Atmospheric distance and value

Plein-air study painted on the north rim of the Grand Canyon, looking west.

Blog reader Rockhopper asked about how tones get lighter in the distance as a result of atmospheric perspective:

"How do you define value as the image recedes? I know it gets lighter and I know it goes bluer. Is there a rule in place to define distance with the value? So for example a mountain is 10 miles away, is the mountain at 20 miles way half the value of the 10 mile mountain?"

Here’s a preliminary answer. As you can see, I'm still trying to figure this one out:

I've always assumed it's a linear progression. In other words, if a dark tree near you is a #8 value (with white=0), and the farthest hill twelve miles away is a #2, then dark objects will shift one step lighter in value for every two miles of distance. But the more I think about it, I have a hunch that this assumption may be wrong. I have a feeling there's more to it.

Let’s begin by supposing that the air is evenly distributed with dust and moisture and that the volume of air is equally illuminated throughout.

Would it be reasonable to suppose that looking through atmosphere is like looking through an evenly spaced series of wedding veils? Each parcel of air introduces a fixed amount of additional scattered light.

By this way of thinking, the tonal shift relative to distance would be geometric rather than linear. In other words, let's suppose you suspend wedding veils 20 feet apart from each other in an infinite progression away from the observer. To test the lightening effect, you can have a person with a black velvet coat stand behind the first veil, then run back a little farther and stand behind next, thereby being obscured by two veils. The process continues as the subject goes back in space. Now let's suppose each layer of wedding veil makes the black 50% lighter. The resulting value progression would be #10 @ 20 feet, #5 @ 40 feet,  #2.5 @ 60 feet, #1.25 # 80 feet, etc, continuing toward pure white.

I have a hunch, too, that our perception of steps in value is influenced by features in our visual system just as much as it is by external reality, and that the whole problem is beyond any easy math.

I remember reading that the perceptually based value scale that we know from art school is not as simple as it seems. The change in the actual amount of light coming to the eye at each step in the value series is not a linear progression of constant units, like the markings on a water beaker. Instead, it's a non-linear relationship, with each step representing a much greater volume of light than the last. As David Briggs puts it in his excellent website HueValueChroma, "value has a nonlinear relationship to luminance - a surface that looks visually halfway between black and white reflects only about 18% of the light energy reflected by a white surface."

Perhaps there’s a photographer, meteorologist, mathematician or a vision scientist who can help cut through the confusion here.
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