Summary:

The calculated (and measured) illuminance values for two rooms with the same surface reflectances will vary, depending on the color of the surfaces. The values will be considerably higher for strongly colored surfaces.

Here is a simple experiment with a surprising answer:

1. Model an empty room with room surface reflectances of 20%. Make all room surfaces gray in color. Calculate the floor illuminance values on a grid.
2. Without changing the surface reflectances, change the surface color to red. Repeat the floor illuminance calculation.

Question: Will the illuminance values be the same or different? If you said "obviously the same," it is time to take a closer look at interreflections and color.

An Infinite Room

To simplify the explanation, imagine we have an infinitely large room with an infinite array of direct lighting. (We can think of this as a luminous ceiling.) The light emitted by the luminaires will directly illuminate the floor. Of this light, 20% will be reflected upwards to the ceiling, while the rest will be absorbed. The ceiling will reflect 20% of the indirect light towards the floor, which will reflect 20% back towards the ceiling … and so on, until all of the light is absorbed.

Calculating the total illuminance (ET) of the floor is simple. Given a surface reflectance r, and direct illuminance (ED) due to the luminaires, it is:

ET = (1 + r + r2 + r3 + r4 + …) * ED = ED / (1 - r)

For our example of r = 0.20, this becomes ET = 1.25 * ED. In other words, 20% of the total illuminance of the floor is due to interreflected light between the floor and the ceiling.

It is important to remember that this answer applies only to infinitely large rooms. Consequently, it is not exactly useful for everyday lighting design. However, it is useful when we consider color interreflections.

Seeing Color

We see color because our eyes are sensitive to three separate color bands of the visible spectrum: red, green, and blue. By combining various amounts of these three primary colors, we can simulate the appearance of almost any color. (Look closely at your color monitor or television screen and you will see a mosaic of red, green, and blue pixels.)

Our eyes are more sensitive to green light than they are to red and blue light. In terms of the primary colors produced by color monitors, the values are approximately:

Red: 21%
Green: 72%
Blue: 7%

That is, given a "white" light source that has equal amounts of red, green, and blue light, 72% of its apparent brightness is due to the green component. An additional 21% is due to the red component, and 7% is due to the blue component. (Illuminance meters are designed to have the same spectral response as our eyes.)

Like most computer graphics programs, AGi32 models surfaces as having three separate spectral reflectances - red, green, and blue. For example, we can model a red surface as having 95% red reflectance and no reflectance of green or blue light. Taking the average of these values, this produces a surface reflectance of:

(0.95 * 0.21) + (0.0 * 0.72) + (0.0 * 0.07) = 20%

From the above calculation, you might think that it will not make any difference whether the room surfaces are gray or red. As long the surface reflectance is 20%, the total floor illuminance ET should logically be 25% more than the direct illuminance ED. This, however, is incorrect. The crucial difference is that the room surfaces have different reflectances for red, green, and blue light. We need to calculate the total spectral irradiance for each color, and then calculate the total illuminance according to the sensitivity of our eyes to these colors.

For the three color bands, we have:

ET(red) = ED * 0.21 / (1.00 - 0.95) = 4.20 * ED
ET(green) = ED * 0.72 / (1.00 - 0.00) = 0.72 * ED
ET(blue) = ED * 0.07 / (1.00 - 0.00) = 0.07 * ED

This gives the total floor illuminance due to direct white light illuminance from the luminaires and interreflections from the red surfaces as:

ET = 4.99 * ED

In other words, the interreflections from the red room surfaces contribute 80% to the total floor illuminance, even though the surface reflectances are only 20%. If this seems surprising, imagine a room with white surfaces that have 95% reflectance. Doing the same calculations, we get:

ET(red) = ED * 0.21 / (1 - 0.95) = 4.20 * ED
ET(green) = ED * 0.72 / (1 - 0.95) = 14.40 * ED
ET(blue) = ED * 0.07 / (1 - 0.95) = 1.40 * ED

In this case, the total floor illuminance is 20.0 * ED - the interreflections account for 95% of the total floor illuminance.

As another example, we can consider a room with 95% blue reflectance and no reflectance of red or green light. This gives a surface reflectance of:

(0.0 * 0.21) + (0.0 * 0.72) + (0.95 * 0.07) = 6.7%

and thus:

ET(red) = ED * 0.21 / (1 - 0.00) = 0.21 * ED
ET(green) = ED * 0.72 / (1 - 0.00) = 0.72 * ED
ET(blue) = ED * 0.07 / (1 - 0.95) = 1.40 * ED

In this case, the total floor illuminance is 2.33 * ED - the interreflections account for 57% of the total floor illuminance, even though the surface reflectances are only 6.7%.

An Intuitive Explanation

Ignoring the mathematics above, it is still fairly easy to see what is happening. In the gray room, 80% of the incident white light is absorbed on each reflection. It is reasonable then that the room surfaces contribute a relatively small amount to the total floor illuminance.

In the case of a red room, all of the incident blue and green light is immediately absorbed on the first bounce, but only 5% of the red light is absorbed. On each bounce from the ceiling, it contributes more light to the total floor illuminance. While our eyes are relatively insensitive to red light, there is a far greater amount compared to the room with the gray surfaces.

To answer the question then, the calculated (and measured) illuminance values for two rooms with the same surface reflectances will vary, depending on the color of the surfaces. As shown above, the values will be considerably more for strongly colored surfaces.1

1 ANSI/IESNA RP-16-17, Nomenclature and Definitions for Illuminating Engineering, defines the reflectance of a surface as "the ratio of the reflected flux to the incident flux," where flux may be either radiant flux (measured in watts) or luminous flux (measured in lumens).

AGI32 performs its lighting (that is, radiosity) calculations using radiant flux. However, reflectance is calculated in terms of luminous flux, in order that the calculated values are in accordance with what we perceive.