BY
RUDOLF B HUSAR

RADIATIVE TRANSFER
Mathematical Formulation
Ideally, to understand vision in the atmosphere, we should follow a solar photon as it:

Descends through the atmosphere, possibly being scattered or absorbed;

Strikes a surface, and is reflected or absorbed;

Traverses the atmosphere, possibly being scattered or absorbed; and

Enters the eye, contributing to sensations.
The preceding sections have reviewed the component steps; the present section analyzes their interactions.
The rigorous treatment of visibility requires a mathematical description of the interaction of light with the atmosphere. This description will be based on a simple energy balance, known as the radiative transfer equation, which governs the distribution of light intensity. The intensity is the energy flux (erg m -2 sec -1 ) of radiation in a specific direction, and in general is a function of direction, location, time and wavelength. The intensity field does not fully characterize the radiation field, since it does not specify polarization or phase; it does, however, determine the response of the eye, and so adequately characterizes the radiation field for the purposes of this discussion.
Consider a beam of light transmitted along the x-axis, and let I(x) be the intensity in the x-direction at any point, as shown in Figure 1a. The intensity I(x) decreases with x, as energy is absorbed or scattered out of the beam. Over a short interval, this decrease is proportional to the length, dx, of the interval and the intensity, I(x), of the beam at that point:
 dI
 b ext
Idx .
(1)
(extinction)
The coefficient of proportionality, denoted by b ext
, is called the extinction coefficient .
The extinction coefficient is in general a function of location, time, and wavelength. IN the situations of interest here, where the individual particles and gas molecules are randomly oriented, it does not depend on direction.
Consider now an observer looking at a distant target, as shown in Figure 1b. Just as a beam is attenuated by the atmosphere, the light from the target which reaches the observer is also diminished by absorption and scattering. The reduced brightness of distant objects is not usually the primary factor limiting their visibility, however; if it were, the stars should be visible around the clack, since their light must traverse the same atmosphere night and day. Tin addition to light origination at the target, our observer receives extraneous light scattered into his line of sight by the intervening atmosphere. It is this airlight which forms the diaphanous, visible screen we recognize as haze.

Figure 1 [a] A schematic representation of atmospherical extinction, illustration: (i) transmitted (ii) scattered, and (iii) absorbed light. [b] A schematic representation of daytime visibility; illustrating: (i) residual light from target reaching observer,
(ii) light from target scattered out of observer’s line of sight, (iii) airlight from intervening atmosphere, and (iv) airlight constituting horizon sky. (For simplicity, diffuse illumination from sky and surface is not shown..) The extinction of transmitted light attenuates the “signal” from the target at the same time as the scattering of airlight is increasing the background “noise”.
The intensity of airlight in the x-direction depends on the distribution of intensities in all directions. Consider a beam of light – direct sunlight, diffuse skylight, or surface reflection – incident upon an atmosphere volume. The energy lost from the beam is proportional to b ext
; a fraction of the lost energy, given by the single scattering albedo
0< w <1, is reradiated in all directions; and increment o f this scattered energy, determined by the angle, θ , of the abeam to the x-axis, is scattered into the observer’s line of sight
(Figure 2). The total intensity in the x-direction added by airlight over a short interval is obtained by spherical integration over all directions v (Chandrasekhar, 1950): dI
 b ext w

Q v
(
 v
) I ( v ) d
 v dx .
(airlight)

Figure 2 Geometry of the spherical integral for the airlight. Diffuse radiation from all direction, v, is scattered into the observer’s line of sight, the x-axis.
The overall evolution of intensity along the given line of sight is governed by the extinction of transmitted light and the addition of airlight: dI = -dI + dI
(extinction) (airlight)
  b ext
Idx
 b ext
 
Q (
 v
) I ( v ) d
 v dx , or
1 dI b ext dx
 
I
  
Q (
 v
) I ( v ) d
 v
.
(2)
Equation 2, known as the radiative transfer equation , forms the basis for our understanding of atmospheric visibility. Its general solution is difficult, because the spherical integral on the right-hand side couples the intensity of the radiation in one direction with the intensities of radiation in every other direction.
The remainder of this section will discuss some important idealized situations in which equation 2 admits simple solutions. These analyses are intended to convey a sense of what equation 2 means in some concrete settings, and should be understood as illustrative examples, not as comprehensive treatment, nor as a framework for regulation.
A later section will resent the results of detailed computer models, based on \numerical integration of equation 2, which take into account such complicating factors as spatially inhomogeneous haze and the earth’s curvature.
Visual Range: the Koschmieder Theory
The visual appearance of distant objects has long been studied by artists and naturalists, many of whom have made perceptive observations. IN the early part of this century, there began to emerge an understanding of the quantitative relationship between visibility and atmospheric optics, culminating in 1924 with the formulation, by H
Koschmieder, of a satisfactory theory for the limiting visual range of dark objects viewed against the horizon. In its simplest form, this theory applies exactly only under somewhat restrictive conditions. We shall see, however, that it gives satisfactory results in a wide

variety of non-ideal applications. Given the complexity of the radiative transfer equation, the simplicity and success of Koschmieder’s approach are quite remarkable.
As indicated in Figure 1b, an observer’s view of a distant object is obscured by airlight from the intervening atmosphere. We recall that the airlight contributed by a small interval along his sight path is proportional to b ext
: dI
 b ext
Adx,
(airlight) where
A
   fQ (
 v
) I ( v ) d
 v
.
The basic simplification involved in Koschmieder’s theory is the assumption that A, the coefficient of proportionality, is independent of x. This is a valid assumption in a uniform atmosphere, uniformly illuminated; it is often a good approximation even in a nonuniform atmosphere, so long as the variability shows up only in b ext
(determined by both composition and concentration of optically active material) and not in ω Q
(determined by composition only). Under these conditions equation 2, governing the evolution of intensity along the observer’s line of sight, reduces to
1 b ext dI dx
 
I

A , whose solution is given by
I (

)
 e
 
I ( 0 )

( 1
 e
 
) A ,
(transmitted from target)(airlight) where
  
( x )
 x

0 b ext dx '.
(3)
Equation 3 gives the intensity received by our observer as a function of optical distance
, τ, the dimensionless integral of the extinction coefficient along his sight path.
If airlight is uniformly proportional to extinction, as assumed, then the intensity received from the direction of the target depends only on the total extinction along the sight path, b ext

1 x x
0
 b ext dex '

 x
.
As indicated below equation 3, the term e
-τ I(0), which decreases with τ , represents the residual light reaching the observer direct from the target; it is, in fact, the integral of equation 1. The term (1-e
-τ )A, which increases with τ, represents the airlight reaching the observer from the intervening haze. With increasing extinction or distance, the intensity transmitted from the target decreases while that contributed by airlight increases; simply put, the observer’s visual target is increasingly masked by haze. In the limit of arbitrarily great optical distance, our observer receives only airlight, and the apparent intensity of the target becomes that of the horizon sky, I
H
; thus,

I
H
 lim
  
I (

)
 lim
 
0
[ e
 
I ( 0 )]

( 1
 e
 
) A

A .
With this interpretation of A, we can put equation 3 into a more easily applied form:
I (

)
 e
 
I ( 0 )

( 1
 e
 
) I
H
.
(4)
The visibility of sufficiently large targets is determined by their apparent contrast with the background. From equation 4, the apparent contrast of a distant target with the horizon sky is given by dI
I

I
H
I
H

I
 e
 
(
I
H

I
H
I ( 0 )
).
(5)

I
H
 e
 
I ( 0 )

( 1
 e
 
) I
H
I
H
The apparent contrast of a target with the horizon sky thus decays exponentially with optical distance.
The limiting distance at which a target can be distinguished from the horizon sky is that at which its apparent contrast (I
H
-I)/I
H
) drops to the observer’s threshold of detection,
ε. From equation 5, the limiting optical distance I
   log
  log(
I
H

I
H
I ( 0 )
), which corresponds to a geometric distance of x

(
 log
  log(
I
H

I
H
I ( 0 )
)) / b ext
.
(6)
The limiting distance at which any large target can be seen against the horizon sky is thus inversely proportional to the extinction coefficient of the atmosphere, which is an objective, physical quantity.
The scaling factor K=-log ε + log ((I
H
-I)/I
H
) in equation 6 depends on the observer’s contrast threshold and the target’s intrinsic contrast. Since log 1=0, the second term of K is small for dark targets; Koschmieder defined the Visual Range , V, to be the limiting distance at which a black (I(0) = 0) target can be distinguished against the horizon sky, thereby setting K = -log ε. Based on the data available to him, he took ε = 0.02 to be representative of daylight viewing, and thus arrived at the Koschmieder Formula:
V

3 .
9 / b ext
The Koschmieder formula is surprising in its simplicity, relating a fundamental optical parameter of the atmosphere to a quantity easily determined with the naked eye. It states, among other things, that the visual range is independent of the level of illumination, over the several orders of magnitude which an observer’s contrast threshold ε is constant. This invariance of the formula derives from the choice of target and background. AS the airlight between the observer and target increases, so does the airlight beyond the target;

the apparent intensity of the horizon thus increases in the same proportion as the apparent intensity of the black target, and their ratio remains constant.
At sea level in a perfectly clean Rayleigh atmosphere, the Koschmieder formula predicts a visual range of 260 km at 0.55μm. This result is somewhat academic, however, since the visibility of most targets at such a distance is limited by the curvature of the earth, which has been neglected in the foregoing analysis. The main practical value of the Koschmieder formula lies in its application to conditions of suboptimal visibility.
The observer’s contrast threshold is of course no universal constant. It varies with apparent target size and overall illumination (Figure 3), and it varies with observer
(Figure 4). Fortunately, however, it enters equation 6 only logarithmically, so that the relationship of visual range to extinction is not unduly sensitive to the psychophysiology of the observer. A sevenfold increase is ε, for example, does not quite double –log ε.
Figure 3 Necessary correction ΔK to the Koschmieder formula due to the varying contrast threshold of the eye, calculated from Blackwell’s data. On the abscissa the varying angular size of the target is platted. The intensity of the horizon sky is given as a parameter (3500lx: bright day – 3.5 lx: twilight. (Horvath, 1971).

Figure 4 Measured apparent contrast of farthest visibility marker identified in 1000 determinations of visual range by 10- observers. Scatter is due to both the variability of observer thresholds and the discrete nature of the marker set. The corresponding value for the Koschmieder constant, K = -log ε, is 3.2 ± 0.6. (After
Middleton, 1952).
Little error is introduced in the determination of visual range by the use as targets of such non-black features as dark forests and deep shadows. For a target whose intrinsic intensity I (0) is as much as 30% that of the horizon sky, for example, the limiting distance given by equation6 is within 10% of the visual range. On the other hand, the intrinsic intensity of sunlit objects can approach that of the horizon sky, in which case their use as targets can lead to large underestimates of visual range (Figure 5). It is interesting to note from equation 6 that a target whose intrinsic intensity is more than twice that of the horizon – the sun’s reflection on a window, for example – can be seen at distances exceeding the visual range.
Figure 5 Necessary corrections (K) to the Doschmieder formula for non-black targets. Curves are derived from simultaneous measurements of the intrinsic intensity, I = I(0), of a brick wall, a mortar covered wall, and the horizon during a sunny day. The geometry of the sun, wall, and direction of observation (→) are shown at top. With reasonable care in the selection of targets, visual range can be estimated within 10%. (Horvath, 1971).

The effect on visual range of inhomogeneous illumination, such as that under scattered clouds, is difficult to analyze by the elementary methods employed in this section.
Experimental evidence indicates that it is not great however. Visual range is consistently found to correlate with the reciprocal of the scattering coefficient, b scat
= ωb ext
, as illustrated in Figure 6. (The scattering coefficient is easier to measure than the extinction coefficient.) The correlation coefficients are commonly in the neighborhood of 0.9, with values for the dimensionless product b scat
V in the range 2-4. The scatter in the data is commensurate with that expected from the known variability of ω, ε, and I(0), leaving little room for scatter attributable to other sources.
Figure 6 Inverse proportionality between visual range and the scattering coefficient b scat
= b ext
measured at the point of observation. The straight line shows the
Koschmieder formula from non-absorbing (b ext
= b scat
) media, V = 3.9/b scat
. The linear correlation coefficient for V and 1/b scat
is 000.89. (Horvath and Noll, 1969).
In summary, visual range – the limiting distance at which a large black object can be distinguished from the horizon sky – is inversely proportional to the atmospheric extinction coefficient, the constant of proportionality depending on the observer’s contrast detection threshold. The theoretical derivation of this relationship is usually based on rather restrictive assumptions concerning the homogeneity of the atmosphere, the uniformity of its illumination, and the nature of the visual target. In practice, the relationship is found to be robust, in the sense that it is relatively insensitive to the approximations underlying its theoretical derivation.
THE COLOR OF THE SKY AND DISTANT OBJECTS
The color of the sky is an important input to our subjective assessment of air quality.
IN locations where visual limits are established by nearby buildings or topography, sky color is often the most sensitive indicator of haze available to the unaided observer. IN the expansive panoramas typical of may Class I areas, the sky is the dominant feature of the landscape.
Consider an observer looking at the sky. What he sees is scattered sunlight:

“I say that the blue which is seen in the atmosphere is not its own color but is caused by … minute and imperceptible atoms on which the solar rays fall rendering them luminous against the immense darkness of the region of fire that forms a covering above them.”
Leonardo da Vinci, ca. 1500
The airlight contributed by a small interval along his line of sight is proportional to b ext
: dI (

)
 b ext
(

) A (

) dx ,
(airlight) where
A (

)
 
(

)

Q (


,

) I (

,

) d
 v
.
Maintaining the elementary approach begun earlier, we will continue to assume that A(λ), the coefficient of proportionality, is independent of X. This is a good approximation for very clean (Rayleigh scattering) atmospheres, in which airlight and extinction are roughly proportional to density; it can be a good approximation as well for very hazy atmospheres, in which airlight and extinction are roughly proportional to aerosol concentration.
The spectrum of airlight is determined primarily by the spectral extinction coefficient, b ext
. It was shown earlier that A(λ), when spatially uniform, is just the intensity of the horizon sky at the appropriate sun angle, so that we can write dI (

)
 b ext
(

) I
H
(

) dx .
(airlight)
As noted earlier, the variation of b ext with wavelength is generally substantial. ranging from b ext
(λ)~λ -4
in a Rayleigh atmosphere to b ext
(λ)~λ
-2
or λ
-1
in a typical smog. The variation of I
H
with wavelength is much weaker; in clean air, and often in dirty air, the horizon appears white (more precisely, the color of the sun). For the moment, we shall confine our attention to such “white hazes.”
The decrease in b ext with wavelength means that the light scattered by a small volume of air tends to appear blue, just as a filament of smoke from the burning tip of a cigarette appears blue when viewed against a dark background. Sine the visible daytime sky is just scattered light, one might expect it always to appear blue. The sky does not always appear blue because we do not see all the light that is scattered in our direction; the light reaching us from a given volume of air has been filtered by the intervening atmosphere.
Just as the shorter wavelengths of sunlight are more likely initially to be scattered into our line of sight, the shorter wavelengths of the scattered light are more likely subsequently to be scattered out of our line of sight. The same filament of cigarette smoke which appears brown against a bright one, because smoke, and the atmosphere generally, act as blue-minus filters for transmitted light.

As Leonardo observed, the sky may be regarded as a black target –outer space- partially or completely veiled by the earth’s atmosphere. According to equation 4, the spectrum of a patch of sky is therefore given by
I (

)

( 1
 e
 
(

)
) I
H
, where τ is the slant optical depth of the atmosphere along the sight path. The slant optical depth can be written as the product,

(

)
 b ext
(

) H m , of the spectral extinction coefficient b ext
(λ), the height H of an equivalent uniform atmosphere, and the relative airmass m along the sight path.
Both the spectral variation and the absolute magnitude of λ affect the spectrum of the sky. The influence of magnitude is most apparent in clean air:
… where a greater quantity of [the atmosphere] comes between the eye and the sphere of fire, there it appears much whiter. This happens toward the horizon. And the less extent of atmosphere between the eye and the sphere of fire of so much the deeper blue does it appear …’
Leonardo da Vince, ca. 1500
Looking up through a Rayleigh atmosphere, τ is small (τ (0.55)
≃
0.1), and
I (

)

( 1
 e
 
(

)
) I
H
 
(

) I
H
 b ext
(

) HmI
H
~ λ -4
; the sky above is blue, especially in the mountains, where τ is least. AS the gaze is lowered toward the horizon, m (and hence τ) becomes large, and the spectrum
I (

)

( 1
 e
 
(

)
) I
H

I
H becomes flat. (Leonardo noted that this loss of color with increasing optical depth shows that the blue of the sky is not, as is the blue of glass or sapphires, due to absorption.)
Haze modifies both the spectral variation and absolute magnitude of τ. The optical depth along a given sight path is increased, which in a clean atmosphere would correspond to lowering the sight path toward the white horizon. Moreover, the preferential scattering of the shorter wavelengths is diminished, as more of the scattering is contributed by particles above the Rayleigh size range. Both effects work to wash out the blue of the overhead sky (Figure 7).
In polluted atmospheres, the horizon sky can appear brown, which is to say a poorly saturated, low intensity yellow or orange. The spectrum of the horizon sky is given by:
I
H
(

)
 
(

)

Q (


,

) I (

,

) dw

, and both albedo ω and phase function Q tend to increase with wavelength in polluted air.
The spectral character of the spherical integral depends on the nature of both the aerosol

and the illumination, and lies outside the scope of our elementary treatment. The spectral character of ω results primarily from strongly selective absorption by NO
2
. Since
  b
SCAT
 b
ABS , AEROSOL
 b
ABS , NO
2
),
The contribution of NO
2
to color depends on the ratio b
ABS , NO
2
/ b
SCAT

[ NO
2
] / b
SCAT
, not an absolute concentration; it takes less NO
2
to discolor the horizon in clean air than in dirty air.
This section has treated the sky simply as outer space viewed through the atmosphere, the only characteristic of space relevant to the analysis having been its blackness. All of the proceeding comments on the color of the sky therefore apply equally well to the apparent color of any distant black target. If, instead, we are interested in the apparent color of a non-black target viewed through the atmosphere, we must return to the full solution of the radiative transfer equation:
I (

)
 e
 
I ( 0 )

( 1
 e
 
) I
H
,(8) where τ is the optical distance between observer and target.
The first term on the right-hand side of equation 8 represents transmitted light reaching the observer from the target. Since τ (λ)= b ext(
λ)x decreases with wavelength, the transmittence of the atmosphere increases with wavelength, and the transmitted image therefore appears somewhat redder than the target itself. The second term on the righthand side represents airlight reaching the observer from the intervening haze. We have just seen how this tem also gives the spectrum of the sky (equation 7), which we know tends to appear blue at small to moderate values of τ.
The apparent color of a non-black target depends on the target’s intrinsic spectrum as well as the optical characteristics of the atmosphere. As shown in Figure 7, the apparent color of any target fades toward that of the horizon sky as τ increases and the target recedes toward the limit of its visibility. At a given distance, the magnitude of the ratio
I(0)/I
H
determines whether “reddened” transmitted light or “bluish” scattered light dominates the radiation received by the observer. A brilliant snow-capped mountain peak may appear a dirty yellow or red when viewed through light haze, where a dark pine-forested range would have a bluish cast under similar viewing conditions.

Figure 7 (a) Apparent chromaticity of a black (I(0)=0) target at various distances
(in km.) (The depth of an equivalent homogeneous atmosphere is about 8 km.)
(b) Apparent chromaticity of a bright (I(0)=2I
H
) white target at various distances
(in km.)
Solid loci correspond to pure air (b ext
(0.55) = 0.15 x 10 -4 m -1 ), dotted loci to light haze
(b ext
(0.55) = 0.28 x 10 -4 m -1 , b ext
(λ)=b scat
(λ)=0.008λ -2.09
km -1 , (λ in μm), both illuminated by CIE source C (simulating overcast sky). Ellipse is the MacAdam locus of colors just noticeably different from source C. (Adapted from
Middleton, 1952.)

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