact about light and not about vision

The claim is deliberately made at the colorimetric layer, before any question of appearance arises, because the phenomenon is usually discussed at the wrong layer.

The usual account of blue shadows involves simultaneous contrast — the shadow looks blue because it is surrounded by warm sunlit ground — or chromatic adaptation, with the visual system adapted to the average and rendering the shadow as the residual. Both of those effects are real, both are on this site, and both are additional to what is computed here.

The measurement above involves no observer’s opinion. A spectroradiometer pointed at the sky and then at the sun reports two different spectra; a colorimeter integrating them reports two different chromaticities. The shadow is bluer before anybody looks at it. Appearance effects then act on top of a difference that is already there, and the common framing — that the blueness is a perceptual illusion — has the causality backwards.

This site’s own dividing line applies. Matching is not appearance, and this page stays firmly on the matching side of it.

What was computed, and how

The radiator. A 5800 K blackbody from Planck’s law, computed rather than tabulated. That is roughly the sun’s effective temperature, and the essay it comes from records that no thermal radiator gets half its power into the visible band at any temperature, peaking at 49.6% near 6950 K.

The extinction. Rayleigh only. Aerosol extinction, ozone absorption in the Chappuis band and water vapour in the near infrared are all absent, and all three matter for a real sky.

The normalisation. The two components are scaled so that the sky carries a stated fraction of the horizontal illuminance, because their absolute ratio depends on geometry this model does not have. Every figure names the fraction it used.

The assertion. The generator refuses to draw unless the shadow’s xx chromaticity is lower than the open ground’s. That is a weak check on a strong effect, and it is there because the failure it guards against — a sign error in the exponent, or the complement taken the wrong way round — would produce a perfectly plausible figure with a yellow sky.

What a low sun does

The model has one free parameter with an obvious physical meaning, and turning it is the cheapest test of whether the mechanism is right.

The two components of daylight, computed from one radiator and one scattering law. A 5800 K radiator through the atmosphere at air mass 3. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 20.5 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.
Fig. 3 The same calculation at air mass 3 — a low sun, three atmospheres of path. The direct beam has moved from (0.3384, 0.3485) to (0.3627, 0.3720): substantially yellower, because the longer path removes more short-wavelength power. The sky moves far less.
The two components of daylight, computed from one radiator and one scattering law. A 5800 K radiator through the atmosphere at air mass 2. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 16.1 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.
Fig. 4 And a half-clouded sky at two atmospheres, where the diffuse component carries as much as the direct one. Between the three settings the shadow’s illuminant moves by more than the difference between two standard daylights.

Two more settings say that the shadow’s illuminant moves continuously with both of the sky’s arguments rather than switching between named cases.

The two components of daylight, computed from one radiator and one scattering law. A 5800 K radiator through the atmosphere at air mass 2. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 23.3 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.
Fig. 5 Two atmospheres with a clearer sky. Less diffuse light means a bluer shadow and a warmer sun, which are two halves of one number and are usually quoted as though they were independent.
Sunlight, skylight, and the two whites an outdoor scene has. The direct beam, the scattered sky, their sum in the open, and the sky alone in a shadow — all computed from one radiator at 5800 K and the Rayleigh λ⁻⁴ law at air mass 1. The shadow's white and the open ground's white differ by ΔE00 = 21.4, which is a fact about the light and involves no surface and no observer's opinion. Nothing here is tabulated: the sky is blue because τ ∝ λ⁻⁴ and for no other reason.
Fig. 6 And the four whites of the opening case on the chromaticity diagram. A shadow’s white is not on the daylight locus, which is why a camera told to assume daylight gets a shadow wrong in a stated direction.

Raising the air mass from 1 to 5 moves the direct beam’s chromaticity from (0.3384, 0.3485) to (0.3862, 0.3913) — a long way towards the orange, and the reason a setting sun is the colour it is. The sky moves comparatively little, from (0.2403, 0.2440) to (0.2569, 0.2711), because the sky is the complement and is already saturated at short wavelengths where the extinction is nearly complete.

So the gap between sunlit and shadowed ground widens as the sun gets lower. Late-afternoon light is not merely warmer; it is warmer in a way that leaves the shadows where they were, which is why the contrast between a sunlit wall and its own shadow is so much more strongly coloured at the end of the day than at noon.

The shadow is not only the sky

There is a second contribution the two-source model above leaves out, and it belongs to this field rather than to atmospheric physics.

A shadow on the ground beside a wall is lit by the sky and by the wall, because the wall is in sunlight and is sending some of it sideways. That contribution is an interreflection — the sun’s spectrum multiplied by the wall’s reflectance — and it is often the larger of the two. A shadow beside a red brick wall is not blue at all; it is a mixture of sky and brick, and the brick term carries the wall’s ρ(λ)\rho(\lambda) exactly as the first rung of this ladder describes.

So the honest statement of what lights a shadow is: whatever it can see that is itself lit. In open snow that is the sky and a great deal of sunlit snow, which is why snow shadows are the cleanest demonstration — the ground’s reflectance is high and nearly flat, so the interreflected component is nearly the sun’s own spectrum and the blue of the sky survives it. In a narrow street it is mostly masonry, and the answer is warm.

That also explains a discrepancy anybody who has tried to measure this will have hit: field measurements of shadow chromaticity scatter enormously, far more than the atmospheric model predicts. The scatter is not measurement noise. It is the interreflection term, and it is a property of the site rather than of the sky.

What a grey card reads

The practical consequence is easiest to see on a surface with no colour of its own.

Take a neutral card of albedo 0.6 and put one in the sun and one in the shade. The two are the same object with the same reflectance under the same sky on the same afternoon, and a colorimeter reports two different chromaticities — separated, at the settings used here, by 21.4 units of ΔE00\Delta E_{00}.

There is no correction that fixes both. A white balance is one transform applied to a whole image, and the transform that neutralises the sunlit card leaves the shaded one strongly blue, while the transform that neutralises the shaded one leaves the sunlit one strongly yellow.

This is the reason the shadows in a photograph are the part that has to be graded by hand, and the reason automatic white balance has an easier time indoors under one lamp than outdoors on a bright day. The estimators that guess the light are all answering the question “what was the illuminant” — and outdoors the question has two answers that vary across the frame.

Late in the day the air mass is several times what it is overhead, and that is where the shadow’s own white is furthest from the open ground’s.

Sunlight, skylight, and the two whites an outdoor scene has. The direct beam, the scattered sky, their sum in the open, and the sky alone in a shadow — all computed from one radiator at 5800 K and the Rayleigh λ⁻⁴ law at air mass 3. The shadow's white and the open ground's white differ by ΔE00 = 20.5, which is a fact about the light and involves no surface and no observer's opinion. Nothing here is tabulated: the sky is blue because τ ∝ λ⁻⁴ and for no other reason.
Fig. 7 The direct beam, the sky, their sum in the open and the sky alone in shadow, at air mass 3. The shadow’s white and the open ground’s differ by ΔE00 20.5, which is a fact about the light and involves no surface and no observer’s opinion.

Where the model stops

Single scattering, no ground albedo. A real sky is brighter and less saturated than this, especially near the horizon where multiple scattering dominates and where light reflected from the ground re-enters the beam. The chromaticities here are therefore more extreme than a measured sky’s, and the direction of the error is known.

No aerosol. Haze scatters much less selectively than air does, so a hazy sky is whiter. The blueness computed here is the clean-air limit.

No spatial structure. The sky is treated as one uniform source. A real sky is strongly graded — deep blue at the zenith, pale at the horizon, and dramatically different near the sun — so a real shadow’s illuminant depends on which part of the sky it can see. A shadow under a tree is a different colour from a shadow beside a building.

The band is 380–780 nm. Ozone’s Chappuis absorption sits inside that band and is absent from the model; the near-infrared, where extinction is weak and a real camera sensor is sensitive, sits outside it entirely.

A larger diffuse share at the same air mass is the overcast case, and it is where the two components come closest to each other.

The two components of daylight, computed from one radiator and one scattering law. A 5800 K radiator through the atmosphere at air mass 1. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 18.0 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.
Fig. 8 A 5800 K radiator at air mass 1 with four tenths of the light diffuse. The direct beam is the radiator times e−τme^{-\tau m} and the sky is what that extinction removed, so the two are complements and neither needs its own model.

What the pictures cannot show

Two limits are worth naming explicitly, because the figures are more confident-looking than the model underneath them.

Every swatch here is a screen showing a computed colour, and the sky is outside the gamut of most screens. The deep-blue skylight at (0.2403, 0.2440) is a saturated colour at low luminance, and where the figures cannot reach it they mark the region rather than drawing the nearest available lie — which is this site’s second figure rule. A reader looking at a hatched patch is looking at an honest report that the display has run out, not at a rendering failure.

Nothing here is a claim about what a shadow looks like. The chromaticities are what a colorimeter reports. A person standing in that scene is adapted to something between the two illuminants, is applying constancy machinery that discounts a great deal of it, and will not perceive anything like a 21-unit difference. The gap between the measurement and the perception is not an error in either — it is the subject of a whole field on this site, and the point of separating them is that the measurement is the part that can be checked.

A hotter radiator is the north-sky case, and the construction should not care which temperature it is handed.

The two components of daylight, computed from one radiator and one scattering law. A 6500 K radiator through the atmosphere at air mass 2. The direct beam is the radiator times e^{−τm}; the sky is what that extinction removed, so the two are complements and neither needs its own model. The sky is steeply blue because τ ∝ λ⁻⁴, and a surface in shadow is lit by that component alone. Open ground and shadowed ground therefore sit under two illuminants differing by ΔE00 = 21.0 — which is why a photograph of snow has blue shadows and why no single white balance fixes both halves of it.
Fig. 9 A 6500 K radiator at air mass 2. The sky is steeply blue for the same reason at either temperature — τ ∝ λ⁻⁴ — so what the radiator’s temperature moves is the direct beam and not the mechanism.

The generalisation

Two things transfer.

A named illuminant is a claim about a point, not about a scene. The whole apparatus of colour management — a profile, a white point, a rendering intent — takes one illuminant for one image. Outdoors that is false at every pixel boundary between sun and shade, and the failure is not small. What a single white balance can and cannot do about it is an essay near the top of this ladder.

A complement is cheaper than a second model. The sky here costs nothing beyond the extinction already computed for the beam, and the fact that it needs no fitting is the strongest evidence that the mechanism is the right one. Where a phenomenon and its complement are both modelled independently, at least one of the two models is doing work that conservation would have done for free.

Who found it, and when

Rayleigh gave the λ−4\lambda^{-4} dependence in 1871, arguing from the scattering of light by particles small compared with the wavelength, and it settled a question that had been open since Leonardo — who had noticed that distant mountains are blue and had proposed a mechanism involving fine particles of moisture.

The colorimetric consequence took much longer to be treated as ordinary. The Impressionists put blue into shadows in the 1870s and were accused of distortion; Monet’s snow scenes are the standard example, and the accusation was that the shadows were painted from theory rather than observation. The instrumentation that would have settled it — spectroradiometry of daylight in the field — did not become routine until the 1930s, which is the same decade that produced the colour-matching functions used to compute the numbers on this page.

Judd, MacAdam and Wyszecki’s 1964 study of daylight, which produced the D-series reconstruction, measured a great many daylight spectra and found them to lie essentially along one dimension. The sun-and-shade split is a different axis from that one: the D series is a family of mixtures, and this page is about what happens when a surface sees only one of the components.

That is worth stating carefully, because it is a place the standard vocabulary misleads. Shade is often quoted as a correlated colour temperature — “open shade is about 7500 K” — and the number is not wrong so much as it is one coordinate of a two-coordinate quantity. Skylight sits well off the Planckian locus, so a correlated colour temperature quoted without a Duv beside it has thrown away the information that distinguishes it from a lamp of the same temperature. Two illuminants can share a colour temperature, sit on opposite sides of the locus, and disagree visibly about a surface; sun and shade are not merely warm and cool versions of one thing.

It is also why “shade” appears as a separate white-balance preset on every camera ever built, rather than as a colder point on the daylight slider. The preset exists because the manufacturers found the daylight family did not reach it.

Where the ladder goes next

The rung above uses the same two-illuminant scene to ask what happens when a camera has to choose one of them: a scene lit by two lights has no single white point, and no estimator can fix that because the problem is not estimation.

Beside this one on the same rung, a highlight turns out to carry the lamp’s spectrum rather than the surface’s — which is exactly the cue a white balancer needs, and which an outdoor scene supplies twice over with two different answers.

Where this goes next

Colour temperature, and the number nobody quotes beside it. The Planckian locus in the 1960 UCS diagram — the only diagram on which correlated colour temperature is well defined — with four sources and the perpendicular from each to its nearest point. The temperature is where the foot of the perpendicular lands; Duv is how long the perpendicular is. halophosphate sits 0.0246 off the locus at 4513 K, which is a visible green cast that its colour temperature does not mention. The sun is not one illuminant Daylight is not a blackbody and is not one spectrum. It is a one-parameter family reconstructed from three measured basis functions, its chromaticities lie on a curve that is near the Planckian locus and not on it, and the difference is the reason a colour temperature needs a second number beside it. five greens, no neutral, under D65 — four guesses at the illuminant. The scene's surfaces as they reach the eye, and each estimator's angular error in degrees against the illuminant that actually lit them. Grey-world 43.1°, Max-RGB 27.7°, Shades-of-grey (p = 6) 31.4°, Grey-edge 34.4°. The best here is Max-RGB, which is best because this scene happens to satisfy its assumption — something reflects fully in every band — and not because it is the better algorithm. Any hatched patch is a surface this display cannot show under this light. The algorithms that guess the light What reaches a sensor is an illuminant multiplied by a reflectance, and no arithmetic separates a product into its factors. Every white-balance algorithm therefore works by assuming something about the world — and the interesting content of each is not its formula but the assumption, because a scene can violate it. One reflectance, two illuminants, two colours. A reflectance peaking near 580 nm, and the colours it produces under D65 and A. The object has not changed. The light has, and colour is a property of the pair. Constancy is the default A sheet of paper looks white in daylight and white under a tungsten lamp, although the light reaching the eye differs enormously. The visual system is solving one equation with two unknowns, and it solves it by assumption.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the index of named objects makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

ChromaticityColour constancyColour managementCorrelated colour temperatureIlluminantPlanck's lawRayleigh scatteringSpectral power distributionStandard observerWhite point