APPENDIX B 

Estimating the Probability of the Public 

Perceiving a Decrease in Atmospheric Haze
Estimating the Probability of the Public Perceiving a 

Decrease In Atmospheric Haze 

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Ronald C. Henry

Department of Civil & Environmental Engineering, University of Southern California, 

3620 South Vermont Avenue, Los Angeles, CA 90089-2531. 

ABSTRACT 

Regional haze regulations require progress towards reducing atmospheric haze as 

measured by particle scattering coefficient of visible light. From a practical perspective, 

this raises the question “Given a decrease in extinction, what is the probability that 

people will notice an improvement in visibility?” This paper proposes a quantitative 

definition of the probability of a perceptible increase in visibility given a decrease in light 

extinction and a general method to estimate this probability from perception 

measurements made in the field under realistic conditions. Using data from a recent 

study of visibility perception by 8 observers, it is estimated that a 2 to 4 deciview change 

gives a 67 percent maximum probability of detecting the improvement. Stated another 

way, the odds of seeing a difference are at most 2:1 for a change of 2 to 4 deciviews. A 

90% probability requires a change of at least 3.5 to 7 deciviews. The limitations and 

possible bias in the results of this study are discussed. These results may have a major 

effect on the cost benefit analysis of regulatory actions to improve visibility. 

INTRODUCTION 

Reduction in levels of airborne fine particles can have many benefits to the public 

good. Among these is improved visibility caused by decreased atmospheric scattering of 

visible light by fine particles. However, because visibility is not directly a human health 

issue, the cost of decreasing particulate levels for improved visibility must be balanced 

against the amount of improvement in the public good. Thus, the visibility debate centers 

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on how much improved visibility is worth and how much it is going to cost1,2. The work 

reported in this paper is directed toward assessing in a very fundamental way the worth of 

improved visibility. 

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This is a multidisciplinary study encompassing concepts of air quality, physics, and 

perception. A glossary of specialized terms that may not be familiar to some readers is 

provided. The reader should be aware that several ordinary English words as used in this 

paper have precise meanings. For example, lightness refers to a perceived sensation of 

light and dark and shades of gray. Terms such as lightness that may cause confusion are 

listed in the glossary. 

Before reduced levels of airborne particles can contribute to the public good of improved 

visibility, the public must be able to actually see the improvement in visibility. For 

example, decreased emissions of sulfur dioxide may decrease the burden of particulate 

sulfate in a region; but if the resulting decrease in particulate scattering is small, the 

general population may not notice the effect on visibility. So the question arises, “What 

is the probability that the general population will actually see a change in visibility given 

a relatively small change in particle extinction?” 

A regulatory response to this question provided in the regional haze air quality 

regulations is that a decrease of 1 deciview will result in a perceptible change in visibility 

regardless of whether the site is heavily polluted or near pristine. As defined in the 

glossary, a deciview is a measure of light extinction defined as 10ln(bext/10) where the 

extinction coefficient bext of the atmosphere is given in Mm-1. A previous paper found 

that the 1 deciview threshold is not supported by actual perception measurements, but 

rather that a change of 1 deciview is always below the perceptible threshold and the 

threshold depends on the existing level of haze3

. However, the data used in this previous 

work had several limitations, the foremost being reliance on two highly trained observers 

that had advance knowledge of the purpose of the study. Thus, there is a significant 

chance of bias in the results due to the small number of observers and the possibility of 

unconscious bias in the observers themselves. A new visibility perception study was 

2
designed to address this and other limitations; in particular, the latest study had 8 

observers of which 7 were naïve subjects

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. The current paper presents a new definition of 

the probability of the public actually perceiving an improvement in visibility and, using 

data from these 8 observers, estimates the probability of seeing improved visibility for a 

given decrease in extinction. In addition to these specific results, the methods presented 

in this paper are general and can be used to estimate the effect of haze on a wide range of 

measures of visibility perception, as shown next. 

DEFINITION OF THE PROBABILITY OF A NOTICEABLE 

DECREASE IN HAZE 

Rather than attempt to define a threshold of perceptible decrease in haze levels, which 

would be subject to endless debate, this paper defines the probability of a noticeable 

improvement in visibility. Consider the following thought experiment. Let C be a 

measure of visibility perception such as perceived contrast, colorfulness, or, as in this 

work, perceived lightness difference between bright and dark areas of a target. Assume 

that an increase in C is an improvement in visibility. Let an arbitrarily large number of 

identical observers make a measurement of C. Then, decrease the atmospheric extinction 

and let each observer make a second measurement of C. The probability of seeing an 

increase in C is based on the outcome of this experiment as follows. Let α equal the 

fraction of observers that report an increase in C, then 1-α equals the fraction of observers 

that do not see an increase in C. Define P the probability that a noticeable decrease in 

haze has occurred as the fraction of the population reporting an increase minus the 

fraction not reporting an increase: 

),1(5.0

and ,12)1(

P

P

+=

−= − = −

α

α α α

(1)

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where 0.5 ≤ α ≤ 1.0, since only improvements in visibility are of interest. This is a 

reasonable definition, for if there is no change in the haze then, by random variation half 

the observers will report an increase and half a decrease. Thus, if α = 0.5 and P = 0, i.e., 

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the probability that a decrease in haze has been observed is 0. If the change in haze is big 

enough that all the observers report seeing an increase in C, then α = 1 and P = 1, as 

would be expected. 

CALCULATING THE PROBABILITY OF A NOTICEABLE 

DECREASE IN HAZE 

Given the above definition, the problem is to estimate how much extinction (in 

deciviews) must decrease in order that the fraction of people seeing an increase in C be 

equal to some value α between 0 and 1. From this, P is easily calculated using Eq.1. 

This problem is approached in two steps. First, calculate the increase ΔC in C that gives 

α using the distribution of C obtained by repeated measurements under nearly identical 

conditions. Second, the decrease in extinction in deciviews that gives ΔC is calculated 

from the empirical relationship between optical depth and C (see the glossary for the 

definition of optical depth). The second step is made more complex by the fact that the 

change in haze measured in deciviews depends upon two vista-specific parameters, the 

initial value of the optical depth, and C(0), which is the value of C when the scene 

elements are seen close up, i.e., at distance 0. The details of these two steps are given 

next. 

Calculating the Required Increase in C 

The definition of P based on the thought experiment given above, requires each observer 

to make two observations of C, one before the decrease in extinction and one after. The 

calculation of P requires calculating α, the fraction of the observers seeing an increase in 

C. The problem is to find ΔC, the change in C that will make this fraction equal to α. 

This is the same as the fraction of the observers for which the change in C is greater than 

0, ΔC > 0. 

To calculate the probability that ΔC > 0, the probability distribution of ΔC is required. 

For each observer assume that repeated observations of C under unchanging conditions 

are normally distributed with standard deviation σ. By elementary probability theory, if 

4
the two observations are independent, ΔC is normally distributed with mean ΔC and 

standard deviation 2

112 

113 1/2

σ. Next, reduce this distribution to a standard normal distribution 

by the usual transformation t = (ΔC - ΔC)/(21/2

σ). The condition that ΔC > 0 is 

equivalent to 

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t > −ΔC/(21/2σ). Let F be the cumulative standard normal distribution, 

then the fraction of observers for which ΔC > 0 is α and is given by 1 - F(- 

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ΔC/(21/2σ)) = 

1 – (1- F(Δ

116 

C/(21/2

σ))) = F( ΔC/(21/2σ)) = α . Solving this gives ΔC= 21/2σF-1(α). Thus, 

the increase in C necessary to make the fraction of observers seeing an increase equal to 

α is 

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ΔC = 21/2σF−1(α) = 21/2σF−1(0.5(1+ P)), (2)

where F-1 120 (α) is the inverse cumulative standard normal distribution. 

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Calculating the Required Decrease in Extinction 

Assume that observations show that C is related to optical depth τ by the following 

expression: 

τ = −aCC τ )exp()0()( (3)

where C(0) is the visibility perception parameter at optical depth zero, i.e., at distance 

zero. In practice, the constant a in the equation is almost always equal to 1. The 

validity of this should be tested by experimental observations. 

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Given the above relationship between optical depth and C, the decrease in extinction 

given in deciviews required to achieve an increase of ΔC is given by 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ Δ −−=Δ +−

)0( )exp(ln 1 ln10)ln(10 C

C

a

a

dv τ τ (4)

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The Appendix gives the derivation of this equation and the limits of its applicability. The 

above equation depends on two site-specific parameters: the current optical depth τ, 

which depends on the concentration of airborne particles; and C(0), which depends on the 

type of terrain in the scene, tree-covered hillsides, barren hills, snowcapped mountains, 

etc. The graph of Eq.4 is U-shaped, as shown is Fig. 1; in the figure a =1 and ΔC/C(0) = 

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0.09. The optical depth that minimizes it is optimal in the sense that scene-elements at 

this distance require the minimum change in deciviews to produce a change of ΔC. 

For the remainder of the paper assume that the constant a = 1 in Eq.3. The minimum 

value of Eq.4 was found for values of ΔC/C(0) between 0.01 and 0.5. A least squares 

curve fit to the results gives the following relationship for the minimum decrease in 

deciviews dvmin that is required to produce a decrease of ΔC. 

)0(

3.27

)0(

) 22.1

)0(

) 9.11

3 2

min C

C

C

C

C

C dv Δ

+ ⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ Δ − ⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ Δ = . (5)

142 Inverting this numerically over the range of ΔC /C(0) between 0 and 0.5 gives 

143 

dvqdvq C

C

1

2

2 )0(

+=

Δ . (6)

144 where q2= -0.00030264, q1= 0.038401 and 0 ≤ dv ≤ 15. 

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146 

Estimating the Probability of Seeing an Improvement in Haze 

From Eq.2 

.1

2

2

),1(5.0 2

)),1(5.0( 2

2/1

2/1

1

2/1

⎟ −

⎠

⎞ ⎜

⎝

⎛ Δ =

⎟ += ⎠

⎞ ⎜

⎝

⎛ Δ

= +

Δ −

σ

σ

σ

C FP

P C F

F P C

(7)

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Putting Eq.6 into this, the final expression for the probability of seeing an improvement 

in visibility for a given decrease in deciviews is 

P = 2F C(0)

21/2σ

(q2dv 2 + q1dv) ⎛

⎝ ⎜ ⎞⎠ ⎟ −1, (8)

for 0 ≤ dv ≤ 15. As it is based on a curve fit, Eq.8 is a good approximation, but is not 

exact; however, in the deciview range cited, the error in the curve fit is negligible. 

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Of interest later in the paper is the dependence of P on the two experimentally 

determined parameters C(0) and σ as shown by Eq. 8. The cumulative standard normal 

distribution F(x) is a monotonically increasing function of x. Thus for a constant 

deciview value, as C(0) increases the value of F increases and P increases. Similarly, as 

σ decreases, P increases. 

 

Using Eq. 8, all that is required to estimate the probability of seeing an decrease in haze 

is to define a measure of visibility perception C, and make measurements sufficient to 

show that C is related to optical depth as in Eq. 3 (with a =1), and to determine C(0) and 

σ. This is done next. 

EXPERIMENTAL RESULTS 

Perception of light and color of a surface is broken down into lightness, hue, and 

colorfulness or saturation. Hue is the color, colorfulness is the degree to which the color 

is expressed, and lightness is the achromatic sensation of light and dark and shades of 

gray (see the glossary). Standard terminology in psychophysics reserves lightness for 

non self-luminous objects such as a mountain, while brightness is used exclusively for 

self-luminous objects such as the sky. The results below use observations of perceived 

lightness of natural targets seen through semitransparent haze. Using an instrument 

known as a visual colorimeter, the observer looks at the natural scene and matches a 

target with an adjustable display seen in the colorimeter. 

In a recent study, 8 observers made repeated matches of three targets: a nearby bush, a 

rock outcrop, and a distant mountainside4

. The bush was a control target and was too 

close to be affected by haze. The rock outcrop and the mountainside had light and dark 

areas, the difference in perceived lightness between these light and dark areas is the 

perception parameter chosen to estimate the probability of seeing an improvement in the 

haze. As haze increases, the dark and light areas approach each other in lightness. If the 

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haze is dense enough, the distinction between light and dark areas will disappear and the 

two will appear to have the same lightness. 

The physical correlate of this perception parameter is the difference in luminance 

between the light and dark areas of the targets. If the haze is spatially uniform, it is easy 

to show that this difference decreases exponentially with optical depth. Theoretically, 

contrast of the target to the horizon sky also decreases exponentially. However, the 

background sky for the targets was not the horizon, but some degrees above the horizon 

where the luminance of the sky is not constant. Also, the sky background may be 

affected by cloud shadows along the site path or clouds behind the target. The simple 

difference between light and dark areas of a target at about the same distance does not 

have these problems. For example, a cloud shadow passing in front of the target will 

affect the light and dark areas equally leaving the difference of luminance of the two 

unaffected. 

The well-behaved nature of the physical difference of luminance between light and dark 

areas of a target is one of the reasons that the difference in perceived lightness was 

chosen as the perception parameter in this study. The other reason for the choice is that 

changes in hue and colorfulness of a target due to haze have been studied in previous 

visibility perception studies3, 6 and it was felt that a visibility perception parameter based 

on lightness was needed since many targets are not very colorful and the main effect of 

haze is often on the lightness of the target. 

The perceived lightness difference for the rocks and mountainside (the two main targets 

in this study) were found to decrease exponentially with optical depth as shown in Fig. 2. 

The r2

 of the exponential fit is 0.67, C(0) for the curve in Eq.2 is 42. Separate estimates 

of C(0) for the rocks and mountainside were made and the mountainside values were 

normalized to equal 42, the same value as the rocks. Each point in the figure represents 

the average of 5 repeated measurements. Analysis of variance of these repeated 

measurements gives a value of σ of 2.854

 (the perceived lightness numbers are 

dimensionless). 

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In Fig. 2, points for observer NA2 for large optical depths are quite far from the 

exponential curve. While debriefing subject NA2 just after the observations, it was found 

that the instructions had been misinterpreted. Nonetheless, since the observer’s 

misinterpretation was consistent for all observations, these matches were used to estimate 

the variability in the observations. 

Measurements of the physical luminance of the targets were made with an imaging 

telephotometer. Luminance differences were calculated from the photometer images for 

the same light and dark areas as used in the perception measurements. In this case, the r2

of the exponential fit is 0.924 with C(0) = 304

. Thus, the luminance difference given by 

the photometer is well represented by an exponential, as expected from radiative transfer 

theory. The relationship between perceived lightness difference and photometer 

measurements is complex; objects seen through a semi-transparent haze appear darker 

than expected5

. 

The maximum probability of detecting a decrease in haze in deciviews is calculated using 

Eq.8 with the value of σ = 2.85 and C(0) values of 25 and 50. The value of 50 represents 

bright objects such as light colored rocks or a hillside without much vegetation. A C(0)

of 25 is appropriate for darker objects such as tree-covered hillsides. Fig. 3 gives the 

probability of seeing a change in haze using Eq.8 and the experimentally determined 

values of σ and C(0). Note that a decrease of 1 deciview has only a 17 - 35 percent 

chance of being perceived by the general population. 

DISCUSSION 

This section discusses the limitations of the methods and experimental results and 

whether these lead to an overestimation or underestimation of the probability of seeing a 

difference in haze. In this study, the key measurement is the variability in the observer 

matches of the targets, σ in Eq.8. If σ is overestimated, then the probability of perceiving 

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a decrease in haze is underestimated; if σ is underestimated, the probability is 

overestimated. Overall, σ in this study is probably underestimated, as explained below. 

 What is the effect of the experimental methodology itself on estimation of σ? The 

variability measured by σ has two major sources, one is in the perception of the observer 

and the other is in the ability of the observer to use the visual colorimeter for color 

matching. This second source of variability is not necessarily of interest. Assume that 

the observer is perceptually perfect, then if the color-matching procedure is perfect, the 

color matches will all be identical and have no variability (assuming the atmospheric and 

lighting conditions do not change). In reality even a perceptually perfect observer will 

probably not be able to make identical matches since the color matching procedure takes 

time and requires manipulating the visual colorimeter. The amount of the variability in 

the color matches due to the color-matching procedure is not known. However, the 

observers underwent training in color matching to help reduce this source of variability. 

Also, the color-matching procedure itself was designed to be as simple and easy as 

possible, which also reduces this source of variability. A goal of future studies could be 

to quantify this “instrument” variability. 

In reality, the perception of changes in haze requires people to compare current levels of 

haze with their memory of past levels of haze. This introduces a significant source of 

variability since peoples’ memory of past haze levels is imperfect. This memory 

variability should rightfully be included in the estimated σ used in Eq. 8 to calculate the 

probability of the seeing an improvement in haze. However, memory variability is not 

part of σ as estimated in the current study. This leads to an underestimation of σ, while 

the instrument error leads to an overestimation, as discussed above. It is likely that the 

memory variability is greater than the instrument variability. Therefore, σ = 2.85 as 

determined in this experiment is probably an underestimate of the true value and the 

estimated probabilities of seeing a decrease in haze are too high. 

Another potential limitation of this work is the choice of the particular visibility 

perception parameter. Other visibility perception studies carried out by the author have 

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focused on perception of hue and colorfulness rather than lightness, as in this study. The 

variability of the visibility perception parameter may depend on the choice of the 

parameter itself. Future studies should systematically study the effect of choice of 

visibility parameter on the estimated probability of perceiving a change in haze. 

Finally, although this study used many more observers than previous studies, a larger 

study of the general population is recommended. The current study has shown that it is 

possible to train subjects to make good color matches in a short amount of time. A larger 

study with many more observers is certainly practical. 

CONCLUSIONS 

One major conclusion to be drawn from this paper is that a 1 deciview decrease in 

extinction as defined as a just-noticeable difference in the regional haze regulations is not 

likely to be perceived by the general population, having at most only a 17 to 35 percent 

chance of being noticed by the general population. This result is consistent with previous 

work which used perceived colorfulness as the perception parameter.3,6 As seen in Fig. 3, 

a decrease of 2 to 4 deciviews is needed for odds of 2:1 of seeing a decrease in the haze . 

This is equivalent to a 67 percent probability of a detectable improvement in visibility. 

Since these are maximum probabilities of detecting an improvement in haze levels, one 

could argue that a maximum probability of say 90 percent is needed to provide a safety 

factor to ensure a perceptible improvement. This would imply a change of 3.5 to 7 

deciviews before the change would be noticeable to the general population. 

These results can have a major impact on the cost benefit analysis of regulatory actions 

taken to improve visibility. For example, if regulatory action is estimated to result in a 

just-perceptible 1 deciview decrease and if this is truly perceptible, then the cost of 

achieving this decrease in haze must be weighed against the public good of the perceived 

improved visibility. However, if a 1 deciview change is not perceptible (as this work 

implies), then there is clearly no public good and the regulatory action would have zero 

perceived benefit, no matter what the cost. Also, if a 3.5 to 7 deciview reduction is 

necessary to ensure an improvement in visibility, this is likely to be considerably more 

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costly to achieve than a 1 deciview change. Therefore the costs in the cost benefit 

analysis would be much higher while the assumed visibility benefits of a just-noticeable 

improvement would be the same. 

GLOSSARY 

Colorfulness – the degree to which a color is perceived, independent of hue and lightness. 

Deciview – a measure of light extinction defined as 10ln(bext/10) where the extinction 

 coefficient bext of the atmosphere is given in Mm-1, non-dimensional. 

Extinction coefficient – the fraction that light energy is decreased per unit of 

 distance traveled through the atmosphere (units of inverse length, often inverse 

 megameters Mm-1). 

Hue – the perceived color, independent of lightness and colorfulness. 

Lightness – the achromatic perception of light and dark and shades of gray, independent 

 of hue and colorfulness. Standard terminology in psychophysics reserves 

 lightness for non self-luminous objects such as a mountain. 

Optical depth – the product of atmospheric extinction coefficient and distance to the 

 target, non-dimensional. 

Psychophysics – the quantitative study of the relationship between perception and 

 physical stimuli. (It is not physics as practiced at the Bates Motel.) 

Visual colorimeter – an instrument for quantifying visual perception by producing an 

 adjustable color spot that the observer matches to a target. 

ACKNOWLEDGMENT 

This research was funded by the Electric Power Research Institute (EPRI) via contract 

number EP-P5419/C2683, EPRI project manager Dr. Naresh Kumar. 

REFERENCES 

1. Chestnut, L. G.; Dennis. R. L. Economic Benefits of Improvements in Visibility: Acid 

Rain Provisions of the 1990 Clean Air Act Amendments ; J. Air & Waste Manage. Assoc.

1997 47, 395-402.

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2. Kemp, M. A.; Smith, A. E.;Taylor C. L., Valuation of Haze Reductions in Eastern 

National Parks: New Survey Evidence, submitted to Land Economics. 

3. Henry R. C. Just-Noticeable Differences in Atmospheric Haze; J. Air & Waste Manage. 

Assoc. 2002, 52, 1238-1243. 

4. Henry, R.C. Perception of Complex Natural Targets Through Atmospheric Haze by 

Multiple Observers in preparation. 

5. Henry R. C. Psychophysics, Visibility, and Perceived Transparency; Atmos. Environ.

1987, 21, 159-164 

6 .Henry, R. C.; Mahadev S.; Urquito S.; and Chitwood D. Color Perception Through 

Atmospheric Haze; J. Opt. Soc. Am. A. 2000, 17, 831-835. 

APPENDIX 

Derivation of Eq.4. 

 First, write the deciview formula in terms of optical depth τ; if the distance to the 

object of interest is x then τ = bextx, and the deciview is given as 

⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ =

⎠

⎞ ⎜

⎝

⎛ = x

b dv ext

10

ln10

10

ln10 τ (A-1)

The x drops out since we are interested in the decrease in dv as optical depth decreases 

from τ

341 

342 1 to τ2 : 

)ln(10)ln(10 10

10 ln10

10

ln10

10

ln10 1 2

2

1 2 1 ττ

τ

τ τ τ −=⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⎟ =

⎠

⎞ ⎜

⎝

⎛ ⎟ −

⎠

⎞ ⎜

⎝

⎛ =Δ

x

x

x x

dv (A-2)

343 The value of τ2 for an increase in C of ΔC is given by 

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ Δ −= +

Δ+ = −

)0()0(

)( ln 1

)exp()0()(

1

2

1 2

C

C

C

C

a

aCCC

τ

τ

τ τ

 (A-3)

344 but 

13
 , )exp()0(/)( 1 1 τ CC = −aτ (A-4)

345 Then Eq. A-3 becomes 

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ Δ −= +−

)0( )exp(ln 1

2 1 C

C

a

a

τ τ (A-5)

346 Putting this in Eq.A-2 for the decrease in deciviews and gives the desired relationship 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛ Δ −−=Δ +−

)0( )exp(ln 1

1 ln10)ln(10 1 C

C

a

a

dv τ τ (A-6)

347 

348 

Since the logarithm is only real for nonnegative numbers, the decrease in deciviews given 

in Eq.A-2 is only valid if the optical depth τ obeys the following condition: 

))0(/1ln( 1

0))0(/)ln(exp( 1

CC

a

CCa

a

Δ−−≥

− Δ+− ≥

τ

τ

 (A-7)

In order for the logarithm in the second condition be real, the obvious condition that ΔC 

< C(0) must also hold. 

349 

350 

351 

352 

KEYWORDS 

Visibility, Perception, Vision, Air Pollution, Haze, Particles, Deciview 

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353 

354 

355 

356 

357 

358 

359 

FIGURE CAPTIONS 

Figure 1. Example of equation 4 

Figure 2. Perceived lightness difference versus optical depth for 8 observers 

Figure 3. Probability of seeing an improvement in regional haze, the upper curve is for 

C(0) = 50, the lower C(0) = 25 

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