# 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 1
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8 9 10 11 12 13 14 15 16 17 18 19 20

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22 23 24 25 26 Ronald C. Henry Department of Civil & Environmental Engine ering, University of Southern California, 3620 South Vermont Avenue, Los Angeles, CA 90089-2531.


### ABSTRACT
Regional haze regulations require progre ss towards reducing atmospheric haze as measured by particle scattering coefficient of visible light. From a pr actical perspective, this raises the question “Given a decrease in extinction, what is the probability that people will notice an impr ovement in visibility?” This paper proposes a quantitative definition of the probability of a perceptible incr ease in visibility give n a decrease in light extinction and a general method to estim ate this probability from perception measurements made in the field under real istic conditions. Using data from a recent study of visibility perception by 8 observers, it is estimated th at a 2 to 4 deciview change gives a 67 percent maximum probability of de tecting 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 ch ange of at least 3.5 to 7 d eciviews. 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 caus ed 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 leve ls for improved visibility must be balanced against the amount of improvement in the publ ic good. Thus, the visibility debate centers

1on how much improved visibility is wo rth and how much it is going to cost 1,2 . The work reported in this paper is directed toward asse ssing in a very fundamental way the worth of improved visibility. 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 This is a multidisciplinary study encompassing concepts of air quality, physics, and perception. A glossary of specialized terms th at may not be familiar to some readers is provided. The reader should be aware that seve ral ordinary English words as used in this paper have precise meanings. For example, lightness refers to a pe rceived sensation of light and dark and shades of gray. Terms su ch 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 actual ly see the improvement in visibility. For example, decreased emissions of sulfur di oxide may decrease the burden of particulate sulfate in a region; but if the resulting decr ease in particulate scattering is small, the general population may not noti ce the effect on visibility. So the question arises, “What is the probability that the gene ral population will act ually see a change in visibility given a relatively small change in particle extinction?”
A regulatory response to this question pr ovided in the regional haze air quality regulations is that a decrease of 1 deciview will result in a pe rceptible 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(b ext/10) where the extinction coefficient b ext of the atmosphere is given in Mm -1 . A previous paper found that the 1 deciview thres hold is not supported by actual perception measurements, but rather that a change of 1 deciview is al ways below the perceptible threshold and the threshold depends on the existing level of haze 3 . However, the data used in this previous work had several limitations, the foremost be ing reliance on two highly trained observers that had advance knowledge of the purpose of the study. Thus, th ere is a significant chance of bias in the results due to the sma ll number of observers and the possibility of unconscious bias in the observers themselv es. A new visibility perception study was

2designed to address this and other limitati ons; in particular, the latest study had 8 observers of which 7 were naïve subjects 58 59 60 61 62 63 64 65

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68 69 70 71 72 73 74 75 76 77 78 79 80 4 . The current paper presents a new definition of the probability of the public actually perceivi ng 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 PROBAB ILITY 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 pa per defines the probability of a noticeable improvement in visibility. Consider the following thought experiment. Let C be a measure of visibility percepti on such as perceived contrast, colorfulness, or, as in this work, perceived lightness difference between br ight 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)

81 82 83 where 0.5 ≤ α ≤ 1.0, since only improvements in visibi lity are of interest. This is a reasonable definition, for if there is no cha nge 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.,

384 85 86

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101 102 103 104 105 106 107 108 109 110 111 the probability that a decrease in haze has been ob served is 0. If the change in haze is big enough that all the observers re port seeing an increase in C , then α = 1 and P = 1, as would be expected.


### CALCULATING THE PROBABIL ITY 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 Δ C

in C that gives

α using the distribution of C obtained by repeated measurem ents under nearly identical conditions. Second, the decrease in extinction in deci views that gives Δ C

is calculated from the empirical relationshi p 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 depe nds upon two vista-spec ific 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 ex tinction 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 Δ C , 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, Δ C > 0

. To calculate the probability that Δ C > 0 , the probability distribution of Δ C is required. For each observer assume that repeated observations of C under unchanging conditions are normally distributed w ith standard deviation σ . By elementary probability theory, if

4the two observations are independent, Δ C

is normally distributed with mean Δ C and standard deviation 2 112 113 1/2 σ . Next, reduce this distribution to a standard normal distribution by the usual transformation t = ( Δ

C - Δ C )/( 2 1/2 σ ) . The condition that Δ C > 0 is equivalent to 114
t > −Δ C /(2 1/2 σ ) . Let F be the cumulative standard normal distribution, then the fraction of observers for which Δ C > 0 is α and is given by 1 - F(- 115 Δ C /( 2 1/2 σ )) = 1 – (1- F( Δ 116 C /( 2 1/2 σ ))) = F( Δ C /( 2 1/2 σ )) = α
. Solving this gives Δ C = 2 1/2 σ F

-1 ( α

) . Thus, the increase in C necessary to make the fraction of observers seeing an increase equal to

α is 117 118 119 Δ C = 2 1/2 σ F − 1 ( α ) = 2 1/2 σ F − 1 (0.5(1 + P )) , (2)

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

121 122 123 

### Calculating the Required De crease in Extinction Assume that observations show that C is related to optical depth τ by the following expression: )exp()0()( τ τ aCC

− = (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 test ed by experimental observations. 124 125 126 127 128 129 Given the above relationship between optical depth and C, the decrease in extinction given in deciviews required to achieve an increase of Δ C is given by ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ +− −−=Δ )0( )exp(ln 1 ln10)ln(10 C C a a dv τ τ (4)

130 131 132 133 134 The Appendix gives the derivation of this equa tion and the limits of its applicability. The above equation depends on two site-specifi c parameters: the cu rrent optical depth τ , which depends on the concentrati on 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 Δ C/C(0) =

5135 136 137 138 139 140 141 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 Δ C

. For the remainder of the pape r assume that the constant a = 1 in Eq.3. The minimum value of Eq.4 was found for values of Δ C/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 dv min that is required to produce a decrease of Δ C

.
)0( 3.27 )0(

) 22.1 )0(

) 9.11 2 3
min C C C C C C dv Δ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ

− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ
= . (5)

Inverting this numerical ly over the range of C Δ /C(0) between 0 and 0.5 gives 142 143 dvqdvq C C 1 2 2 )0( += Δ

. (6)

where q 2 = -0.00030264, q 1 = 0.038401 and 0 ≤ dv ≤ 15. 144

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### 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 P F C

(7)

147 148 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 = 2 F C (0) 2 1/2 σ ( q 2 dv 2 + q 1 dv ) ⎛ ⎝ ⎜ ⎞ ⎠

⎟

− 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. 149 150

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163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 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 observ ations of perceived lightness of natural targets seen through se mitransparent 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 mountainside 4 . The bush was a contro l target and was too close to be affected by haze. The rock out crop 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 area s approach each other in lightness. If the

7179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 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 spatia lly 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 ho rizon, but some degrees above the horizon where the luminance of the sky is not c onstant. 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 sh adow passing in front of the target will affect the light and dark areas equally leav ing the difference of luminance of the two unaffected.
The well-behaved nature of the physical diffe rence of luminance between light and dark areas of a target is one of the reasons th at the difference in perceived lightness was chosen as the perception parameter in this st udy. The other reason fo r the choice is that changes in hue and colorfulness of a target due to haze have been studied in previous visibility perception studies 3, 6 and it was felt that a visibility perception parameter based on lightness was needed since many targets are no t very colorful and the main effect of haze is often on the lightness of the target.
The perceived lightness difference for the rock s and mountainside (the two main targets in this study) were found to d ecrease exponentially with optical depth as shown in Fig. 2. The r 2 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 ro cks. 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 2.85 4 (the perceived lightness numbers are dimensionless).

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234 235 236 237 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 obs ervations, it was found that the instructions had been misinterpr eted. 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 r 2 of the exponential fit is 0.924 with C(0) = 30 4 . Thus, the luminance difference given by the photometer is well represented by an exponent ial, as expected from radiative transfer theory. The relationship between per ceived lightness difference and photometer measurements is complex; objects seen th rough a semi-transparent haze appear darker than expected 5

. 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 d eciview 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 ove restimation or underestimation of the probability of seeing a difference in haze. In this study, the key m easurement is the variability in the observer matches of the targets, σ in Eq.8. If σ is overestimated, then the probability of perceiving

9238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 a decrease in haze is underestimated; if σ is underestimated, the probability is overestimated. Overall, σ in this study is probably undere stimated, as explained below.
What is the effect of the experi mental 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 observ er to use the visual colorimeter for color matching. This second source of variability is not necessarily of in terest. Assume that the observer is perceptually perfect, then if the color-matchin g procedure is perfect, the color matches will all be identical and have no variability (assuming the atmospheric and lighting conditions do not change). In real ity even a perceptually perfect observer will probably not be able to make identical matc hes since the color matching procedure takes time and requires manipulating the visual colori meter. 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 righ tfully 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 th e instrument variability. Therefore, σ = 2.85 as determined in this experiment is probably an underestimate of th e 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 per ception studies carried out by the author have

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279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 focused on perception of hue and colorfulness rather th an lightness, as in this study. The variability of the visibility perception parameter may depend on the choice of the parameter itself. Future st udies should systematically study the effect of choice of visibility parameter on the estimated proba bility of perceiving a change in haze.
Finally, although this study used many more obs ervers 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 dr awn from this paper is that a 1 deciview decrease in extinction as defined as a just-noticeable diffe rence in the regional h aze regulations is not likely to be perceived by the general populati on, having at most only a 17 to 35 percent chance of being noticed by the ge neral 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 de tecting 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 d ecrease and if this is truly pe rceptible, then the cost of achieving this decrease in haze must be wei ghed 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 visibil ity, this is likely to be considerably more

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323 324 325 costly to achieve than a 1 deciview change . Therefore the costs in the cost benefit analysis would be much higher while the assume d visibility benefits of a just-noticeable improvement would be the same.


### GLOSSARY
Colorfulness – the degree to which a color is perceived, in dependent of hue and lightness. Deciview – a measure of light extinction defined as 10ln(b ext/10) where the extinction coefficient b ext 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 percep tion of light and dark and sh ades 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 Po wer Research Institute (EPRI) via contract number EP-P5419/C2683, EPRI projectmanager Dr. Naresh Kumar.


### REFERENCES
1. Chestnut, L. G.; Dennis. R. L. Economic Be nefits 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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338 339 340 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 Natu ral 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 = b ext x , 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 th e decrease in dv as op tical depth decreases from τ 341 342 1 to τ 2 : )ln(10)ln(10 10 10 ln10 10 ln10 10 ln10 2 1 2 1 2 1 ττ τ τ τ τ −= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛

− ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =Δ x x x x dv (A-2)

The value of τ 2 for an increase in C of Δ C is given by 343

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ + −= − = Δ+
)0()0( )( ln 1 )exp()0()(

1 2 2 1
C C C C
a aCCC

τ τ τ τ (A-3)

344 but

13, )exp()0(/)( 1 1 τ τ a CC

− = (A-4)
345 Then Eq. A-3 becomes ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ +− −= )0( )exp(ln 1 1 2 C C a a τ τ (A-5)

346 Putting this in Eq.A-2 for the decrease in deciviews and gives th e desired relationship ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ +− −−=Δ )0( )exp(ln 1 ln10)ln(10 1 1 C C a a dv τ τ (A-6)

347 348 Since the logarithm is only re al for nonnegative numbers, the decrease in deciviews given in Eq.A-2 is only vali d 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 c ondition be real, the obvious condition that Δ C < C(0) must also hold. 349 350

351

352 

### KEYWORDS
Visibility, Perception, Vision, Air Po llution, Haze, Particles, Deciview

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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 improvementin regional haze, the upper curve is for C(0) = 50, the lower C(0) = 25

15