RESEARCH ON THE OPTICAL STATE OF THE ATMOSPHERE
by
Michael McClintock
Alden McLellan
Leaf Turner
A Report to
The Environmental Protection Agency
on research conducted between
August 1, 1971 and June 1, 1972
Contract No. 68-02-0337
Principal Investigator: Michael McClintock
The Space Science and Engineering Center
The University of Wisconsin
1225 West Dayton Street
Madison, Wisconsin 53706
November 30, 1972
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CONTENTS
Page
I — INTRODUCTION 1
II — CLIMATIC CHANGE FROM DUST AND CLOUDS, by
Michael McClintock 2
The Effect of Clouds on Surface Temperature 3
Interaction Between Dust and Clouds 5
Conclusions 10
References 11
III — REMOTE SENSING OF GLOBAL ATMOSPHERIC POLLUTION,
by Alden McLellan 13
1. Introduction 13
2. Climatic Radiation Changes 13
3. Satellite Detection of Regional Pollution 21
4. Satellite Detection of the Movement of Large-scale
Man-made Pollution 27
5. Urban Modeling for Air Pollution Detection 29
6. Summary and Recommendations 33
References 39
IV — RAYLEIGH-GANS-BORN LIGHT SCATTERING BY ENSEMBLES OF
RANDOMLY ORIENTED ANISOTROPIC PARTICLES, by Leaf Turner 42
1. Introduction 43
2. The Stokes Matrix for Symmetric Media 44
3. The Scattering Amplitude Matrix 45
4. Applications of the RGB Approximation I: Variable
Scalar Refractive Index 48
5. Applications of the RGB Approximation II: Constant
Tensor Refractive Index 51
6. Summary: Implications for Future Research 53
References 55
ii
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I — INTRODUCTION
This is the third report to the Environmental Protection Agency in a
series that deals with research on the feasibility of using satellite-
based instruments to detect atmospheric pollution. The first report was
titled "Studies on Techniques for Satellite Surveillance of Global At-
mospheric Pollution," and was submitted to the National Air Pollution
Control Administration (since incorporated into the Environmental Protec-
tion Agency) in September, 1970. The second report was titled, "Satellite
Measurement of Spectral Turbidity and Albedo, and their Rates of Change,"
and was submitted to the Environmental Protection Agency in July, 1971.
Where discussions in the present report depend on the earlier work, an
appropriate reference is made rather than repeating the prior material.
The report investigates the possibility of climatic change from the
radiative interference from dust and clouds in the atmosphere, provides
an example of satellite detection of large-scale atmospheric pollution
and a laboratory experiment on the non-Lambertian radiative reflection
properties of urban areas, and presents a theoretical treatment of elec-
tromagnetic scattering from randomly oriented anisotropic particles with
the intent of obtaining information about their shape.
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II — CLIMATIC CHANGE FROM DUST AND CLOUDS
If there is disagreement among workers in the field of global air
pollution as to the extent that man's activities may be responsible for
climatic change, there seems to be a consensus that the problem must be
looked into. The possibility is well founded that whether of man-made
or natural origin, changes in atmospheric composition may lead to changes
in the radiative energy balance of the earth, and hence to climatic varia-
tion.
A series of papers have dealt with two of the important variable con-
stituents of the atmosphere: carbon dioxide and particulate matter (dust).
In 1956, Plass [1] directed attention to the greenhouse effect of C02,
and it was widely accepted that the increased C02 content of the atmos-
phere caused by greater combustion was responsible for the gradual in-
crease in mean surface temperature.
But the observation documented most recently by Mitchell [2], that the
mean temperature appeared to be decreasing since some time in the 1940's,
seemed to call for an appropriate effect which exceeded the greenhouse
effect of C02. A phenomenon with a time constant of the same order as
the observed C02 increase was necessary, and McCormick and Ludwig [3]
proposed an increase in airborne dust to account for the temperature de-
crease. Charlson and Pilat [4] pointed out that the optical properties
of the dust are important in determining the sign of the temperature
change, and Lettau and Lettau [5], and later Atwater [6] showed how the
reflectivity of the earth's surface also enters into determining the
change in albedo when dust is present in the atmosphere. More recently
Rasool and Schneider [7] have performed a more sophisticated analysis and
have demonstrated, consistently with Atwater, that if the surface reflec-
tivity is low and particles are small, the initial effect of increasing
dust should be to cool the earth. In a later analysis by Mitchell [8],
however, the predominant effect of a tropospheric aerosol near the surface
is shown to be very likely one of warming rather than cooling.
Although it is not always stated explicitly, all these workers must
surely realize that the part of the problem they have addressed may not
be the most important part from the standpoint of climatic change. Radi-
ative transfer modification by CO 2 and dust may be only the first step
in a series of coupled processes by which the planetary energy balance
undergoes a change. If a change in atmospheric dust results in a change
in mean cloudiness, for example, as recent urban climatic studies sug-
gest [9], this could be the determining factor in a change of the plane-
tary albedo, considering the high reflectivity of many clouds as seen
from the top. Clouds also usually behave as black bodies in the infra-
red, greatly altering the radiative characteristics of the part of the
atmosphere which they occupy. Thus, even if dust only moderately in-
fluences mean cloudiness, the optical effects of changes in cloudiness
are probably large enough that the dust-cloud interaction cannot be over-
looked as a possible determinant of climatic change. Among recent authors
who have dealt with this possibility in climatic terms are Mitchell [8]
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and Robinson [10]. In fact, if one grants that the most important
variable quantities in the earth's atmosphere from the standpoint of the
planetary radiation balance are probably water vapor, cloud, dust, COz,
and 03, and that a change in the amount of one of them may produce
changes in others, it seems clear that an understanding of the problem
has just begun.
We wish to indicate in this section that the interaction between dust
and clouds need not be large to be significant in affecting the tempera-
ture near the earth's surface. We also have constructed a simplified
mathematical model in an attempt to demonstrate one possible functional
dependence of mean cloudiness on particulate matter in the atmosphere.
Since Chagnon's controversial study of precipitation patterns at
LaPorte, Indiana [11], attention has been directed toward the possibility
that air pollution is acting as a seeding agent in clouds downwind of
major industrial centers. Evidence of this is not clear-cut because pre-
cipitation is also induced by the heat island of the city as well as by
prominent geographic features (LaPorte, for example, is situated near
Lake Michigan). On the other hand, there is also evidence that cloud
condensation nuclei may sometimes be so numerous that overseeding of clouds
occurs [12]. Overseeding encourages the growth and thickening of clouds
and thwarts their tendency to dissipate through precipitation. Thicken-
ing of cloud cover can also occur by another mechanism. Recent studies
indicate that the effluent of steel mills [13], and lead from automobiles
combined with traces of iodine in city air [14] may serve as excellent
ice nucleation sites in clouds. Large quantities of latent heat
released by ice formation can cause the explosive growth of the cloud
[9]. This is an important consequence, for it is well known that cloud
albedo, transmissivity and emissivity are sensitive to cloud thickness.
Of course, atmospheric dust itself has the potential to dramatically
alter climate if it exists in sufficient quantities. On the mesoscale,
Bryson and Baerreis [16] have found that the presence of heavy concen-
trations of dust in the atmosphere over the Rajasthan Desert of north-
west India increases the diabatic cooling rate of the midtroposphere by
30 to 50%. This leads to an increased subsidence rate which promotes
the continued aridity of the region, in spite of the fact that sufficient
moisture exists for much more frequent precipitation. On the global scale,
the numerical model of Rasool and Schneider [7] provides a clue to the
radiative effects of increasing background turbidity. The model predicts
that a four-fold (400%) increase in dust would lead to a 3.5° K decrease
in mean global temperature. A decrease of this magnitude is probably
sufficient to increase the long-term ice coverage of land masses signifi-
cantly [17].
The Effect of Clouds on Surface Temperature
As a first step in comparing the relative effect of clouds and dust
on climate, an attempt was made to determine the amount of variation in
mean cloudiness necessary to cause a decrease in surface temperature of
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3.5°, so that this could be compared with the result of Rasool and
Schneider. Work by Lettau and Lettau [5] and Manabe and Wetherald [18]
provided the information necessary to perform two independent calcula-
tions of this sort.
Lettau and Lettau make use of statistical data obtained by Haurwitz
[19] which relates radiation to fractional cloudiness, and a study by
Neiburger [20] based on measurements of shortwave radiation fluxes for
varying stratus cloud thickness. In the Lettau model cloud type was sub-
stituted for cloud thickness. The three parameters of albedo, absorption
and scattering were adjusted so that increases in mean cloudiness repro-
duced Haurwitz's data. The Lettau paper includes the calculated changes
in radiation flux for a stepwise increase in fractional cloudiness (from
0 to 1 in increments of one-third) for both cirrus and stratus over rep-
resentative city, desert and prairie surfaces (Kew, England; Pampa de La
Joya, Peru; and O'Neill, Nebraska, respectively). The data in this
paper were used in our calculation of surface temperature which follows.
In the notation of Lettau, the fraction of incoming radiation that
is absorbed at the surface is (l-a)G* where a is the surface albedo
and G* is the fraction of incident radiation. For steady-state con-
ditions
S0(l-a)G = eaT1* + Q (1)
where So is the solar constant, eaT1* is the radiative heat transfer,
and Q the combined conductive and convective heat transfer from the
surface. T was then allowed to vary from a value of TI = 290°K
(chosen as a typical average daylight temperature over land) to
T2 = 286.5°K, a AT of -3.5°K. As a first approximation, it was as-
sumed that Q did not vary, an assumption open to obvious question but
one which is not likely to change the result by an order of magnitude.
Thus
S0(l-a)(Gi - G*) = ea(Ti - Ta ) = ea(Ti - (Ti-AT)4). (2)
e, the emissivity, was taken to be unity and the right side of the ex-
pression was expanded about TI to obtain:
(l-a)(ct - G*) = 4aTiAT/S0 - 0.014 . (3)
By linear interpolation between the values of fractional cloudiness
listed in Table 8 of Lettau [5], it was possible to determine how much
increase in cloudiness causes a AT of -3.5°K. The results appear be-
low:
Cloud Type
Cirrus
Stratus
Kew
+ 8.6%
+ 3.0%
La Joya
+ 11.9%
+ 3.2%
O'Neill
+ 11.3%
+ 3.2%
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Comparing these results with the 400% change in dust content of the at-
mosphere which Rasool and Schneider calculated for the same change of
temperature, one sees the considerably greater effectiveness of clouds.
This is not a particularly surprising result, of course, but it does
provide the opportunity for quantitative comparison.
The same comparison can be obtained by another method from Manabe
and Wetherald's [18] numerical model of atmospheric radiation for a fixed
distribution of relative humidity. One part of the model computations
involved a series of radiative and convective thermal equilibrium calcu-
lations for varying distributions of cloudiness at low, middle and high
altitudes. The reflectivity of solar radiation was assumed to be 20% for
cirrus clouds, 48% for middle clouds, and 69% for low clouds, based on
observations by Haurwitz. Low and middle clouds were treated as full
black bodies, cirrus as half-black. The average fraction of cloud cover
was taken to be 0.218 for cirrus, 0.072 for middle, and 0.306 for low
cloud. The fraction of each type of cloud was then varied from 0 to 1
while the other two types were held fixed at their average fractional
values, and the resulting surface temperatures were calculated.
If it is assumed that atmospheric dust effectively increases low and
middle clouds only, the fractional change of each type required to lower
surface temperature by 3.5°K can be readily calculated from the results
of Manabe (Table 9 [18]). It is found that an increase of 4.5% of low
cloud or 9.2% of middle cloud is sufficient to lower surface temperature
by this amount. These estimates agree well with those derived inde-
pendently from Lettau's climatonomy model. The agreement is still more
impressive when it is recalled that Lettau's results are based largely
on observational data, while Manabe and Wetherald have used a purely
analytical model.
Dividing the 400% change in dust content required for a 3.5°K tem-
perature change (Rasool and Schneider) by the percentage change in
cloudiness required for the same effect, we see that low and middle
clouds are perhaps some 30 to 130 times more effective than dust in chang-
ing the surface temperature.
Interaction Between Dust and Clouds
Among the activities recommended by the participants of the recent
SMIC conference on inadvertent climate modification [21] was the use of
simplified parameterized climatic models in order to gain insight into
some of the basic factors of climate and climate change. It is therefore
desirable to construct a simplified model which incorporates those
features of cloud formation and depletion appropriate to evaluate
changes in mean cloudiness that may arise from a gradual flux of parti-
cles from the stratosphere to the troposphere. Such a model might be
thought to describe, for example, the gradual fallout over several
years of stratospheric dust from a violent volcanic eruption. If all
other meteorological processes are allowed to operate without modifica-
tion, the minor perturbation of stratospheric fallout would lead to the
following rate equation for the fractional cloudiness in a given area:
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= KiH + K2f + K30f (4)
where df/dt is the rate of change of fractional cloudiness, 0 £ f <. 1.
H is the rate of latent heat energy due to surface evaporation and is
thus proportional to the rate or evaporation of moisture, is the flux
of atmospheric particles which play the role of cloud condensation nuclei,
and the coefficients Ki, K2, Ka are proportionality constants which
contain the physics of cloud formation and depletion. This basic rate
equation for the change of cloud cover over an area is an expression
designed to illustrate the effect of gradual particulate fallout from
stratosphere to troposphere by singling out this process in the last
term of the equation. In the other terms we are interested in establish-
ing only a physically reasonable dependence on other variables which enter
the energy balance equations set forth later for a two-layer atmospheric
model. Thus KiH represents the physically reasonable assumption that
the rate of cloud formation over a sufficiently large area is, on the
average, proportional to the rate of evaporation of moisture from the
surface and is thus proportional to H. The quantity Ki represents
the mean atmospheric conditions which contribute to cloud formation over
the area. The term Kaf likewise represents the physically reasonable
assumption that the rate of cloud dissipation over a sufficiently large
area is, on the average, proportional to the existing cloudiness, with
the constant of proportionality Ka containing the mean atmospheric con-
ditions which contribute to cloud dissipation over the area. To see this,
one imagines the sky over the area to be divided into smaller units of
area, N in total, of which n subunits may be considered covered by
clouds. Then since each of the subunits is independent of the others,
to first order, it is clear that
where k is a constant. Also, since f = n/N, we have immediately the
dissipative term in Eq. (4):
df _ kf (6)
dt ~ kf '
The same logic supports the f-dependence of the term expressing the
effect of particulate flux from the stratosphere into the troposphere.
In addition, we suppose this term to be linearly dependent on the particle
flux, 4>, in the range of particle fluxes appropriate to long-term fallout
from explosive volcanic eruptions. We will refer to Ka as the "seeding
interaction coefficient," although we do not insist that its algebraic
sign be negative for all areas of the earth which one might consider.
In fact, the magnitude and sign of Ka depends upon local conditions.
If the number and size distribution of aerosol is such that seeding oc-
curs, the resultant induced precipitation would presumably cause de-
creases in mean cloudiness. On the other hand, if the dust overseeds
the clouds, increases in cloud cover and attendant decreases in precipi-
tation may result. However, if the dust is so heavy that a pronounced
increase in stagnation occurs (as in the Rajasthan), the dynamic condi-
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tions will not be favorable for cloud formation and the sign of the
interaction will once again be negative. This suggests the existence
of threshold levels of dust concentration which serve as lines of de-
marcation between the seeding, overseeding and stagnation regimes.
The SMIC report summarizes what is known about the first two of these
processes in terms that illustrate present uncertainties [21] :
. . .An increase in the number of cloud nuclei could lead to a de-
creased efficiency of rain formation, especially in regions where
rain is formed predominantly in nonsupercooled clouds and where
the air is naturally low in cloud nucleus content. If the water
vapor input remained constant, an increase in cloud cover or cloud
depth (or both) would result. Another possible effect of particle
pollution works in the opposite sense: if giant soluble particles
or effective ice nuclei are added inadvertently or otherwise by
man's activities, then a possible stimulation of precipitation or
cloud dissipation, accompanied by a reduction in cloud cover,
could result. To sum up, it may be said that mechanisms exist
whereby cloud cover may be increased by pollution. Mechanisms that
are probably less important tend to decrease cloud cover.
Since at steady state df/dt = 0, Eq. (4) can be solved immediately
for the steady state value of f, and, given the probability that
« Kz, can be expressed in the form of a convenient approximation:
KIH _ Kl
K2
Probably Ka > 0, as we have discussed, and given that K2 < 0, we see
that an increase in $ implies an increase in f, as it should, if H
does not change greatly with . Mathematically expressed, we note in
passing that 3f/9 > 0 if 8H/3 > 0, as expected, since
V TJ V1 V A *MJ
KI! Ks _ JL. n _ £ai -> M
Tf 2
K. 2
2 K2 K2 3cj)
The rate of surface evaporation H is a function of the surface
temperature of the earth T and may be expressed conveniently as
H = b (V ) [
where b(V^) is a coefficient proportional to the mean horizontal
winds VH, p (T ) is the (saturated) vapor pressure at temperature
S 3 u lli
T , and p is the actual partial pressure of water vapor in the atmos-
E
phere at the surface. The vapor pressure of water as a function of
temperature may be expressed in a form familiar for most liquids:
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= c
where C is a constant. Thus we have
H = b(VR)[C exp(- T-) - p] . (10)
Ei
The surface temperature will depend on the fractional cloudiness
since clouds modify both the visible and infrared radiative transfer of
the atmosphere. If Tg(f) is determined, H(Tg) can be expressed as
a function of f and Eq. (7) may then be solved to obtain f as a
function of . This will provide some insight into the functional de-
pendence of f upon stratospheric dust content.
In order to determine Tg(f), a set of local energy balance equations
for a two-layer atmosphere can be written:
Earth Surface: F0(l-fa) + [ (1-f )a_+f ]aT,! + t_(l-f)a_aT* = aT* + H
1 11 So r.
(ID
Troposphere: [(1-f)aT+f]oTj + [(1-f)aT+f]agaT^ + H = 2[(1-f)aT+fjoTJJ
(12)
Stratosphere: (1-f ) t^aT^ + [ (1-f )aT+f ]agoT^ = ^a^ (13)
where FO is the diurnally averaged local solar flux; T , T , and T
£j 1 O
are the temperatures of the earth, troposphere and stratosphere, respec-
tively; a and t are the absorptivities and transmissivities of the
troposphere, a the absorptivity of the stratosphere; a is the albedo
o
of the clouds which occupy a fraction f of the sky; H is the rate of
transfer of latent heat of evaporation from the surface, as before.
The above equations are based on a model subject to the following
restrictions:
(i) The atmosphere is considered transparent to visible radia-
tion-,
(ii) For the longwave spectrum, the earth and clouds are
assumed to be black, the atmosphere gray;
(iii) The tropospheric water vapor content is taken as constant,
and no convective contact between the troposphere and
stratosphere is provided for;
(iv) The clouds in the model have an effective albedo a,
but do not absorb in the shortwave.
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At first glance these restrictions may appear to be too unrealistic
to be useful, but since the philosophy of the effort is to determine
small changes in the steady-state value of cloudiness, brought about by
a minor perturbation, errors in the description of the system itself
tend to diminish in importance if observed steady-state data are used as
the point of departure in the final quantitative evaluation.
Equations (11), (12) and (13) may be solved simultaneously for 0T :
F0(4-a_a ) + [F0(a a a+a a -a -4a) + K*+K3(2-a a -a tj]f
it _ _ S T _ si Si S _ Kj _ b 1 b 1
E
[4-2aT-as(l+2aTtT+2t)] + [as(-l+aT+4aTtT-2tT+4t)-2 (l-aT) ]f
[F0oa_(l-0 -aa (l-a_-t_)Jf2
- § - 1 - Ki - § - T__T -
+ [2astT(l-aT-tT)]f2
If longwave backscattering from dust and molecular constituents is
neglected, then a + t = 1, and Eq. (14) simplifies somewhat. Further-
more, we can neglect terms in the product of small fractional quantities
relative to terms of order unity or greater (i.e., terms in the product
a^t,-) to obtain an equation for aT^ which is linear in f:
SI L
F0(4-a-a) + [F0a(a_a -4)
_
S_T - Kj
(4-2aT-as)
The steady-state fractional cloudiness may then be expressed as
C exp
T F0(4-a_aT) + [F0a(a_a -
LI , a 1 o J.
R a(4-2aT-as)
-Pi
(16)
Because of the complexity of this equation, a graphical or numerical
solution for f(<£>) is called for. Values of appropriate to an ex-
plosive volcanic eruption might then be used to calculate corresponding
values of f, provided mean values of the other parameters in Eq. (16)
are known for the region of interest. Current climatological observa-
tions must be used to determine these parameters, with set equal to
zero. The seeding interaction coefficient, K , may be estimated from
cloud physics data extrapolated to the lighter particle fluxes appropriate
for volcanic fallout from the stratosphere. Typical areas chosen might
be, for example, a tropical ocean area, a midlatitude ocean area, a mid-
latitude continental area, a large desert area, an arctic area, etc.
In this way one hopes to establish the variation of f with for
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typical areas, and, by then using Eq. (15), find the corresponding
variation in T .
Likely forms of the solution to this problem can be deduced from
physical reasoning. They present a rich spectrum of possibilities. First,
by simply looking at Eq. (15), one sees that if the predominant effect
of a long term flux of particles from the stratosphere causes an increase
in cloudiness f, then Tg will tend to be decreased by a larger (nega-
tive) first term in the square brackets simply because more sunlight is
reflected by clouds. The second term in f is also negative (since
Ka is intrinsically negative and \K-2\ > \K3$\ ), so a larger value of
of cloudiness will also decrease Tg because more heat is extracted
from the surface by evaporation (recall Eq. (7)). The flux of infrared
radiation from the surface is inhibited by absorption and reradiation of
clouds and by the clear atmosphere; factors expressing this inhibition
appear in terms containing a§ and a^ . This represents the part played
by clouds and air in the real atmosphere in warming the earth, as opposed
to their more obvious cooling effect in blocking incoming solar radia-
tion.
One sees from Eq. (15) that the infrared energy radiated from the
earth's surface is a linear function of the cloud cover, consistent with
the calculations of Schneider (SMIC report, 1971, p. 119) which were
based on a more complex atmosphere than that assumed here. Even without
detailed numerical examination, it appears that the quadratic terms in
the more precise Eq. (14) are small compared with the linear term, a fact
which forms the basis for the simpler equation. This does not appear to
be obvious from the assumptions made in constructing the model, but
arises naturally in the solution of the descriptive equations.
Things, however, are actually more complex than this, since f itself
depends on and atmospheric parameters as indicated in Eq. (16). This
equation expresses quantitatively the way in which a change in cloudi-
ness changes the surface temperature by allowing more solar absorption,
which then changes the saturation water vapor pressure so that more
clouds are made by additional evaporation. The full set of equations
therefore expresses some of the important feedback mechanisms by which
the radiative energy balance responds to an interaction between dust and
clouds. One might wish for simpler expressions, but the natural system
is complex, and any degree of fidelity to this system appears to demand
a certain complexity of mathematical expression. Fortunately the com-
puter holds out the hope that complexity of the necessary degree will not
produce merely confusion and intractibility. Work along these lines is
planned for the future.
Conclusions
1. An increase in carbon dioxide content of the earth's atmosphere
acts to increase the surface temperature by inhibiting infrared radia-
tion to space. An increase in dust content of the atmosphere modifies
both visible and infrared radiative transfer in ways that depend sensi-
tively on the optical properties of the dust, very likely producing a
decrease in surface temperature except for very heavy industrial pollu-
10
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tion and at high latitudes [21] . The present study calls attention to
another mechanism by which atmospheric dust may influence climate: by
interacting with atmospheric water vapor to modify cloud cover. Although
this has been mentioned in the recent literature, no attempt at quanti-
tative treatment appears to have been undertaken. We have shown here
that a change in cloud cover is between approximately 30 and 130 times
more effective (depending on cloud height and type) in causing a change
in surface temperature than a change in dustiness by modification of the
radiative transfer. The work of Rasool and Schneider [7] has shown that
a change in dustiness of more than 100% is required to change the mean
surface temperature 1° by modifying atmospheric radiative transfer. By
comparison we find that a change in mean cloud cover of between approxi-
mately 1-4% will change the surface temperature this amount. Consider-
ing the mean planetary temperature change of approximately -0.3° since
1940, it would seem extremely desirable to have satellite measurements
of cloud cover sensitive to long-term changes of 1%, at least over areas
of greatest particle production if not the entire planet.
2. A simplified mathematical model based on energy balance, and in-
corporating a portion of the hydrological cycle in an ad hoc way, has
been constructed to illustrate one mechanism by which the long-term fall-
out of dust from the stratosphere might bring about a change in mean
cloudiness of an area. Graphical or numerical solution of the equations
based on climatological observations is called for, but even a cursory
analysis indicates the correct sign of the result, i.e., increased parti-
cle flux from the stratosphere to the troposphere (if it produces the
probable effect of increased cloudiness) causes a decrease in surface
temperature. This predicts a long-term cooling trend following large,
explosive volcanic eruptions, and also corresponds to the sign of the
mean global temperature changes noted since the 1940's. Lacking quanti-
tative data, it is impossible to say whether this represents any more
than just an interesting chance correlation, but at the present state
of knowledge this must be said about the relation between almost all
climatological observations and theories of climatic change.
References
1. Plass, G. N.: Tellus,8, pp. 140 (1956).
2. Mitchell, J. M., Jr.: Global Effects of Environmental Pollution,
Springer-Verlag/D. Reidel, New York, pp. 139 (1970).
3. McCormick, R. A. and Ludwig, J. H.: Science, 156, pp. 1358 (1967)
4. Charlson, R. J. and Pilat, M. J.: J. Appl. Meteor., 8, pp. 1001
(1969).
5. Lettau, H. H. and Lettau, K. : Tellus, 21, pp. 208 (1969).
6. Atwater, M. A.: Science,!!!), pp. 64 (1970).
7. Rasool, S. I. and Schneider, S. H.: Science,pp. 173 (1971).
11
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8. Mitchell, J. M.: J. Appl. Meteor., 10, pp. 703 (1971).
9. See, for example, Peterson, J. T.: "The Climate of Cities," NAPCA
Publication No. AP-59, Environmental Protection Agency (1969).
10. Robinson, G. D.: "Long Term Effects of Air Pollution—A Survey,"
The Center for the Environment and Man, Publication GEM 4029-400
(1970).
11. Chagnon, S. A., Jr.: "The LaPorte Weather Anomaly—Fact or Fiction?"
Bull. Amer. Meteor. Soo.3 49, pp. 4-11 (1968).
12. Warner, J.: "A Reduction in Rainfall with Smoke from Sugarcane
Fires—An Inadvertent Weather Modification," J. Appl. Meteor., 7,
pp. 247-251 (1968).
13. Noblis, P. V., Radke, L. F., and Shumway, S. E.: "Cloud Condensa-
tion Nuclei from Industrial Sources and Their Apparent Influence on
Precipitation in Washington State," J. Atmos. Sci., 27, pp. 81-89
(1970).
14. Schaefer, V. J.: Global Effects of Environmental Pollution,
op. cit., pp. 158.
15. Peterson, J. T. and Bryson, R. A.: Proceedings of the First National
Conference on Weather Modification, Albany, New York (1968).
16. Bryson, R. A. and Baerreis, D. A.: Bull. Amer. Meteor. Soc., 48,
p. 3.
17. We are indebted to Dr. John Kutzbach of the University of Wisconsin
Center for Climatic Research for pointing out to us that climatolo-
gists, in a manner of speaking, regard our present era as an ice age.
Paleographic records indicate that Greenland and Antarctica, presently
ice-covered, have been so for less than 0.1% of the time over which
such records extend.
18. Manabe, S. and Wetheraid, R. T.: "Thermal Equilibrium of the Atmos-
phere with a Given Distribution of Relative Humidity," J. Atmos. Sci.3
24, pp. 241-259 (1967).
19. Haurwitz, B.: "Daytime Radiation at Blue Hill Observatory in 1933,"
Harvard Meteor. Studies No. 1 (1934).
20. Neiburger, M.: "Reflection, Absorption and Transmission of Insola-
tion by Stratus Clouds," J. Meteor., 6, pp. 98-104 (1949).
21. Inadvertent Climate Modification—Report of the Study of Man's Impact
on Climate (SMIC), M.I.T. Press, Cambridge, Mass. (1971).
12
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Ill — REMOTE SENSING OF GLOBAL ATMOSPHERIC POLLUTION
1. Introduction
The origins of atmospheric particulates comprise a diverse group of
sources both natural and man-made, and the particulates vary over a wide
range of sizes, shapes, densities and chemical compositions. Large-scale
natural pollution from windswept arid regions, aerosols, meteoric dust
and volcanic activity contribute to the particulate loading of the atmos-
phere as well as man's contribution by the combustion of fuels, incinera-
tion of waste materials, and industrial process losses. Of man's total
quantity of atmospheric pollutants, particulates make up only 10%. How-
ever, because of their widely varying economic and biological effects,
man-made particulates are an enormous problem (Scorer, 1968; Wolkonsky,
1969). Natural atmospheric particulate matter, on the other hand, can
cause local problems (intense dust and sand storms), but these phenomena
will, in general, cover a much larger region and have a much stronger
influence on climate.
In a discussion on the detection and surveillance of atmospheric pol-
lution by remote sensing, one must first define not only the optical
properties of the pollution but also the spatial dimensions and the fre-
quencies of measurement. A knowledge of these objectives is necessarily
inherent in the consideration of what instrumentation is to be used. It
is the purpose of this paper to give a brief description of the most im-
portant types of pollutants in the earth's atmosphere and their foremost
optical properties, as well as the measurements that have been performed.
For convenience, the relative scales of measurement have been divided
into three sections. The first section deals with spatial dimensions on
the global scale and time domains on the order of years; the second sec-
tion considers active remote sensing systems for very short time intervals
(less than a second); and the third section covers the more popular pas-
sive systems used primarily for intermediate times and spatial dimensions.
2. Climatic Radiation Changes
It has been in the interests of physical climatologists to investigate
the history of the earth's energy balance and, in particular, the role
played by the atmosphere. A number of remote sensing instruments were
developed in the nineteenth century in order to record the variations in
sunlight due to atmospheric phenomena (Middleton, 1969). See Figure 1.
These early sunshine recorders primarily made use of photochemical changes
and thermal effects for the conversion of solar radiation energy. Even
though these instruments were more of an attempt to quantify human well-
being than a serious contribution to meteorological theory, they did serve
as the forerunners of modern radiometers and pyranometers. See Figure 2.
Information on atmospheric radiation during this century is often ob-
tained from radiation observation records kept over decades by both
astronomers and meteorologists. Data from one of the oldest remote sensing
optical instruments—the telescope—are presently being used to determine
13
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Figure 1. Nineteenth Century diagram of early sunshine recorders. Note
that refraction takes place upon entering the sphere, but the
draftsman neglected the refraction of light rays upon their
exit.
14
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Figure 2. A crystal ball or a spherical flask full of liquid, placed
centrally in a wooden hemispherical shell so that the sun's
rays will burn a groove throughout the day. The depth of the
groove was supposed to be related to the amount of sunlight.
This idea was proposed as early as 1646.
15
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atmospheric turbidity variation over the past century. A series of
normalized photographic plates of the same sky area taken over a period of
time can give an indication of the turbidity variation.
For the most part, climatologists have relied on a network of sampling
stations for data on the gaseous pollutants of the atmosphere. To obtain
information on the electromagnetic radiation reaching the earth's surface,
a network of pyranometric ground stations measures the visible and near
infrared radiation from the sun and sky. There are about 100 of these
stations in the continental United States that use the Eppley pyranometer.
Most of them have been in operation since the early 1950' s. The pyrano-
meter 's 180° receiving surface has essentially two sensing elements under
a glass hemisphere, one blackened to absorb a large proportion of the inci-
dent radiation, and the other coated white to reflect the visible radiation.
If we assume, for simplicity, that the black element absorbs all incident
radiation and the white element absorbs only the infrared radiation, then
we have the energy equation for each element:
(black) Q + aT1*. = S, . . + oT* .
glass black black
(white) aT*. - S . .. + aT",
glass white white
(1)
where Q is the shortwave solar radiation plus the shortwave diffuse sky
radiation, a is the Stefan-Boltzmann constant, S is the storage rate,
and T is the temperature. We have assumed the infrared emissivity of the
glass cover to be 1.0. Since the storage rate is a function of the physi-
cal and thermal properties of the sensor, and is proportional as well to
its surface temperature, we have from Eqs. (1):
Q = k(T, , . - T , . ) (2)
x black white v
where the constant k is a function of the physical properties and is only
weakly dependent on the temperature of the sensor. The temperature differ-
ence between the two ring sensors is measured with a large number of thermo-
junctions. The Eppley pyranometer has 16 to 50 thermo junctions (depending
on the required precision) which are in thermal contact with the lower sur-
face of two thin, flat concentric silver sensor rings. The assembly is
enclosed in a blown spherical glass dome that is filled with dry air. This
limits the radiation to wavelengths between 0.3 - 0.4y to 3.0 - A.Ou. See
Figure 3.
The family of pyranometers receive solar radiation from the whole
hemisphere. Other related remote sensing instruments, such as pyrhelio-
meters, measure the direct solar radiation at normal incidence; pyrgeo-
meters measure the whole hemisphere infrared radiation; pyrradiometers
measure both solar and infrared radiation from a single hemisphere;
and net radiometers determine the radiation balance between upward and
downward directions. All these large-bandwidth instruments are helpful
in determining overall local atmospheric turbidity conditions over long
time periods.
16
-------
Figure 3. An Eppley pyranometer. This instrument, the basic type of
which is in widespread use throughout the world, measures the
total 2ir ster radiation reaching a point on the earth's
surface.
17
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Using the same physical principle as described above, radiometric in-
struments have been placed on board space platforms to determine the global
distribution of the reflected solar and long wavelength radiation leaving
the earth. The reflected solar radiation and the earth's emissive long
wavelength radiation occupy two separate spectral regions with little over-
lap. Approximately 98% of the reflected solar radiation lies in the wave-
length region below 3 microns, whereas approximately 94% of the radiation
emitted by the earth into space lies above 3 microns.
The sensor array shown in the photograph (Figure 4) was developed by
the University of Wisconsin as adapted for flight on the TIROS Operational
Satellite (TOS). The two-sensor configuration consists of a flat plate
radiometer with unrestricted field of view and a flat plate radiometer set
within a cone to restrict its field of view and to eliminate sensor response
to direct solar radiation. This array is mounted perpendicular to the
spacecraft spin axis so that the sensors scan the earth once per satellite
revolution. Radiation measurements of this nature are critical to the
determination of the energy balance of the earth-atmosphere system (Sellers,
1969).
The sun photometer is an optical instrument for remote sensing of
atmospheric turbidity that has come into widespread use during the 1960's.
This small, simple photometer was first developed by F. E. Volz (Volz,
1959). See Figure 5. It is used by the observer to make a rapid determina-
tion of the haze attenuation by measuring the direct solar radiation at a
wavelength of 0.5 micron with a bandwidth of about 0.06 micron. The in-
strument consists of a small wooden box which contains diaphragms, filter,
a bubble level, a selenium photocell sensing element whose current is
measured with an enclosed microammeter, and a movable diopter mounted on
a pivot outside the box. When the pivoted diopter is moved against the
stop, being aligned parallel to the top of the box, it is used to aim the
photometer directly at the sun. But with the instrument level and the
diopter directed at the sun as before, one can read the air mass M from
the scale on the inner side of the diopter.
In the determination of the haze attenuation coefficient from the
measurements, the attenuation due to atmospheric molecules and the weak
absorption of ozone is taken into account. Water vapor does not absorb
in the 0.5 micron region. However, in more recent sophisticated versions
of the sun photometer, various combinations of filters and detectors
which extend the measurements into the infrared are used.
The intensity of direct solar radiation at a point on the earth's
surface may be expressed in the form
I . it 10 IL 'u (3)
_ -(a.0 K-f- + B)/M .
Il in L P°
Consider the expression
Alog I = log I0 - log I - log F . (4)
By substituting expression (4) into (3), we have
18
-------
M
Q)
4J
(1)
e
o
•H
0)
•U
CD
r-t
P.
4-1
01
0)
VJ
00
19
-------
RADIATION
•:•:•:••
'•:'• :'•:'•:'• '•: '.'•:'•: x:i:: : ::::x£
:•:•:•:•:•:•:•:•:•:•:•:•:•:•:•:•
:•:•:•:•:•:•:•:•:•:•:•:•:•:•
•.'.V.'.V
••
1. i I 1
0| 10 20 30 -V
2
v^
1
3 50
^
\
=r
o
\,
i
•
L
LEVEL. INDICATOR
DIOPTER 2>CALE-
METER.
Figure 5. A schematic of a typical sun photometer.
20
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Alog I - a^K-E-M = BM (5)
L po
where
I = the measured value of the solar radiation
Id = extraterrestrial value of the solar radiation. It is
obtained by instrument calibration.
B = haze attenuation coefficient per unit optical path length
at the particular wavelength that is being detected
a° = attenuation coefficient of pure air at the wavelength
being sensed per unit optical path length at sea level
M = optical path length
K = factor to allow for the differences
p,po = ambient and sea level pressure, respectively.
This section has described a few of the simple instruments that have
proved extremely useful in remote sensing the atmosphere. These instru-
ments are currently in use at many observations throughout the world.
3. Satellite Detection of Regional Pollution
We have discussed above some of the early systems whereby man began
to record the contents of the atmosphere by remote-sensing techniques.
Almost all early attempts made use of radiation that was transmitted and
scattered by the sun. Other more modern methods not only use the sun as
a source, but also starlight, reflected light from the moon, and even the
natural infrared emissions directly from the aerosol constituents of the
atmosphere. The above methods may be classified as passive systems, in
that the source of radiation is from natural phenomena. However, remote
sensing detecting systems employing their own source of radiation, such as
laser radar techniques, are generally referred to as active systems.
To date all remote sensing systems on board satellites are passive.
Active systems for satellites have been designed, but they have yet to be
proven operational in a satellite environment. In any case, the most in-
triguing feature of satellite remote sensing of particulates is that by
the proper data analysis one can choose local areas and short periods or
large areas and long periods for investigating the properties of atmospheric
pollution, and one may in this way get a handle on man-made and natural
variations of turbid atmospheres. By the proper use of mathematical model-
ing one may then be able to estimate the strength of the source, the dis-
persive aspects of the pollution, and thus be in a position to forecast,
at least in a crude way, global, synoptic and even mesoscale pollution
variations.
21
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It is by the use of instruments that are able to detect various broad
bands of polarization radiation from the near ultraviolet (.30y) through
the near infrared (1.5y) that one can then obtain an estimation of the
particle size distribution of the turbid atmosphere. Most aerosols and
particulate pollutants are readily transported and dispersed by winds.
Rain-out will often remove the larger particles (> ly) from the troposphere,
but the smaller particles (<_ In) essentially become long-term residents of
the atmosphere, although they may agglomerate into large particles. Given
suitable conditions, many of the primary pollutants (gases and particulate
matter ejected directly into the atmosphere by man-made or natural means)
can participate in chemical, photochemical, and/or nonchemical reactions
in the atmosphere to produce secondary pollutants. The result is that the
particle size distribution, and thereby the radiative spectral absorption
and reflection properties will not be constant but will be extremely
variable, as in the pollutant concentration, from one urban area to another,
from one land mass to another—indeed, from one ocean to another and even
between hemispheres. Hence, one might like to see a feasibility integrated
program to develop the pilot instrumentation for satellite remote sensing,
the mathematical computer modeling, and the procedures necessary to combine
the observational and theoretical results into a system of data analysis
for the purpose of forecasting pollution levels on global and synoptic
scales.
A number of models describing various standard atmospheres have been
proposed in order to aid in the determination of those atmospheric condi-
tions which vary from the standard or background atmosphere. Elterman
(1964) recognized that a clear standard atmosphere is a helpful concept
for the interpretation of the optical properties of the atmosphere, and it
can be conveniently utilized if the parameters are an array of attenuation
coefficients and optical thickness values varying with altitude for each
wavelength band of interest. Such a model would comprise a molecular at-
mosphere with a standard aerosol component. The model concept is readily
workable for a molecular atmosphere and, in fact, has been computed by
Deirmendjian (1955). Sekera (1956) has computed a family of curves showing
the variation of the optical thickness of the molecular atmosphere with
height and with wavelength. Extending this concept to include a standard
particulate component is much more difficult, primarily because it varies
enormously with the meteorological conditions over the polluting source.
Flower, McCormick and Kurfis (1969) analyzed five years of turbidity
measurements made with a Volz sunphotometer from a network of stations in
the United States. They found among other things the mean turbidity to be
higher in the eastern regions than in the western regions, and an annual
cycle of low winter turbidity and high summer turbidity. McCormick and
Ludwig (1967), from other ground-based data, showed a large worldwide in-
crease in turbidity over the last 50 years. Hanson (1969), in studying the
use of Rayleigh scattering viewed by a geostationary satellite as a measure
of various atmospheric parameters, concluded that for small values of
satellite-sun angles and for relatively large wavelengths (A = . 7y), the
radiance for aerosols appears to be about three times greater than from
molecular scattering.
22
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As far as remote sensing from above is concerned, the problem is com-
plicated by the variation in albedo of the surface from which the pollution
originates and over which it moves. Another difficulty which besets this
method of aerosol pollution surveillance, is the non-Lambertionality of the
reflecting earth's surface. These are the two most important problems
which have to be solved before one can begin an attack on the development
of a method of detecting the variation of particulates on a local and on a
global scale by satellite remote sensing.
Sellers (1969), in modeling the global climate, based his work on the
concept that the steady-state average latitudinal distribution of surface
temperature, to a first approximation, should depend greatly on the incom-
ing solar energy, the transparency of the atmosphere, and the planetary
albedo. From Figure 5 of Sellers' paper we see strong dependency of the
mean annual temperatures on various assumed polar albedos.
There is little doubt that the constituents of the atmosphere have an
enormous impact on the climate of the earth. Water vapor, carbon dioxide
and other major components of the atmosphere have the greatest impact on
global climatological parameters, whereas particulate matter such as dust
and aerosols, which is indeed a major component of the atmosphere, has an
impact on climate both on a global scale as well as regional. The magni-
tude and cause of this impact is being debated in the scientific literature
(Bryson, Mitchell, Rasool and Schneider).
Whether or not global long-term climatic changes are due to the increas-
ing man-made particulate loading of the atmosphere or due to natural par-
ticulate loading from volcanic eruptions, it is certain that regional
changes in particulate loading do indeed affect the temperature, rainfall
and radiation reaching the earth's surface (Barrett et al., 1970).
This phenomenon is borne out in a number of special situations. The
strong decrease of atmospheric optical radiation observed in the northern
hemisphere after volcanic eruptions of Krakatoa, 1883, and Katmai, 1912
(Volz, 1965), as well as the more recent explosion of the Agung volcano,
1963 (Volz, 1970) located at Bali Island in the southern hemisphere, in-
jected dust into the atmosphere that spread over the entire hemisphere and
remained airborne for a number of months.
Mohr (1971) correlated two ESSA 8 satellite pictures of western and
central Europe to show that a satellite could locate an existing low-level
inversion over a large regional area in a situation that is cloud free and
that a certain minimal amount of pollution is present. In Figure 6 the
atmosphere is relatively clear (marked "+"), whereas in Figure 7 the at-
mosphere over the same area (marked "C") at a later date is polluted.
Both pictures were received by automatic picture transmission (APT) by
Deutscher Wetterdienst, Offenbach/Main, Germany. Figure 6 shows a large
clear area over northcentral and western Europe on 18 June 1970. The
coastlines along the Baltic and North Seas are sharp and distinct, and most
of England is visible. Central Europe and southern Scandinavia are under
the influence of the southeastern edge of an anticyclone centered in the
Norwegian Sea. A very dry northeasterly airflow dominated the area
(marked "+"). The range of dry-bulb/wet-bulb temperature separations was
23
-------
o
30%
o
Figure 6. ESSA 8 photograph at 1001 GMT on 18 June 1970. Note the clear
skies over northcentral and western Europe.
24
-------
SON
Figure 7. ESSA 8 photograph at 1046 GMT on 5 August 1970. Note the
extensive area of atmospheric pollution located at C.
25
-------
(CM.MAP9
,4 S* **""*Mr-^ a* "•&** *
"
«*»*s.-. " » * **r
Figure 8. A geographical map of the central Europe region.
26
-------
between 15° and 20CC. Visibilities in northern Germany were reported in
the 25- to 40-kilometer range, and maximum temperature reached 25°C.
Note the fog forming in the easterly flow over the northern part of the
North Sea and the clear area in the lee of northern Scotland and the Shet-
lands.
However, the situation on 5 August 1970 was quite different. (See
Figure 7.) A ridge of relatively high pressure lying between a low over
the Bay of Biscay and a weak low over the Baltic Sea (B) dominated central
Europe. The coastlines that showed so clearly in Figure 6 are now obscured
in Figure 7 by haze in the moist maritime air that was moving onshore. The
band C extending from the frontal zone at A has the appearance of dense
cirriform cloudiness. Figure 8 is a geographical map of the area. Although
only a few stations in this area reported cirrus, all stations recorded
dust or haze. Temperatures reached a maximum of about 27°C, relative
humidity values were between 70% and 80%, and visibilities ranged beteen
1.5 to 6 kilometers. Therefore, one may assume that the band C represents
a large regional area of air pollution with haze and dust trapped below
the temperature inversion. This band lies over a heavily industrialized
region where the effluents emitted by factories and the unfavorable at-
mospheric conditions of the low-level inversion and light surface winds
combined to produce this regional pollution situation. This situation is
not unlike that which often occurs over the northeastern part of the
United States.
4. Satellite Detection of the Movement of Large-scale Man-made Pollution
An example of large-scale man-made particulate loading is exemplified
by agricultural slash burning operations carried out throughout the world
in certain seasons. The most striking operations occur annually in Africa
and in South and Central America. In Guatemala, the annual slash burning
operations usually begin in February and last until the onset of the rainy
season in May. Parmenter (1971) has pointed out that the airborne pollu-
tion from this large-scale burning operation in Central America can be
viewed from satellite, and he showed that the smoke pattern can be followed
in space and time by means of Applications Technology Satellite III (ATS-
III) spin-scan camera system. Figure 9 shows a series of ATS-III pictures
that show the smoke for a 25-day period from 18 April 1971 to 12 May 1971.
From these photographs, the smoke appears to originate in two areas:
Tabasco, Mexico and the lowlands along the Gulf of Honduras. This entire
area was located in the southwest quadrant of a high for the first 12 days
of ATS coverage. The resulting southerly flow carried the smoke from the
land northward across the Gulf of Mexico as far north as New Orleans,
Louisiana. The smoke was only slightly visible in the first picture as
outlined by the dotted line. On the next day, 19 April, the smoke from the
two sources extended offshore and was more dense. On 22 and 23 April, the
smoke covered a large area of the lowlands north of the Sierra Madre range
and extended northward to the front located near the southern coastline
of the United States. To the end of April, the anticyclonic flow con-
tinued to carry smoke northward and eastward in advance of the front to
the point that it covered the western half of the Gulf of Mexico.
27
-------
29
30 MAY
Figure 9. A series of ATS-III pictures showing the tracking of large-
scale man-made pollution. Smoke from agricultural slash
burning operations in Central America from 18 April to 12 May
1971.
28
-------
On 1 May a strong frontal system moved into the Gulf with a change in
fog and stratus clouds, with the result that the smoke moved southward
across southern Mexico and out over the Pacific Ocean. A subsequent change
in low-level flow which occurred on 4 May once again caused the smoke to
drift northward across the Gulf of Mexico while it gradually diminished
over the Pacific. A large covering frontal system brought moderate rain
to the region on 12 May, thereby ending the burning season in Central
America. Even at this time in which the source was being cut off, smoke
could still be seen over the Pacific. This is an interesting example
showing the feasibility of satellite tracking of pollution originating from
what is essentially a point source, then moving over thousands of kilometers.
5. Urban Modeling for Air Pollution Detection
In detecting pollution between a detector and a surface by means of
remote sensing, it is often necessary to consider in some detail the physi-
cal structure of the surface. This is especially true if one is concerned
with polarization studies, but it is also true for a remote sensing study
involving any wavelength of light, even into the infrared where the surface
itself is the emitter. In the investigation of the intervening atmosphere
between a surface and observer, we must first be concerned with the light
intensity emerging from a portion of a surface S. The surface may not be
a real surface, such as a case where one has a dense hazy atmosphere with
an extremely large optical depth in which multiple scattering predominates,
or the surface may be the actual radiating source, or an illuminated sur-
face. For our particular case we will consider the reflected light from
an opaque surface.
Let P(x,y,z) be a point on the reflective surface where the coordi-
nates (x,y,z) refer to any point on this surface. Even though we are con-
sidering a truly two-dimensional surface, we will describe it with the
three Cartesian coordinates, most suitable for a city with flat planar
streets and roof tops and with buildings with high vertical sides. The
average amount of energy which is reflected per unit of time from an in-
finitesimal surface element 6S at the point P and within an element 6ft
of solid angle around a direction specified by the polar angles (a,3), can
be expressed in the form
J = B(x,y,z;a,3) cos0 6S 6ft , (6)
where 6 is the angle which the vector (a,3) makes with the normal to
the surface 6S (see Figure 10). The term cos8 is introduced into
Eq. (6) because it is the projection of 6S onto a plane normal to the
direction (a,3) which is the physically significant quantity, rather
than 6S itself. The reflected brightness B is a very complicated func-
tion of the five coordinates (x,y,z; a,3) with the constraint
f(x,y,z) = 0 (7)
which specifies the two-dimensional reflective surface. For a complete
solution to the problem of remote sensing atmospheric pollution by means
of reflective radiation, B must be known as a function of the above five
29
-------
VERTICAL TO 8s
Figure 10. Geometry of model for radiation scattered from urban areas.
Symbols are defined in the text.
30
-------
coordinates as well as a function of wavelength. Polarization parameters
may be included for completeness. However, in many practical cases when
only limited results need be known, a number of approximations can be
employed.
The photometric intensity is defined as
61 = 6f/6ft = B cos6 6S . (8)
Upon integrating, we have
I(a,3) = / B cos6 ds . (9)
The brightness B at the point P depends, of course, on the nature of
the surface, whether it is rough or smooth, whether it is self-radiating
(infrared), or whether it is reflecting other light (multiple scattering
within the atmosphere). In the analysis of remote sensing data over oceans,
forests and flat desert areas, it is often permissible to assume, to a good
approximation, that the radiation is isotropic, that is, that B is indepen-
dent of direction. If we also assume the radiating surface to be a plane,
Eq. (9) becomes
I = I0 cos6, (10)
where
I0 = / B ds . (11)
Equation (11), Lambert's cosine law, states that the photometric intensity
in any direction varies as the cosine of the angle between that direction
and the normal to the surface. The analysis of the Los Angeles ATS-III
data assumes essentially that the reflecting surfaces of the ocean, desert
and city are identical, which of course they are not.
A flight was made over the downtown areas of Madison and Milwaukee,
Wisconsin, and Chicago, Illinois, in order to determine the urban ratio of
total rooftop area to total street area, the urban building height to
street width ratio, as well as the overall surface albedo of building
sides, building tops and street-sidewalk areas. The flight was made
in a 4-seat 172 Cessna Skyhawk. William Kuhlow and David Cadle photographed
the urban regions with a Bronica S 2 1/4 by 2 1/4 still camera and a Bolex
16 mm Rex movie camera with various filters.
Efforts are underway at the present time to model the reflectivity of
a typical city, that is, to determine B for various source angles, observ-
ing angles, and wavelengths. Construction has been completed on a model
city with an appropriate building-height to street-width ratio and with
appropriate street, building-side and rooftop albedos. The cardboard
model city consists of 18 blocks, each of which may be viewed as a city
block or an individual building, located on an area of 55 cm by 60 cm
(3300 cm2). Each block covers an area of about 8 cm by 10 cm (80 cm2).
The height of each block (building) is approximately 8 cm, and the distance
between each block (street width) is 4 cm. All rooftops are flat. The
total rooftop area is 1440 cm2 and the total street area is 1860 cm2. The
31
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ratio of the total rooftop area to total street area is .78, the rooftop
area is about 45% of the total citywide area, the street area is about
55% of the total area, and the ratio of the height of each building to the
width between buildings is 2. The streets were painted a flat beige color,
the sides of the buildings were painted with flat gray enamel, and their
tops were painted semiflat black. The detail specifications were as
follows:
1) Streets: Brand name KRYLON (Borden)
Beige enamel No. 2504
Pigment 4.90%
Telanium Dioxide 83.30%
Iron Oxide Yellow 14.10%
Iron Oxide Red 2.10%
Carbon Black .50%
Nonvolatile 8.9%
Cellulose Nitrate
Oil-Free Alcohol Derived Alkyd
Coconut Oil Alkyd
Dioctyl Phthalate
Volatile 8.9%
Ketones
Esters
Alcohols
Aromatic, Aliphatic and Halogenated Hydrocarbons
2) Sides: Brand name KRYLON (Borden)
Smoke Gray No. 1608
Federal Color Standard 595 No. 16187
Pigment 2.58%
Titanium Dioxide 94.68%
Carbon Black 2.58%
Iron Blue 1.54%
Iron Oxide Yellow 1.20%
Nonvolatile 13.03%
Cellulose Nitrate
Oil-Free Alcohol Derived Alkyd
Coconut Oil Alkyd
Dioctyl Phthalate
Volatile 84.38%
Ketones
Esters
Alcohols
Aromatic, Aliphatic and Halogenated Hydrocarbons
32
-------
3) Tops: Brand Name KRYLON (Borden)
Semiflat Black No. 1613
Pigment 1.55%
Lampblack 71.5%
Silicates 28.5%
Nonvolatile 8.80%
Cellulose Nitrate
Dioctyl Phthalate
Secondary Amines
Volatile 89.65%
Ketones
Esters
Alcohols
Aromatic, Aliphatic and Halogenated Hydrocarbons
The laboratory setup for this urban scattering experiment is shown
in Figure 11. The spectrophotometer is viewing the model city at an
angle 30° south from the nadir. The light source, which was a Sylvania
No. 2 Superflood EBV 110-120 V, was located southwest of the model city.
The angle of the sun above the horizon, ip, as seen from the model city,
was set at three different positions for each experiment. These positions
were 20°, 40° and 60°. Each experiment consisted of varying the filters
in the spectrophotometer. Figure 12 shows the results of the visible
radiation, Figure 13 the infrared, and Figure 14 the polarized visible. The
data were taken in increments of 10° as the model city was rotated in the
horizontal plane. For isotropic scattering off a flat horizontal plane,
the results would graph as a circle, the radius of which would be propor-
tional to the intensity. Deviations from a circle show the effect which
the orientation of the streets (alignment of buildings) has upon the angular
variation of the reflected radiation, with respect to the source-detector
angles. As can be seen in all three graphs, the higher the source is over
the horizon, the more intense the reflected radiation. Also, in the visible
and infrared studies, the lower the source is to the horizon, the larger
the percentage change of angular variation. However, this is not the case
in the polarization studies. In the situation of observing the reflected
visible polarized radiation, there is a strong dependency on street orien-
tation with respect to the source-detector angle with an increase in
source altitude. This point emphasizes the care which must be taken in
interpreting polarization studies of non-Lambertian reflectors.
6. Summary and Recommendations
We have outlined in this paper the early methods and instruments that
man used for remote sensing the state of the atmosphere. The most recent
development, that of satellite observations, differs markedly from all
earlier methods, since the remote sensing observations are made from
above the atmosphere. The scope and range of satellite observations in
detecting atmospheric pollution were shown to vary from urban areas to
the global scale. A modeling experiment was carried out to determine the
33
-------
variation in radiation reflected from a city through an optically thin
atmosphere.
From this work we have seen that there is a profound difference be-
tween viewing atmospheric pollution with older instruments from the
ground and with satellites from above. It was not that the difference
was in the more modern sophisticated instruments. Indeed, some of the
early satellite radiometers, even though they were quite primitive in
operation, yielded startling results. Rather, the satellite observa-
tions required a knowledge of the optical properties of the earth's
surface below the pollution for proper interpretation of their results.
In the study of cloud motions by analysis of satellite data, the earth
surface areas are used only for navigation, and this could be done be-
cause the optical depth through most clouds is small. (However, this
is not true for high cirrus shields.)
Thus for further study of atmospheric pollution from satellite data,
we recommend that an investigation of the optical properties of various
earth surfaces be undertaken. When the properties of reflected light
through a clean atmosphere from a particular surface are known, it will
be possible then to say something about a) observed deviations from this
norm and b) the particulate loading of the intervening atmosphere.
34
-------
g
i-t
cd
3,
•H
fn
35
-------
VISIBLE
\
\
\
\
90°
\
\
\
\
Figure 12. Reflection amplitude of visible radiation for 0-360° street
angles for three source positions.
36
-------
270°
I80<
90°
Figure 13. Reflection amplitude of infrared radiation for 0-360° street
angles for three source positions.
37
-------
100
POLARIZED
VISIBLE \
\
\
270'
20-40
40-60
80 -100
Figure 14. Reflection amplitude of polarized visible radiation for 0-360°
street angles for three (unpolarized) source positions.
38
-------
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41
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IV — RAYLEIGH-GANS-BORN LIGHT SCATTERING BY ENSEMBLES OF RANDOMLY
ORIENTED ANISOTROPIC PARTICLES*
ABSTRACT:
A formula for the electromagnetic scattering amplitude is presented.
Using the Rayleigh-Gans-Born approximation, we apply it to the calculation
of Stokes matrices for ensembles of randomly oriented anisotropic parti-
cles. Two types of anisotropy are treated in detail: particles of any
shape with a variable scalar refractive index n(r)6 . and spherical
particles with a constant tensor refractive index nj.' For a monodis-
perse ensemble of the second type, we describe a method of extracting from
data a shape parameter y One may be able to generalize this method to
yield structural information for a wider class of ensembles.
Paper accepted for publication in Applied Optics.
42
-------
1. Introduction
Ever since the original investigations by Tyndall (1869) [1] and
Rayleigh (1871) [2] of the effects of small particles upon incident light,
scattering of electromagnetic radiation has been a potent tool for scien-
tists. Chemists and biologists have used the scattered radiation to study
the structures of colloidal suspensions, macromolecules and polymers [3]—
astronomers to study interstellar dust and to deduce from polarization
interstellar magnetic fields [4].
The effects of aerosols on man and, conversely, the effects of his ac-
tivities on the aerosol content of the atmosphere are now undergoing
intensive scrutiny by environmental scientists. The influence of aerosols
upon global climate is one of the most debated issues in atmospheric
science [5]. Will the aerosols produced by volcanic activity, and by man's
slash-and-burn agricultural, industrial and automotive activities, tend to
increase or decrease the mean surface temperature of the earth? Which have
the greater influence upon climate— volcanic or man-made aerosols? To
answer these questions, one must understand the light scattering and ab-
sorbing properties of different kinds of aerosols [6].
Knowledge of the optical signature of each aerosol may enhance legis-
lative control of air quality, since one may be able to identify violators
by tracing an aerosol to its point of origin by remote sensing from satel-
lites or from a network of monitoring stations.
Finally, just as studies of the optical properties of interstellar dust
have provided information about interstellar magnetic fields, so also a
deeper understanding of the optical properties of the earth's aerosols
(i.e., atmospheric dust) might lead to further knowledge about the motions
of the earth's atmosphere and about its electromagnetic fields.
In solving radiative transfer problems, most atmospheric physicists
treat an aerosol as if it were an ensemble of homogeneous spheres differ-
ing in size and refractive index [7]. Their motivation in choosing such
a model is that it is the only one for which an exact solution exists for
the scattering amplitude (Mie, 1908)[8], even when the size of the particle
is approximately equal to the wavelength of the radiation.
This model's flaw is that details of the structure of the particles
forming the aerosol are omitted. Thus, one cannot describe correctly the
polarization of the scattered light [9]. Holland and Gagne have shown
that Mie theory is not only unable to fit the polarization data of a known
aerosol, but also is unable to fit the backscattered intensity [10].
Understanding backscattering is crucial for the interpretation of aerosol
data accumulated by satellites [11]. We, therefore,are interested in study-
ing the effects of anisotropic particles upon an incident beam of electro-
magnetic radiation.
43
-------
A suitable method for describing the optically measurable quantities
for a scattering process is to employ the Stokes matrix S [12]. After
briefly reviewing the properties of S for symmetric scattering media, we
shall derive the matrix elements for two classes of monodisperse ensembles.
A monodisperse ensemble is one that has all its particles identical, in
contradistinction to a polydisperse ensemble. We shall first derive S,
using the Rayleigh-Gans-Born approximation, for a monodisperse ensemble of
randomly oriented particles of arbitrary shape with a variable scalar re-
fractive index n(r) 6.. . Using the same approximation, we shall secondly
derive S for a monodisperse ensemble of randomly oriented, spherically
shaped particles with a constant tensor refractive index nij • For an
ensemble of this kind, we shall outline a procedure for obtaining a shape
parameter y f°r tne constituent particles. We shall suggest that one may
be able to modify this procedure to yield information for more general kinds
of ensembles.
2. The Stokes Matrix for Symmetric Media
The experimental procedure for measuring the Stokes matrix elements has
been presented elsewhere [13]. In this section, the well-known properties
of the Stokes matrix are reviewed.
If one knows the scattering amplitude for a particular scattering pro-
cess, one can obtain the Stokes matrix [14].
Consider an incident plane wave traveling in the H-z direction. Let it
be scattered by a particle located at the origin of our coordinate system.
We are interested in determining the amplitude matrix of the outgoing
spherical wave. This matrix is generally a function of the polar angle 0
and the azimuthal angle . Because of the linearity of Maxwell's equations,
one obtains:
BO \
E« \ ikr //A2(e>*) A» , , ,
(1>
A.,(e,c|>) Ai(0,cf>)/ V El/
where E? , and E.I i are the components of the initial and final elec-
11 y J— > ~^~
trie vectors measured parallel and perpendicular to the scattering plane.
(the plane passing through the z-axis at an angle ).
In the next section, we shall present a method for calculating the
amplitude matrix. We shall confine our attention here to determining the
Stokes matrix, given the amplitude matrix.
The elements of the Stokes vector (I,Q,U,V) are defined by
1 =
-------
is a 4 x 4 matrix that transforms the initial Stokes vector (1°,Q°,U°,V°)
into the scattered Stokes vector (isc} QSC, use, vsc); i.e.,
(2)
The structure of S follows from Eqs. (1) and (2) using the definition of
the Stokes vector:
f T ^
QSC
Usc
vsc
S
= II
T7
'i° ^
Q°
u°
v°
S(6,cj>)
Rat + RSI
+ 131
R2it - RSI
R23
R23 -
Rai + Rat
lai +
-I23 -
-123 +
R2i -
(3)
& "h
where M = \k±\2, I.. = Im(A A.), and R E ReCA^A.).
We shall be concerned with media composed, not of one particle in the
scattering region as assumed above, but of many particles incoherently
scattering the radiation. We shall assume media that are optically thin so
that multiple scattering is negligible. For such media the composite Stokes
matrix Sc is given by the sum of all the particles' Stokes matrices.
All the optical information about the medium contained in the scattered
light is specified by the Stokes matrix. As originally demonstrated by
Perrin [15], the imposition of various symmetry conditions simplifies the
structure of S.
If one assumes that the scattering region remains invariant under re-
flection about any plane through its center, then R2if, I2i», RSI, Isi,
R23» 123, RII*> and Im vanish. If, additionally, the scattering region
is isotropic and satisfies the conditions for optical reciprocity, then
SC(6) =
-ai(8)
bi(6)
0
0
bi(6)
a2O)
0
0
0
0
a3(8)
-b2(6)
0 '
0
b2(6)
a., (6)
j
45
-------
If, furthermore, we impose the condition that spherically symmetric, i.e.,
isotropic, particles are incoherently scattering the radiation, we find
that:
3i(0) = a2(8), 33(6) = 3^0) . (5)
If the spheres are identical, then the constraint
33(0)3^(6) + b2(0)
R(6) E
ai(8)az(8) - b?(6)
a§(8) + bice)
a! (8) - b?(6)
1, (6)
is also satisfied. Equation (6) follows immediately from the property
satisfied by the Stokes matrix elements for one particle (see Eq. (3)):
Sn(e,4>)s22(0,<{>) - Si2(
Therefore, if a\ £ a2 or 33 £ ai», as in the study of Holland and
Gagne [16], one knows immediately that the scattering medium is composed
of snisotropic particles. The converse of this statement need not be
valid.
We shall now present the conventional formalism for deriving the
smplitude matrix for the scsttering of electromagnetic radiation by a
single particle. We shall use the notation of Newton [17].
3. The Scsttering Amplitude Matrix
In the absence of any scattering, the electric vector for a plane
electromagnetic wave of unit amplitude traveling in the direction of its
wave vector k (|k| = k) may be represented at point r by:
-»• ->
E(k,v,?) = E°(k",V,?) E xv(k)eik'r; k«xv(k) = 0 , (7)
where x (k) is the unit polarization vector of the incident wave. The
time dependence has been suppressed. If a particle is then placed at the
origin of the coordin3te system, the electric vector is then modified by
the scsttering of the same incident wave:
E(k,v,?) = E°(k,v,r) + ESC(k,v,?), (8)
where, asymptotically
i£-r 1
lim ESC(k,v,r) -•L— I X.(£')A(k',vI;k,v) (9)
r -*• °° v' = ll
46
-------
and, of course, via Eq. (7), k'«x ,(k') « 0. Equation (9) states that
the scattered wave is an outgoing spherical wave. The amplitude for the
->
scattering from the incident direction k and polarization V to a final
direction k' and polarization v' is given by A(kr ,v' ;k,v) . One may
choose v and v' to be polarization states parallel and perpendicular
to the scattering plane, the plane normal to k x k' .
In order to relate Eq. (9) to Eq. (1) , we let the incident wave travel
in the +z direction and look at the radiation scattered in the (6,)
direction. We must then have:
k = kz; k' = k(sin6 costf) x + sin9 sln y + cosSz) , (10)
where x,y,z are unit vectors along the x,y,z axes. Since the scattering
plane is now determined (being the plane containing the z-axis and oriented
at an angle with respect to the -he-axis) , the polarization vectors can
also be determined :
X|,(k) = cost}) x + sincj) y; Xll(k') = cos6 cost}) x 4- cos6 sine)) y - sin6 z;
X|(k) = -sin x + cost}) y; Xl(k') = -sine}) x + cost}) y . (11)
We also can determine the elements of the scattering amplitude matrix of
Eq. (1):
,l;k,i); A2(6,) = A(k', II ;k, I);
= A(k?, || ;k,JJ; A*(0,)
By use of Maxwell's equations, the right-hand sides of (12) may be
expressed as integrals over the volume V of the scattering particle:
o
A(k',v';£,v) =£; I / d3r E° *(k',v',?)p4,(?)E,(k,v,?), (13)
i,j=l V
13
.
3
P1;J(r) = pj±(r) = (n2() -1)^, [*2l±j " e±j () . ,
where ^ is the unit tensor, e (r) and a (r) are, respectively, the
values of the dielectric constant tensor and conductivity tensor at point
r, and to(=ck) is the angular frequency of the incident wave. The tensor
n (r) is the refractive index tensor. We are assuming in Eq. (13) that
the magnetic permeability y of the scatterer is unity. The subscripts
i and j refer to the coordinate axes. E(k,v,r) refers to the exact
solution for the electric field vector as defined in Eq. (8). Equation (7)
47
defines E°(k',v',r).
-------
We shall approximate the exact solution for E(k,v,r) by substituting
in its place the solution with no scattering, E°(k,v,r):
2 3
A(k',v';k,v) =7- I J d3r E°*(k' ,v' ,r")p (?)E° (k,v,?). (14)
* i,j=l V 1J J
Equation (14) is the Rayleigh-Gans-Born (RGB) approximation [18] for the
scattering amplitude and is expected to be valid when |p .| « 1 and
kajp..| « 1, where a is a typical dimension of the scatterer.
4. Applications of the RGB Approximation I; Variable Scalar Refractive
Index
An isotropic or spherically symmetric particle has a scalar refractive
index satisfying n..(r) = n(r)6 . Water- or ice-coated spherical
particles are examples of inhomogeneous isotropic particles found in the
atmosphere. However, most aerosol particles are anisotropic.
We shall use the RGB approximation to study scattering of electromag-
netic radiation by two classes of monodisperse ensembles of randomly oriented
anisotropic particles.
First we shall treat the scattering by a monodisperse ensemble whose
particles are defined by the refractive index n . (r) = n(r)6 . Since
this refractive index is generally not spherically symmetric, this defini-
tion applies to each particle in a certain orientation with respect to the
coordinate axes. Plugging this refractive index into the RGB approxima-
tion, Eq. (14),yields:
A(k',v';k,v) = ,(k').J (k){J ep(?)d3r}, (15)
HM V V ^
where Ak = k - k' and p(r) = n2(r) - 1. Using Eqs. (11) and (12), we
obtain:
-*••*•
2 -•••
A /Q A\ K r iAk»r /^xjs
Ai(6,4>) = T- J e p(r)d3r,
HII ^
2 •*•-*•
A2(e,) = T-{J eiAk'r p(?)d3r}cos6, (16)
V
A3 (6, ())) = A,, (6 ,<(>) = 0 .
The Stokes matrix for the scattering by one particle of the ensemble fol-
lows using Eq. (3):
48
-------
5(0,4,)
iAk-r /»\j3
e p(r)d;>r
l+cos26 cos26-l
cos20-l l+cos2(
0
0
2 cos0
0
0
0
0
2 cos0
(17)
To obtain the composite Stokes matrix for the ensemble, Sc, we sum all of
the particles' Stokes matrices. If there are N identical randomly oriented
particles, this sum is equivalent to averaging over all possible orienta-
tions of Ak. The composite Stokes matrix is therefore given by:
SC(8,4>) = SC(6)
f(0)
where
f(6) = -r-
4lT
l+cos20
cos20-l
0
0
cos20-l
l+cos20
0
0
0
0
2 cos0
0
0
0
0
2 cos0
32TT
lAk'r
P(r)d3r|
(18)
(19)
and Jd^Ai - integration over solid angle of Ak. To simplify the right-
hand side of Eq. (19), we use the well-known plane wave decomposition:
iAk'r
I
I
£=0 m—«,
where the Ym are the spherical harmonics, j_ are the spherical Bessel
functions of the first kind, and * denotes unit vectors. In our case
|Akj = 2k sin -jQ, where 0 is the scattering angle. We then find that:
f(0)
d3rd3r'p(r)p(r')j£(2kr
,(2kr'
-------
The integral over dftf^ reduces to <50(,,<5 , . Therefore, we finally ob-
. /Aic A/A/ nun.
tain:
')| . (20)
u" £,m V * '
Note that the 8 in the argument of the spherical Bessel functions is not
an integration variable; it is the fixed scattering angle determined by
the magnitude of Ak .
Unlike Eq. (19), Eq. (20) lends itself to physical interpretation. For
example, if we are dealing with spherically symmetric inhomogeneous parti-
cles, i.e., p(r') = p(r'), then only the £ «= 0, m = 0 term contributes:
K
f(8) = ^H / r'2p(r')jo(2kr' sin^6)dr'|2 . (21)
V
Furthermore, if p(r') = n2 - 1 for r' < a and p(r') = 0 for r' > a,
then we may perform the integration to obtain the well-known result for in-
coherent scattering by N identical homogeneous spheres:
Nlr2 if ji(2ka slnie)
f(6) = ^~- (n2-!)2 5-=— . (22)
sin2 ^
Expanding Eq. (22) in powers of ka and keeping only the lowest order term
in ka, we obtain the Rayleigh scattering formula for f(8):
f(6) = ^j- (n2-!)2 . (23)
The terms in Eq. (20) with £ / 0 reflect the anisotropy or departure
from spherical symmetry of the individual scattering particles.
If each particle of our monodisperse ensemble has an axis of rotational
symmetry, we then can orient our coordinate system so that p(r') in Eq.
(20) has a rotational symmetry axis pointing along the z-axis; i.e.,
p(r') = p(r',y'). Only the m = 0 terms will then contribute to the right-
hand side of Eq. (20), yielding:
f(6) = ^|-£(2£+l)| //dr'dy'rl2p(rf,y')j (2kr' sin|e)P (y')|2 , (24)
£ V
th
where P.(y ) is the £ Legendre polynomial. If, in Eq. (24),
p(r', y') = p(r',-y'), then only the even £ terms can contribute to f(8).
50
-------
Using Eq. (24), one can reproduce the results summarized by Van de
Hulst for the scattering of electromagnetic radiation by monodisperse
ensembles of randomly oriented, infinitely thin homogeneous rods or disks
[19]. In the case of rods of length L, one sets
p (r' , y') « (n2-!) ' for 0^r'_<-r , -1 _< y' _<_ 1, and
p(r',y') = 0 elsewhere. In the case of disks of radius a, one sets
p(r' ,y') °c (n2-!) ^u|* for 0 <_ r' <_ a, -1 <_ y' <_ 1, and p(r',y') = 0
elsewhere. The functions <5(y'2-l) and 6(yf) are Dirac delta functions.
To summarize: in this section we have derived the Stokes matrix,
Eq. (18), for a monodisperse ensemble of randomly oriented anisotropic
particles defined by the refractive index n (r) = n(r")6 . We have
derived Eq. (20) without any additional assumptions. We have noted that
if the particles have an axis of rotational symmetry, then Eq. (20) re-
duces to Eq. (24).
We conclude this section by generalizing Eq. (20) to the case of a
polydisperse ensemble of randomly oriented particles. If we have m kinds
of particles defined by p. (r), i = l,2,...,m, and if there are N.
th
particles of the i kind, then for the polydisperse ensemble, f(6) is
given by:
, it m
f(0) = f I Ni |d3r'Pl(r')j^(2kr' sine)Y(f')|2 . (25)
5. Applications of the RGB Approximation II; Constant Tensor Refractive
Index
In this section, we shall use the RGB approximation, Eq. (14), to
compute the composite Stokes matrix for a monodisperse ensemble of randomly
oriented anisotropic particles defined by the refractive index
n (r) = n (r) = n for r < a and n (r) = 6 . for r > a. This
•i-J J ~^J -^. J ^. -^J
refractive index implies that p .(r) = p..(r) = p for r < a and
p = 0 for r > a. This definition applies to each particle for a cer-
tain orientation with respect to the coordinate axes. As the orientation
of the particle changes, the values of the tensor components p . vary in
a well-known fashion [20]. ^
The amplitude for the scattering by a single particle in a fixed
orientation is then
51
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A(k',v';k,v) = j-
ji(2ka
sinj
Using Eqs. (11) and (12), we find that;
Ai(6, <(>-0) = g(6)p22,
A2(6, <}>=0) = g(6)(cos6 pu - sinB p3i),
A3(6, (j)=0) = g(6)(cose P12 - sin6 p32),
A^(e, <})=0) = g(6)P2i .
One may compute the composite Stokes matrix for an ensemble of N of these
particles randomly oriented [21]:
(26)
f g2(6)3
2(l+Y)+(Y-l)sin20 (y-l)sin26 0
(Y-l)sin26 (1-Y)(l+cos26) 0
0 0 2(l-Y)cos6
000
0
0
0
2(l-3y)cos6
(27)
Note that, as in Eq. (18), S (6,cj>) is (^-independent.
The parameters 3 and y define those measurable quantities of this
monodisperse ensemble that reflect the optical anisotropy of the particles.
If pi, p2, and ps are the eigenvalues of the matrix {p..}, then
3 = ^(pi+pi + pi) + B
_3_
15
+ P2P3 + P3Pl),
(28)
Y
153
2 , _2
[(Pi + P2 + P3) - (P1P2 + P1P3 + P3P1)]
The physical meaning of these eigenvalues is made transparent by noting
52
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1/2
that (p.+l) for i = 1,2,3 are the lengths of the principal axes of
the particle's Fresnel ellipsoid [22]. If the ellipsoid reduces to a
sphere, as in the case of isotropic particles, then Pi = Pa = P3 so that
y = 0. In this case, Eq. (27) must and does yield the matrix given by
Eqs. (18) and (22).
When Y = 0 in Eq. (27), we obtain Su = 822 and 833 = S^ , which,
as we have noted earlier, is a necessary condition for Stokes matrices of
isotropic particles.
One should note that if g(6) is expanded in powers of ka and only
the lowest order term is kept, then Eq. (27) gives the usual result for
scattering by randomly oriented anisotropic dipolar particles [23].
One may obtain the Rayleigh result, Eqs. (18) and (23), from Eq. (27),
by keeping only the lowest order term in ka in the power series expan-
sion of g(6) and by choosing j = 0 .
The results of this section may be generalized.
If one extends the derivation of the composite Stokes matrix presented
in this section to the case of particles having the more general tensor
refractive index n..(r) = n (r) = F(r)n , r < a (where F(r) is an
arbitrary function depending solely on jr|), one finds that g(8) is no
ji(2ka sin|e)
longer proportional to - : - ; otherwise the composite Stokes
matrix is unaltered. Even though the size of the Fresnel ellipsoid for this
refractive index F(r)n . is a function of r, the shape of the ellipsoid
is constant throughout the particle. The parameter Y remains the shape
parameter of the Fresnel ellipsoid; e.g., y = Q implies isotropy.
For the even more general tensor refractive index n (r) = n (r)
= n (r) , r < a, the parameters 3 and Y become 6-dependent. In this
latter (rather artificial) case, the size and shape of the Fresnel ellip-
soid of n . (r) are functions of r .
Finally, as does Eq. (20), Eq. (27) admits generalization to the case
of polydisperse ensembles of randomly oriented particles with tensor re-
fractive indices.
6. Summary: Implications for Future Research
Crucial for the understanding of the effects of real aerosols in any
radiative transfer problem is a knowledge of features that distinguish
electromagnetic radiation scattered by real aerosols from radiation scat-
tered by an ensemble of homogeneous spheres. Such knowledge is essential
for understanding the polarization of radiation scattered by real aerosols,
since polarization is very sensitive to the shape and structure of the
scatterer.
53
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Using the RGB approximation, we have studied above the departures from
Mie scattering, i.e., the scattering by homogeneous spheres. We have de-
rived the Stokes matrices for two classes of ensembles of randomly oriented
anisotropic particles and have discussed possible generalizations.
If the elements of a Stokes matrix do not satisfy Eq. (5), we know that
anisotropic particles are scattering the radiation. If the elements do not
satisfy Eq. (6), we know that the scattering particles do not constitute a
monodisperse ensemble of spheres. These conditions are well known. The
violation of Eqs. (5) and (6) should be studied as a function of particle
and ensemble structure. For example, computer experiments could be per-
formed on polydisperse ensembles of homogeneous spheres, using Mie theory,
to isolate the effects of polydispersity upon R(6) as defined by Eq. (6):
833(6)8^(6) = 53^(8)8^3(6)
R(6) = . Perhaps polydispersity effects are
Sii(6)si2(6) - 812(6)821(6)
significant at some scattering angles and are always negligible at others.
One then may be able to disentangle the effects of particle shape and
structure from the effects of polydispersity when analyzing R(6) .
For a monodisperse ensemble of anisotropic particles with constant ten-
sor refractive index, we note from Eq. (27) that in the RGB approximation,
R(Q) m (l-3Y)cos2e ^ (29)
Y + cos26
For this kind of ensemble, R(6) is a function of only 6 and y Using
Eq. (28), we see that 0 _< y ^ T • For Fresnel spheres (pi = p2 = ps),
J i
Y = 0 and R(8) = 1. For Fresnel "needles" (pi £ 0, p2 = ps = 0), Y = T
and R(9) =0. At 6 = 0° or 0 = 180°, R = ~* . Finally, R = 0 at
6 = 90° for Y t 0 •
For a monodisperse ensemble of anisotropic particles with scalar refrac-
tive index n(r)6.., we note from Eq. (18) that R(6) = 1 in the RGB ap-
proximation. (Remember that R(6) =1 is a necessary but insufficient
condition that the scatterers be isotropic.) If we relax the RGB constraint
that nz(r) - 1 be small, we must also consider terms of second order
in n2(r) - 1 in the scattering amplitude. Then the ratio R(6) is no
longer unity. For example, consider the case of scattering by a mono-
disperse ensemble of randomly oriented homogeneous ellipsoids. In the
quasi-static limit ka « 1, we may expand the components pi, p2, and ps,
of the diagonalized polarization tensor in powers of n2 - 1. We then find
that these components differ from each other in the second order term in
n2 - 1 [24]. Equation (30) then might be qualitatively a parameterization
for R(6), with Y now related to the particles' geometric shape.
54
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(Remember that Eq. (30) was originally derived not for particles with
ellipsoidal boundaries, but for particles with n (r) = n for r < a
_^, •'•J "^-J
and n . . (r) = 1 otherwise.)
This admittedly heuristic and qualitative discussion shows the need
for further analysis. Limitations upon the validity of various theoreti-
cal models should be examined by laboratory studies of simple mono-
disperse and polydisperse ensembles of particles whose size, shape and
structure are carefully controlled. Further theoretical work must be done
in order to understand and construct scattering models where the RGB ap-
proximation is inappropriate and Mie theory fails. First, one must im-
prove upon the plane wave assumption E = EO utilized in obtaining the
RGB approximation, Eq. (14) . Finding the solution for scattering by
realistic aerosols represents a formidable challenge.
55
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56
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17. Roger G. Newton, Scattering Theory of Waves and Particlest (McGraw-
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