Scattered sunlight in the atmosphere, from the middle ultraviolet through the near infrared


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Scattered sunlight in the atmosphere, from the middle ultraviolet through the near infrared
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vii, 164 leaves : ill. ; 28 cm.
McPeters, Richard Douglas, 1947-
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Light -- Scattering   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis--University of Florida.
Includes bibliographical references (leaves 160-163).
Statement of Responsibility:
by Richard Douglas McPeters.
General Note:
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University of Florida
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All applicable rights reserved by the source institution and holding location.
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aleph - 000168740
notis - AAT5140
oclc - 02884814
lcc - QC911 .M17 1977
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The author wishes to thank the members of his super-

visory committee for their guidance in his graduate program.

In particular, he wishes to thank the chairman of his

committee, Dr. A.E.S. Green, for his assistance and advice

in all phases of his research. Dr. G. Ward provided much

helpful advice on aspects of the atmospheric light scat-

tering problem. Dr. A. Smith and Hans Schrader were help-

ful in our development of good photographic techniques.

He is also indebted to many other members of the faculty

and staff of the Department of Physics and Astronomy.

The author's parents provided much needed support at

several critical points in his graduate studies.

The author also acknowledges helpful discussions with

and suggestions of George Findley, Tsan Mo, and Bob


Finally, it should be noted that the National Science

Foundation provided financial support for much of this

work through grant number NSF-GA4479, A.E.S. Green, princi-

pal investigator.






2.1 Aerosol Size Distribution
2.2 Physical Properties of Aerosols
2.3 Aerosol Altitude Distribution

3.1 Units
3.2 The Extraterrestrial Solar Flux
3.3 Geometry of the Atmosphere

4.1 The Attenuating Components
4.1.1 Rayleigh Scattering
4.1.2 Mie Scattering
4.1.3 Ozone Absorption
4.2 Standard Bouguer-Langley Analysis
4.3 Ultraviolet Solar Radiometry
4.4 Instrumentation
4.5 Results

5.1 Aerosol Models
5.2 Mie Phase Functions
5.3 The Single Scattering Calculation
5.3.1 Geometry
5.3.2 Wavelength Integration
5.3.3 Ground Albedo Integration
5.3.4 Rayleigh Multiple Scattering
5.4 Comparison of Measured and Calculated
Solar Aureoles
5.5 Chromaticity Diagrams





6.1 Equipment 124
6.2 Densitometric Analysis 128
6.3 Sensitometry 131
6.4 Analytic H and D Curves 136
6.5 Conclusions 145


List of References 160

Biographical Sketch 164

Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy





Richard Douglas McPeters

August, 1975

Chairman: A. E. S. Green

Major Department: Physics and Astronomy

Scattered sunlight in the atmosphere was studied

through computer modeling and comparison with aureole

measurements. An analytical technique was developed to

determine the aerosol optical depth from solar radiometer

measurements in the ultraviolet. A photographic technique

was used to measure the scattered light of the solar aureole

for comparison with a curved Earth modified single scat-

tering calculation.

Because of the rapid spectral variation of the ozone

absorption coefficients below 350nm, the bandwidth of a

filter radiometer (sunphotometer) is sufficient to inval-

idate the standard Bouguer-Langley analysis used to deter-

mine aerosol optical depth. By carefully modeling our

instrument with 0.lnm wavelength intervals, we were able to

reproduce the observed deviation from straight line Langley

plots. Using this model we found an analytic function to

characterize the deviation from linearity, thus allowing

direct computation of the aerosol optical depth even in

the ultraviolet.

A modified single scattering calculation was used to

predict sky brightness near the sun as accurately as pos-

sible without going to the formalism of a full multiple

scattering calculation. The calculation uses spherical

Earth geometry but does not include refraction. It includes

the contribution to sky brightness from sunlight reflected

from the ground and contains an empirical correction term

for Rayleigh multiple scattering. Using the tables of

Coulson, Dave, and Sekera for total scattering in a pure

Rayleigh atmosphere, correction terms for multiple scat-

tering were derived for the Rayleigh component as a function

of wavelength and solar zenith angle. Analytic functions.

were used to represent the altitude distribution of atmos-

pheric constituents based on the measurements of Elterman.

The photographic technique for aureole study is de-

scribed. The sun is blocked by a neutral density 4.0

occulting disc, allowing the sun to be photographed simul-

taneously with the surrounding sky for accurate determina-

tion of the sky to sun intensity ratio. Film sensitometry

is discussed, including the use of a new analytic function

to describe the characteristic curves of film.

A series of graphs of calculated and measured aureoles

show the comparative accuracy and range of validity of the

modified single scattering calculation. The calculation

appears to accurately describe the sky brightness except

when aerosol multiple scattering becomes dominant, such as

at middle wavelengths near sunset. At short wavelengths

Rayleigh scattering is dominant and the Rayleigh correction

factor aids accuracy. The ultraviolet (309nm) aureole is

compared with a Monte Carlo multiple scattering calculation,

and with our modified single scattering calculation, to

show that the latter is fairly accurate even in the ultra-





The main features of the daytime sky brightness were

explained long ago by Rayleigh as scattering of sunlight

by molecules (actually by molecular density fluctuations)

in the atmosphere, and an idealized atmosphere consisting

only of air molecules is known as a Rayleigh atmosphere.

But the complexities of observed sky radiation depend

to a large extent on the atmospheric aerosols, the partic-

ulate matter suspended in the atmosphere including water

droplets, dust, smoke, and other air pollutants solid or

liquid. In addition absorption is an important factor

in the middle ultraviolet and in the infrared, but in the

range 400nm to 800nm radiation reaching the ground under-

goes extinction primarily by scattering. Thus the study

of light scattering by atmospheric aerosols is of great

interest in this wavelength range.

Typical aerosols include particles with radii ranging

from 10nm to 10,000nm, and although the range of sizes

may extend well beyond these limits, the fact that the

optical wavelengths occupy the range 400nm to 700nm makes

these particles of similar size important in light scat-

tering. An important parameter used widely in Mie scat-

tering calculations is the size parameter x = 2wr/A

where r is the particle radius and A is the wavelength of

the scattering radiation. For x > 1.0 the departure from

Rayleigh scattering is generally considerable.

The theory of scattering of electromagnetic radiation

from spheres of arbitrary size and index of refraction

is generally credited to Mie (1), although N.A. Logan (2)

observes that Ludwig Lorenz published a similar solution

to Mie's eighteen years earlier in 1890. The best known

book on Mie scattering is one by Van de Hulst (3). The

Mie solution consists of a series whose terms contain

Bessel functions of half-integral order (spherical Bessel

functions) with complex argument, and first and second

derivatives of the Legendre polynomials. The number of

terms required for reasonably accurate convergence of the

series is on the order of the size parameter x. Only since

the advent of the high speed computer has it been possible

to perform the extensive calculations needed to describe

the wide range of sizes present in the atmospheric aerosol.

The Mie scattering calculations for this dissertation were

done using a subroutine developed by Dave (4) which uses

a downward recurrence procedure that accurately gives

Mie scattering intensities even for very large particles

of x=200.

An additional complexity of observed sky brightness

is multiple scattering, which will be important whenever

the intensity of scattered radiation is a non-negligible

fraction of the intensity of direct sunlight entering a

unit volume. Such a condition will occur when the aerosol

optical depth is large, near sunset when light travels

over a very long path through the atmosphere, and for the

middle ultraviolet wavelengths where the Rayleigh optical

depth is large. Chandrasekhar (5) developed the theory

of multiple scattering for a Rayleigh atmosphere and in

1957 Sekera (6) used this theory to calculate the distri-

bution of Rayleigh sky radiation. Application of multiple

scattering calculations to aerosols is more difficult, but

Plass and Kattawar (7) have used a Monte Carlo technique

in which the computer does a light scattering"experiment."

The computer injects photons into the atmosphere and fol-

lows them down, deciding when scattering will occur and

what kind by generating a random number and applying it to

a weighted probability for scattering, non-scattering, or

absorption. By following a large enough number of photons

for good statistics, a sky brightness distribution results

which is quite accurate. A Monte Carlo program based on

that of Plass and Kattawar was written by Furman (8) of

our group, and this program was used for the multiple scat-

tering results in this work.

The composition and size and altitude distribution

of the atmospheric particulate matter is studied by a

number of techniques. A good survey of the different

methods is presented in a book by Allen (9) with most

experimental methods falling into two major categories,

direct sampling and remote sensing.

Direct sampling involves moving a collector through

the volume of air of interest and analyzing the particles

collected. Ground based sampling systems can study only the

lowest layer of the atmosphere, while airborne systems

generally cannot sample continuously and have problems

with contamination and calibration. It is interesting to

note that during coordinated studies performed at the

University of Florida in 1971 a serious discrepancy

between the size distribution obtained by our ground based

aureole measurements and that obtained by an airborne

particle counter was resolved with recalibration of the

particle counter.

Remote sensing determines parameters of the atmos-

pheric aerosols by observing the effects of aerosols on

natural or manmade radiation fields. This dissertation

concerns measurements of natural sunlight, direct and

scattered at several different wavelengths, from which we

determine the aerosol optical depth, number density, size

distribution, and to some extent composition and altitude

distribution. Experimental work concentrated on measure-

ment of the attenuated direct solar radiation as a function

of wavelength and solar zenith angle to determine aerosol

optical depths, and on measurement of the solar aureole,

the region of intense forward scattered light in the

vicinity of the sun, to determine the aerosol size distri-




The characteristics of atmospheric aerosols are

important because they affect light scattering in the

atmosphere, especially the scattered light of the solar

aureole. Unfortunately aerosols may vary in a number of

ways: in index of refraction, chemical composition, size

distribution, and shape and structure. And because aerosol

particles are constantly being produced and removed, all

these parameters may vary with time. In order to draw

conclusions about aerosol particles from the limited

information available from optical study of the solar

aureole, it is necessary to identify the important

parameters and realistically model them using the best

information available.

2.1 Aerosol Size Distribution

The natural aerosol consists of all solid or liquid

colloidal particles present in the atmosphere, ranging in

size from the molecular complexes of the small and inter-

mediate ions to the giant nucleii 20 microns or more in

radius. (Here I shall use the micron as the unit of aerosol

radius, following the convention of aerosol study.) The

radius interval of greatest interest is that of the large

nucleii, from 0.1 micron to 1.0 micron. Particles in this

size range are most effective in scattering light.

The very small particles, the Aitken nucleii, are not

very important because of their relatively low concen-

tration and because they scatter light ineffectively.

The theory of Smoluchowsky describes the process by which

small particles coagulate to form large particles. Small

particles coagulate more quickly than large because of

their greater numbers, so that, for instance, 0.01 micron

particles will be reduced to a negligible number in only

half a day.

For particles larger than the Aitken nucleii, the num-

ber of particles per unit volume in the size range r to

r+dr generally decreases with radius as r where the

exponent of the decrease v is found to be 1.3 as a world-

wide average. Such a distribution ia known as a Junge (10)


Sedimentation, direct fallout, and perhaps washout

by rain decrease the number of giant nucleii to the point

that they are not important in atmospheric light scattering.

Thus, to a good approximation, scattering calculations may

be performed using a simple Junge distribution with an

imposed lower limit of 0.04 micron or so and an imposed

upper limit of 10.0 microns.

Many aerosol particles are produced through conden-

sation. According to Bullrich (11), combustion processes

are responsible for the formation of the very small

particles, with the gaseous combustion products condensing

upon cooling to form smoke. Subsequent coagulation then

produces the larger atmospheric particles, which are mainly

man made. Particles of condensation origin may be either

solid or liquid. Hygroscopic materials such as ammonium

chloride may undergo a phase transition from solid to

liquid, depending on the humidity of the medium.

Other aerosol particles are produced by dispersion,

the process of atomization of solids and liquids and their

injection into the atmosphere by the action of winds or

vibration. Dispersion aerosols are generally solid and of

varied shape.

Specific mathematical models for the aerosol size

distribution will be discussed in Chapter 5.

2.2 Physical Properties of Aerosols

Inasmuch as Mie scattering calculations assume that

all particles are spherical, the effect of irregular

aerosol particles must be considered. As mentioned above,

the larger aerosol particles produced by dispersion are

more likely than the small condensation particles to be of

irregular shape. In addition, ice crystals are known to

form as hexagonal rods and platelets, and salt crystals

formed by evaporation of sea spray are cubic. The conclu-

sions drawn by Holland and Gagne (12), that scattering

from irregular particles deviates from that of spherical

particles only for large particles and only for large

scattering angles, indicate.that particle irregularity

will not be an important factor for this dissertation,

which concentrates on light scattered at small angles.

To quote Holland and Gagne, "...the unpolarized mass scat-

tering coefficient for a polydisperse system of irregular

but randomly oriented particles shows a remarkable similar-

ity to the corresponding coefficient for spherical parti-

cles, but ... predicting backscattering for such particles

on the basis of the spherical model can lead to serious

errors" (p. 1120).

The index of refraction is a further parameter that

affects light scattering. For years an index of refraction

of 1.33, that of water, was assumed for lack of any better

estimate; but more recent data indicate that an index of

refraction of 1.50 to 1.55 is more accurate. Volz (13)

has estimated the index of refraction from distillation

residue of rainwater and obtained a value of 1.53, but

this method is applicable only to the water soluble

component, which typically constitutes only about 15%

of the total aerosol. Estimates have also been made based

on analysis of chemical composition of collected aerosol.

Recently Bhardwaja (14) has used integrating nephalometer

measurements of total scatter and hemispheric backscatter

for an in situ measurement of the refractive index, and

obtained an index of refraction of 1.55 0.03 with an

imaginary component of 0.03.

Another method for determining the index of refraction

is use of bistatic laser scattering to measure the polar-

ization of light over a large range of scattering angles.

This technique was used by G. Ward (15) at the University

of Florida in 1972, who obtained an index of refraction

of 1.50 + 0.005i.

Because of the insensitivity of forward scattering to

index of refraction, no attempt should be made to infer

it from study of the solar aureole. On the other hand,

this insensitivity means that assumption of a reasonable

value for the index of refraction, 1.50+0.01i, will not

lead to serious error in aureole calculations.

Parameters that affect the index of refraction of

aerosols are the chemical composition and the humidity. As

noted earlier, condensation aerosols are frequently hygro-

scopic and undergo a phase transition from solid to liquid

with increasing humidity. According to data presented by

Bullrich (11) such a particle with an index of refraction

of 1.5 will show little change until a humidity of 80%

is reached, but between 80% and 100% humidity the index

of refraction rapidly decreases to that of water, 1.33.

Simultaneously the particle grows in size, but this is a

smaller effect. At 100% humidity a typical hygroscopic

particle is only 1.3 times larger than in its solid phase.

2.3 Aerosol Altitude Distribution

The aerosol number density is found to decrease very

rapidly with altitude, an order of magnitude in the first

three kilometers, but is found also to have a high altitude

layer at approximately 20 kilometers. The bulk of all aero-

sols are found in the first few kilometers and they are

generally as described in the preceding sections. The

microstructure of the tropospheric aerosol distribution

has been studied by Rosen (16) who found that the altitude

distribution can be far more complicated than a simple

exponential falloff, especially when temperature inversions

are present. Layering and variation in size distribution

as well as number density were found. In some cases the

mixing layer model, in which particles are assumed uniform-

ly distributed up to some mixing height, may be an accurate

model. To a large extent, however, the fadt :that light

from the sun must traverse the entire atmosphere will

average out the effects of these small scale variations.

The stratospheric dust layer has been studied by

Junge (17) using balloon and aircraft borne detectors.

He found that particles less than 0.1 micron decreased

rapidly in number with height and were of tropospheric

origin. The peak of the particle size distribution was

between 0.1 and 1.0 microns with a number density of about

1.0 per cubic centimeter. Chemical analysis showed that

they were mostly sulphates of ammonia and were probably

formed in the stratosphere by oxidation of gaseous sulphur

compounds. Particles larger than 1.0 microns were very

rare and were distributed according to a different power

law, suggesting that they were of extraterrestrial origin.

Volz (18) has used twilight measurements to make long

term studies of the stratospheric aerosol layer. As the

sun sets the shadow of the edge of the Earth sweeps

upward through the atmosphere,,and the presence of a dust

layer will be revealed as a sudden change in sky brightness.

His studies show the importance of volcanic eruptions in

injecting large quantities of dust into the stratosphere,

dust that may persist for more than a year.

Several workers have used active source remote sensing

systems, high intensity laser beams or searchlight beams,

to study the aerosol altitude distribution. Elterman (19)

directs a searchlight beam vertically into the atmosphere

and observes the scattered light emanating from the common

volume formed by the intersection of the searchlight beam

and the observation cone. By scanning along the beam he

obtains information on aerosol density at progressively

greater altitudes, as high as 50 kilometers. The altitude

distributions of air, ozone, and aerosol used for compu-

tations in this dissertation are based on the standard

atmosphere tabulated from the measurements of Elterman.



3.1 Units

In view of the confusion that has traditionally

existed in the fields of photometry and radiometry, a

review of radiometric units currently being used might be

useful at this point.

The confusion, inconsistency, and duplication in the

definitions of fundamental quantities arise in part from

the fact that the fields of radiometry and photometry

were for years independent and not clearly connected.

Radiometry deals with measurement of electromagnetic

energy regardless of wavelength, while photometry is

restricted to visible light wavelengths. Conversion of

radiometric values to photometric must take into account

the response of the human eye. This response, called the

luminous efficiency curve, is set to 1.0 at 555 nanometers

and decreases to 0.001 at 410nm and 720nm. The conversion

factor between photometric and radiometric units is that

one watt of monochromatic 555nm light is equivalent to

685 lumens. Measurements at other wavelengths must be

weighted by the luminous efficiency curve.

The radiant quantities which are employed in this

dissertation follow a summary of basic radiometric quanti-

ties defined by McCluney (20) in 1968. These quantities

are summarized in Table 3.1.

The radiant flux may be defined as the radiant energy

per unit time and is measured in watts. The radiant energy

per unit time per unit wavelength interval is called the

spectral radiant flux and is measured in watts per nano-

meter. The term "spectral" will always denote per-unit-

wavelength measurements. The photometric quantity corres-

ponding to the radiant flux is the luminous flux, which

is measured in lumens. If P is the spectral radiant flux

and K is the luminous efficiency curve, the luminous flux

F in lumens is given by

F = 685 fm K(X) P(A) dA (3.1)

Intensity is a term that is often misused and over-

worked. The radiant intensity is defined as the radiant


Radiometric Units

Name (spectral-)

Radiant flux

Radiant intensity





watts/(nm sr)
watts/(nm sr cm2)
watts/(nm sr cm2
watts/(nm sr cm

Photometric Units

Luminous flux

Luminous intensity







flux per unit solid angle from a point source. A similar

term applicable to extended sources is the radiance, which

is radiant flux per unit solid angle per unit area of

emitting surface. Corresponding photometric terms are

luminous intensity and luminance, measured in terms of the

candle, which is one lumen per steradian, and candles per

unit area respectively. These are calculated in a similar

fashion to equation 3.1.

Finally, the irradiance is defined as the radiant flux

incident upon a surface per unit area. The magnitude of

irradiance from a point source follows an inverse square

decrease with distance; the irradiance from an extended

source may in some cases decrease more nearly linearly.

The cosine law states that if either an emitting or an

irradiated surface is other than perpendicular to the

direction of the light, the irradiance will vary as the

cosine of the angle. Irradiance is measured in watts per

square centimeter and the corresponding photometric unit,

the illuminance, in lumens per steradian per unit area.

By these definitions a sunphotometer, which measures

the intensity of sunlight over a narrow wavelength band

after transmission through the atmosphere, should more

properly be called a solar radiometer, especially when

the instrument operates at non-visible wavelengths.

3.2 The Extraterrestrial Solar Flux

Since this work concerns remote sensing of the atmos-

phere using the sun as the light source, it is appropriate

at this point to discuss some of the features of this light


Viewed at a distance of one astronomical unit, the

average distance from the Earth to the sun, the sun sub-

tends thirty-two minutes of arc, little enough for the

finite extent of the source to be neglected for many radio-

metric calculations. Light intensity is fairly constant

across the face of the sun in this wavelength region,

varying about 30% due to limb darkening and with some

variation due to sunspots, small scale granularity, and

possible solar flares.

Because the Earth's orbit is elliptical, there will

be a variation in intensity amounting to about 3.4% at

most because of the varying sun-Earth distance, with the

Earth being closest to the sun in December.

With the exception of the very short wavelengths, the

energy output of the sun is quite stable. The total

irradiance of the sun, the solar constant, was measured

by Moon (21) in 1940 to be 1323 watts per square meter and

by others since to be 1374 in 1950, 1395 in 1954, and 1361

in 1968. These uncertainties are due mostly to difficulties

in making measurements through the highly variable dust,

haze, and cloud cover of the Earth's atmosphere. In 1969

Thekaekara et al. (22) reported the results of a series

of solar constant measurements made from an aircraft at an

altitude of 11.6 kilometers, above 80% of the permanent

gases of the atmosphere and above 99.9% of the highly

variable water vapor, dust, and smoke. They report an

average solar constant of 1351 watts per square meter.

The spectral distribution of sunlight may be approx-

imated in the visible by a smooth 7200K continuum with

superimposed Fraunhofer absorption lines, and as a 6000K

greybody for wavelengths beyond 1250nm. On a microscopic

scale sunlight below 600nm is extremely complicated, so

that determining the continuum is difficult. This is

especially a problem between 300nm and 450nm and is one

source of the greater uncertainty in the solar flux in

this region.

Probably the most accurate determinations of the

extraterrestrial spectral solar irradiance were made on

the same 1968 aircraft flights that the solar constant

measurements mentioned above were made. Although Thekaekara

(22) and Arvesen et al. (23) made irradiance measurements

from the same airplane at the same time, Thekaekara using


Solar spectral irradiance at selected wavelengths.

F9() w/m2 nm






a Perkin-Elmer monochromator and Arvesen et al. using a

Cary 14 monochromator, there are discrepancies between the

two solar spectra. Integrating over his spectral measure-

ments Thekeakara finds a solar constant of 1351 watts/m2

while Arvesen et al. find 1390 watts/m2. We have chosen to

use the spectrum of Arvesen et al. because that of Thek-

aekara fails to give the correct wavelengths for several

well known Fraunhofer dips, while that of Arvesen et al.

correctly gives the Fraunhofer structure. No explanation

for the discrepancy is given, but it might be caused by

scanning with a large slit width.

We calculate the apparent position of the sun in the

sky using ephemeris data, the exact latitude and longitude

of the observation site, and the local time. In Figure 3.1

we show the time at which the sun rises, crosses the

meridian, and sets throughout the year in Gainesville,

neglecting refraction. Also shown are the azimuth angle

of the sun at sunrise and the noon solar zenith angle.

Q 0 0C
(6ap) YZ E-
0 0 0 M N N
N wD Hr

o 4


-0 H




1 S En $1

S .4

*r-I H 0U

l l I,- %DOa,

,-4~rc 1-4 ** **** rr

(59p) qqnumTzv

qWTI plepueS uaaqsva

3.3 Geometry of the Atmosphere

An assumption common to atmospheric optics calcula-

tions is that of horizontal homogeneity, so that optical

properties of the atmosphere depend on only one coordinate,

the height above ground level. This assumption is justified

for the molecular component of the atmosphere, but less so

for the aerosol component. As point sampling by particle

counters shows, there can be large variations in aerosol

content from place to place depending on local sources.

The fact that optical measurements are made over long path-

lengths somewhat alleviates problems of horizontal homogen-

iety through averaging, but the experimentalist must be

aware of the problem.

The atmospheric air mass is a measure of the amount

of air traversed by sunlight before reaching the ground.

If light propogating vertically through the-atmosphere to

sea level is defined to have penetrated one air mass, then

to a first approximation light propogating at zenith angle

0 traverses an air mass of m(O) = sec(e). This is a result

of the flat Earth approximation, which, because the atmos-

phere is a thin layer on a very large sphere, gives very

accurate air mass values for zenith angles from zero to

about seventy-five degrees. As the zenith angle approach-

es ninty degrees the flat Earth sec(O) approximation goes

sun ^-

h s

(a) Flat Earth approximation.


(b) Spherical geometry.

Figure 3.2

to infinity, while actually due to curvature the air mass

only increases to thirty or so. Spherical Earth corrections

for the air mass for an exponential atmosphere were calcu-

lated by two men, Bemporad, whose work was tabulated by

Schoenberg (24), and Chapman (25). Rozenberg (26) has

proposed an empirical expression for the air mass.

m(e) = l/(cos(8) + 0.025 e-1cll s(0)) (3.2)

Bemporad's results along with Rosenberg's empirical formula

are compared with sec(8) in Table 3.3.

The air mass is defined as the ratio

m(e) = N(s) ds (3.3)
J(oN(h) dh

where N(h) represents the height distribution of absorbing

or scattering material, dh is a differential element of

vertical path length, and ds is a differential element of

actual path length.

Figure 3.2b shows the geometry of an incoming ray

traversing a spherical atmosphere for which


Air mass as a function of solar zenith angle.

8 secO Bemporad Rozenberg

























seqR8 seqM6

1.00 1.00

1.15 1.15

2.00 2.00

2.91 2.91

5.61 5.66

10.4 10.7

18.7 20.8

24.3 30.2










seq T










ds = dh sec(6) (3.4)

Snell's law written in spherical coordinates appears as

r n(O) sin(6)
sin(O) = (3.5)
(r+h) n(h)

where n(h) is the index of refraction as a function of

height, r is the radius of the Earth, h is the altitude,

0 is the zenith angle of the ray at height h, and 6 is the

zenith angle of the ray at the ground. According to

Kondratyev (27) a first order expression for the index of

refraction is

n(O) 1 + 2a
nl 1 + 2a{p(0)/p(h)}

where a=2.9x104. We may now write

ds = 1 (3.7)

1 f r+h )x(n- 2 xsin() n2

We may now use this expression in eqn. 3.3 to give a

general formula for the air mass.

S(h) dh
m(9) = / (3.8)

{ 1 (- -) n(O)2 xsin2) 1/2

By the usual definition

T = f/ 0(h) dh (3.9)

where the altitude distribution, the volume scattering

coefficients 8(h) give the attenuation per kilometer for

a given altitude. Analytic expressions for the 8(h) will

be given in the next chapter. Using those distributions we

calculated the total air mass and also the individual air

mass contributions for the Rayleigh, Mie, and ozone compo-

nents. Where Bemporad and Chapman assumed an exponential

distribution, our analytic distributions are based on the

measurements of Elterman and accurately model the atmos-


Using our calculated air masses, we represented each

of the constituent air mass functions by an analytic

modification of the secant law introduced by Green (28)

and called a seq function, defined as

seqi(8) =

q1 = 1.0018

q2 = 1.0074

q3 = 1.0003


( 1 sina(e)/qi)1/2





Attenuation through the atmosphere is thus accurately

given for the three species by

I = 10 exp(-I Ti seqi(O) )


The individual and total air masses for the atmospheric

species are compared to secant and Bemporad air mass

results in Table 3.3.




The techniques usually used in the analysis of

sunphotometry and radiometry data are not applicable below

a wavelength of 350nm because of the spectral structure

in the ozone absorption coefficient. We have developed (29)

experimental and analytical techniques for extending the

range of solar radiometry to the middle ultraviolet.

4.1 The Attenuating Components

As sunlight traverses the atmosphere three species

remove energy from the direct beam: air molecules Rayleigh

scatter light, aerosols remove light by Mie scattering,

and gases such as ozone absorb light. In the infrared other

gases such as water vapor and carbon dioxide replace

ozone as the absorbing component. The total optical depth

is the sum of the individual optical depths.

T = R + (4.1)

Because the optical depth due to aerosols is highly variable

and has a large effect on visibility, it is the term of

greatest interest in solar radiometer studies. The Rayleigh

and ozone terms are much less variable and are subtracted

from the total optical depth to obtain the aerosol optical


4.1.1 Rayleigh Scattering

Rayleigh scattering is the scattering of electromag-

netic radiation by particles very much smaller than the

wavelength of the incident radiation, i.e. the scattering

of light by air molecules. The intensity of radiation

scattered from a volume of air molecules depends on the

intensity, polarization, and wavelength of the incident

radiation, on the molecular number density and index of

refraction of air for the particular wavelength, and on

the angle between the incident and scattered radiation.

The equation for Rayleigh scattering is written

R( ) = 2 (m )2 (1 + cos2(*)) (4.2)
N A0

where R is the Rayleigh scattering coefficient for angle i

and wavelength X. NO is the molecular number density and

m is the index of refraction. If for a given wavelength

the intensity of an unpolarized incident beam is I0, the

scattered intensity is

I(*) = R(*,A) I0 (4.3)

Equation 4.2 displays the 1/A4 dependance that makes

Rayleigh scattering very important in the blue and ultra-

violet. The strong scattering of blue light out of direct

sunlight produces the diffuse "blue sky" background.

Rayleigh scattering does not precisely obey a 1/A4 law

because it also depends on the index of refraction of air,

which is also wavelength dependent. According to Edlen (30)

the index of refraction of air at standard temperature and

pressure is given by

m 1 = 0.000064328 + 0.0294981 +0.00025540 (4.4)
146 1/A2 41 1/A2

where A must be in microns.

The Rayleigh attenuation coefficient is the product

of the molecular optical cross section and the number of


8R(h) = aR(A) NR(h) (4.5)

The cross section is given by

8 ir (m2 1)2 6 + 3 6
aR(X) = (4.6)
3 4 N2 6- 7 6

e R = Rayleigh cross section (cm2)

A = wavelength (cm)

m = index of refraction of air

NO = molecular number density for a

standard atmosphere (cm-3)

NO is usually given as 2.547x1019 particles per cubic


The altitude distribution of air is approximately

exponential, but it is more accurately represented by a

form of the generalized distribution function (31)

(1.312)2 eh/6"42
NR(h) = N e (4.7)
R 0 (0.312 + eh/6.42)2

where h is the altitude in kilometers. A comparison of the

altitude distribution calculated using eqns. 4.5 and 4.7

with the altitude distribution given by Elterman is shown

in Figure 4.1.

4.1.2 Mie Scattering

Mie scattering is the scattering of electromagnetic

radiation by spherical particles of arbitrary size;

Rayleigh scattering is simply the very-small-particle limit

of Mie scattering. Recently progress has been made on

calculating the scattering from arbitrarily shaped parti-

cles, but since, as explained earlier, this work is

concerned with small angle scattering, the classic Mie

solution for spherical particle scattering is adequate.

According to the Mie solution the scattering from a

particle is a function of the size of the particle relative

to the wavelength of the incident light, so the scattering

depends on the size parameter x = 2rr/A. A good treatment

of Mie scattering is given by Van de Hulst (3) whose

notation we follow.

The problem is formulated as a plane electromag-

\ Elt. calc.
t* -- -- ---- Rayleigh
+ .--- Aerosol
40 o Ozone

+ \




10-5 10 10-3 10-2 10-1
Attenuation Coefficient (km-1)

Altitude distributions of the 500nm Rayleigh,
aerosol, and ozone attenuation coefficients
according to Elterman (1968) compared with
analytic functions.

Figure 4.1

netic wave incident on a dielectric sphere of index of

refraction m. Beginning with Maxwell's equations, the form

of the solution is postulated in terms of spherical Bessel

functions for the interior and exterior of the sphere with

boundary conditions applied at the surface. The solution

is in terms of two scalar constants composed of Ricatti-

Bessel functions.

a n(mx) *n(x) m *n(mx) In(x)
a = (4.8)

b = m n(mx) n(X) n(mx) n(X) (4.9)
m n'(mx) n (x) n (mx) n'(x)

where primes indicate differentiation with respect to r

the particle radius. The equation for the electromagnetic

wave scattered from the particle is

El ik(z-r) S2 0 El
Se- 2 (4.10)
E2 ikr 0 S E20

where El and E2 are the components of the electric field

parallel and perpendicular respectively to the scattering

plane. The S1 and S2 terms are expressed in terms of the

parameters a and bn-

SP l(cos6) d
S = 2n+l ( a n + b -- P(cose)) (4.11)
1n=l n(n+l) n sine n d

a P (cose) d
S = I 2n+l ( h n + a Pl(cos8)) (4.12)
2 n=l n(n+l) nsine n de n

The differential scattering cross section is

ae(x,m,e) = x IS112 2 (4.13)
8 { 1 S2 (4.13)

When multiplied by the irradiance incident on a particle,

a, gives the intensity of light scattered in the direction

0. The units of Oa as defined are area per unit angle.

The total cross section, which is a measure of the total

energy which will be removed from the incident beam by

one particle,is found by integrating over all angles.

o(x,m) = f aO(x,m,0) dw (4.14)

In order to apply Mie theory to atmospheric scattering

work the scattering for individual particles must be

integrated over all particles found in the atmosphere. It

would be more accurate to allow the size distribution to

vary with altitude, but we simplify the problem by assuming

that the size-altitude distribution is separable. The Mie

extinction coefficient is found by integrating over the

size distribution.

B(A,m,h) = fo a(x,m) NM(h) n(r) dr (4.15)

For our size distribution n(r), the fraction of

particles in the radius range r to r+dr, we use a regular-

ized power law distribution which corresponds to the Junge

distribution for large r.

v r-1
n(r) = -. (4.16)
a { 1 + (r/a)v }2

The parameter v is the power of the number decrease and

a is approximately the radius of maximum frequency.

The altitude distribution we represent as the sum of

two generalized distribution functions, the second term

being included to represent the high altitude aerosol


1.135 eh/1.18
N (h) = N (0) -
M(() = 0.0661 + eh/1.18)2

8.347 eh/3.3
+ (4.17)
(80.0 + eh/3.3)2

NM(O) is the aerosol number density at ground level as

determined by a particle counter for instance, and h is

the altitudein kilometers.

4.1.3 Ozone Absorption

We now consider the third attenuating component, the

ozone absorption, which occurs in three wavelength bands

the Hartley band, the Huggins band, and the Chappius band.

The Hartley band extends from 200nm to 300nm but is

not very interesting for the ground level observer because

the optical depth is so large, exceeding 100 at 255nm,

that no photons reach the ground below 290nm.

The Huggins band is of greatest interest because it
is the dominant attenuator in the middle ultraviolet.

The band extends from 300nm to 360nm decreasing approx-

imately exponentially over this range. If the structure of

the spectral absorption coefficient is not important, the

following simple form is useful:

T ) = oz 10.0 e-(A-300)/8 (4.18)

where A is in nanometers and woz is the equivalent thick-

ness of ozone at standard temperature and pressure expressed

in centimeters.

The Chappius band extends from 440nm to 850nm, reach-

ing a maximum at 600nm. This band is ordinarily only a

small part of the total optical depth, but it is necessary

to explain certain twilight phenomena. In Figure 4.2 we

show the Huggins and Chappius absorption coefficients.

The ozone attenuation coefficient is given by

8oz(h) =Ao (A) oz Noz(h) (4.19)

where the Aoz in cm are the Vigroux ozone absorption

coefficients (32),.woz is the total ozone thickness in cm,

and Noz in km-1 is the ozone altitude distribution. No
oz oz
is also represented by a sum of generalized distribution


Wavelength (nm)



600 700
Wavelength (nm)


Ozone absorption coefficients in Huggins
band (300-360) and Chappius band (440-850).

Figure 4.2







0.2139 exp{(h-23.0)/4.44}
N (h) =
N hoz ( 1 + exp{(h-23.0)/4.44} )2

+ 0.0096 e-h/5"78 (4.20)

We used the absorption coefficients of Vigroux for -440C,

but Waltham (33) indicates that for our latitude an

average ozone layer temperature of -550C would be more


The optical depths of all three major attenuating

components are shown as a function of wavelength in

Figure 4.3, the standard atmosphere of Elterman being

used as the model.

In addition to the attenuators discussed there will be

absorption by oxygen, water vapor, carbon dioxide and other

minor gases. None of these absorb light in our wavelength

range except H20 which has bands, which we avoid, as listed

in Table 4.1.

1.0 -


- Rayleigh



300 400 500 600 700 800 900
Wavelength (nm)
A comparison of the three attenuators.

Figure 4.3


Water vapor absorption bands.


Wavelength (nm)

Centroid (nm)

700 740

790 840

930 980

1095 1165

1319 1498










4.2 Standard Bouguer-Langley Analysis

The standard technique for analyzing solar radiometer

data is based on the attenuation law attributed variously

to Bouguer, Beer, and Lambert.

I = 10 e-T (4.21)

where the total optical depth is the sum of the three terms

given in eqn. 4.1. Unless otherwise stated the optical depth

is for ground level and an air mass of one. For non-zero

solar zenith angles the optical depth in eqn. 4.21 is

replaced by mT.

The turbidity coefficient used here is that of

Volz (34). The turbidity B is the decadic version of the

aerosol optical depth for the wavelength 500nm. The solar

radiation J observed by a Volz sunphotometer is defined

by the equation

J.s = J 10-(TR + To + B)m (4.22)

where s, the square of the sun-Earth distance in AU,

corrects for variations in the distance and JO is the

extraterrestrial constant for the instrument, correspond-

ing to the reading the instrument would give at the top

of the atmosphere.

Sunphotometers are frequently calibrated by the

Langley method. A series of meter readings J are taken

over a wide range of air mass values (m=l to m=6). If the

total optical depth remains constant over the period of

observations, and this requirement is the weakness of the

Langley method, then a graph of logJ versus m will be a

straight line and extrapolate to J0 at m=0.

Assuming that J0 has been determined and that the

Rayleigh and ozone base 10 optical depths at 500nm are

known ( TR = 0.0634 and Toz = 0.004) equation 4.22 may be

solved for the turbidity.

B l og -- ( T + To (4.23)

The turbidity was originally defined for 500nm only,

but the definition may be extended to other wavelengths.

The turbidity has been found usually to vary with wave-

length according to

B(A) = B(A0) (A/A0)-a (4.24)

The parameter a, which to some extent characterizes the

aerosol size distribution, varies from 0 to 4 with a world-

wide average of 1.3.

4.3 Ultraviolet Solar Radiometry

We now consider certain difficulties encountered in

the ultraviolet and not in the visible. The equation for

the signal from an instrument with bandwidth AA centered

at 0 and with an acceptance angle Afl is

J(A0) = C(A0) f f R(A) I(A,n) dA dn (4.25)

where J is the observed signal, R(X) is the relative

spectral response of the instrument, and I(X,n) is the mono-

chromatic intensity at wavelength A from direction Q.

C(XO) is the calibration constant which converts the signal

to absolute irradiance.

J (A0)
C(A = c 0 (4.26)
f R(X) H (A) dA
AX c

Here H is the spectral irradiance of a standard lamp.

The light entering a sunphotometer is composed of the

directly transmitted sunlight and the scattered sunlight.

I(A,n) = Id(X,s) + I s(X,n) (4.27)

Usually the instrument acceptance angle is one or two

degrees and is centered so as to contain the sun's half

degree disc, so the direction integration for Id is not

necessary. The scattered intensity is variable, but

normally it is so much less intense than the direct compo-

nent, by three or four orders of magnitude, that it can be

neglected. At ultraviolet wavelengths scattering is much

more important. But aureole measurements, which give the

ratio of scattered intensity to direct intensity, show that

for a well collimated instrument the contribution to the

total signal made by scattered light at 309nm is less than

1% and may be neglected.

Inserting the solar direct intensity at ground level

into eqn. 4.25 gives

J(AO) = C(X0) f R(A) H0(X) e-1 mi 'i() dA (4.28)

where H0 is the extraterrestrial solar flux and the m. are

the individual air masses for the three attenuating compo-

nents. The limit m=0 gives the extraterrestrial constant

for the instrument.

JO 0() = C(AO) / R(A) HO(A) dA (4.29)

For wavelengths below 350nm the rapid increase and

structure in the ozone spectral absorption coefficients

lead to significant change in the spectral quality of

attenuated sunlight with air mass, producing deviations

from straight line Bouguer-Langley (BL) plots. Equation

4.28 accurately models the radiometer to allow for this

effect. Solar irradiance data together with optical depth

data were input at O.lnm intervals into eqn. 4.28. A family

of curves is generated by varying TM and plotting the log

of the expected instrument reading against the air mass.

Curves generated for the 313nm channel are shown in Figure


A form of Langley extrapolation to find J0 is possible

using such a family of curves by superimposing data for a

constant turbidity day over the calculated curves and

adjusting J0 to find the best fit. This method gives results

as good as calibration with a standard lamp.

Because of deviations from straight line BL plots the

simple sunphotometry equations of the last section can no

longer be used; yet we wish to be able to calculate the

aerosol optical depth from air mass and instrument readings.

Since the optical depth is the slope of the straight line

of a BL plot, we shall consider the derivative of J(X0)




1.0 0





1 2 3 4 5 6


Theoretical Bouguer-Langley plot for 313nm

Figure 4.4

with respect to m (= sece) and designate it the effective

optical depth.

r (m) =dJ(Xn) (4.30)
J( 0) dm

( dmi
f R(X) H0(A) ( i dm e-Timi dX
Te(m) = -(4.31)
T emM) = f R(A) H0() e-imi dA (4.31)

The dependence of T (m) on m is shown in Figure 4.5 for
four values of the aerosol optical depth. The degree to
which the curves are parallel suggests that within certain

air mass limits the aerosol optical depth is independent

of air mass.

Te(m) = Te0(m) + TM (4.32)

Te0(m) is Te(m) calculated for zero optical depth for
the aerosol.
We investigated several analytic forms for TeO and

for the 313nm and 340nm channels found the following form
the most useful:



1.0 -


, 0.9

- 0...8 -

o 0.7

" \ 0.46

0.6 -


0.5 -


0.4 I I I
0 1 2 3 4 5

m = sec(O)
Variation of total effective optical depth
with aerosol optical depth at four values.

Figure 4.5

TeO(m) = (4.33)
( 1 + mtl)

where to is Teo at m=0 and tl and y are constants for a
particular channel determined using a least squares program
to fit the curve generated by eqn. 4.31. The upper curve

in Figure 4.6 shows the best fit for the 313nm channel,

obtained for t =0.686, t1=0.293, and Y=0.488.

This equation is integrable when inserted into
eqn. 4.30.

InJ(X 0) -nJ(A0) = mTM -

x { 1 (l+mt1) 1-y (4.34)

Solving for the aerosol optical depth gives the equivalent

to eqn. 4.23 which we were seeking for use in the ultra-

TM n( J(Ao ) t0{((+mt-1) 1}
mM Jps ml(1-(
m Mt (l-Y)










Best fits

2 3 4 .5

to Te0(m) at two wavelengths.

Figure 4.6



For the 330nm channel another analytic form for Te0

was found useful.

Te (m) = to y+ (4.36)
0y+emt 1

This is the lower curve in Figure 4.6, the best fit being

given by values of t0=0.319, tl=0.204, and y=6.35.

When we investigated the error involved in approxi-

mating the effective optical depth as the sum of a zero

aerosol term containing all variation with m plus a constant

aerosol optical depth term, eqn. 4.32, we found that

the error was small for the 313nm channel: about 3% for

an air mass of 2.0. But we found that the error for the

330nm channel was almost 10%, so a simple correction term

for the residual m dependence in TM was included to reduce

the error to less than 1%.

Te(m) = Te0(m) + TM(0.96 + 0.02m) (4.37)

This form, when inserted into eqn. 4.30 and integrated,

gives the equation

1 J00) to (1+y)
S= In(l )J -
(0.96m-0.01m2) (A0)-s

{ m + 1in( 1+ Y (4.38)
tl y+emtl

This equation was used for analytic calculation of aerosol
optical depths for the 330nm channel.
Clearly the equations developed here are for our
specific instruments, but similar equations with variations
in the analytic forms as required will be applicable for
ultraviolet radiometers in general.
The term we chose to study, the derivative of J(A0)

with respect to m, is not unique. A different definition
for Te(m) could have been made that would contain all m
variation yet not require re-integration to find the aerosol
optical depth. Such a procedure would have been simpler in
some ways, but it might have made the analytic fits more
difficult to achieve.

4.4 Instrumentation

As part of the NOAA (EPA) national turbidity network

(35) we have made routine measurements with a two channel

selenium cell sunphotometer. The channels of this instru-

ment are centered at 381nm and 498nm with halfwidths of

llnm and 13nm. In addition we have a six channel instrument

kindly provided by Dr. F. E. Volz which has channels at

313nm and 341nm with halfwidths of 12nm and 13nm. The

two visible channels overlap the first instrument, and

we have not made use of the two infrared channels, which

are designed to observe a water vapor band.

Selenium cell sunphotometers are widely used because

of their simplicity, low cost, and portability. They do

have two possible drawbacks: the photocell may not be

completely linear throughout its range, and the sensitivity

and spectral response of the instrument may change with

age. One can correct for nonlinearity, but changes in

spectral response and sensitivity amount to a change in

the extraterrestrial constant of the instrument. Such

changes necessitate periodic recalibration of the instrument

either by the Langley method or by comparison with an

infrequently used standard instrument. We have observed

some drift in our 498nm and 381nm channels over the past

year, while the 313nm and 341nm channels have been fairly


In addition we have constructed a research quality

solar radiometer for use in conjunction with our field

instruments. The instrument was first used with an RCA 7102

photomultiplier tube operated at 680 volts. This is a low

gain photomultiplier tube which has as S-1 photocathode

of maximum sensitivity at 850nm but still useful at 330nm.

The photometer can usefully span three decades in intensity

and is linear to within 3% over this range (36). It is

used in conjunction with interference filters of 10nm half-

width centered at wavelengths of 330nm, 400nm, 500nm, 600nm,

and 700nm. The field of view of the instrument is limited

by a two aperture collimator with an acceptance angle of

68' of arc, sufficient to contain the 32' solar disc while

admitting little scattered light.

Recently this instrument was modified, replacing the

photomultiplier tube with an RCA 1P28 which has an S-5

photocathode of maximum sensitivity at 350nm and useful at

700nm. Due to damage, the 400nm and 500nm interference

filters were replaced with 420nm and 505nm absorption

filters with bandwidths of 50nm and 30nm. Such large band-

widths would normally be avoided, but use of eqn. 4.28

Multichannel photomultiplier solar radiometer
mounted alongside camera and occulting disc
used to photograph the solar aureole.

Figure 4,7







0) 4


14 .-4
. 0
r- (
M *
M .-


to model these channels allowed us to use them with no

problems. In addition the five filters were installed in a

motor driven filter wheel, allowing rapid observations to

be made and use of a chart recorder if desired. The photo-

multiplier solar radiometer is shown in Figure 4.7.

4.5 Results

We made several series of observations in 1974, and

of these three days exhibited sufficiently constant aerosol

optical depth for the BL method to be useful: February 4,

February 10, and November 22. The February observations

were made in developing our analytic technique for calcu-

lating ultraviolet aerosol optical depths, while the

November data were taken in conjunction with the solar

aureole study discussed in Chapter 5.

These three days were all characterized by very low

optical depths as an anticyclonic mass of cold Canadian

air moved into Florida. The more usual atmospheric condition

is a low to medium dust level with water droplet haze,

because of the warm humid air. This leads to a very varia-

ble aerosol optical depth. A common pattern might be a low

morning optical depth of 0.1 increasing to 0.2 by noon,

then decreasing somewhat in the afternoon.

In Figures 4.9 through 4.12 we show photometer readings

as a function of air mass for various of these days along

with calculated BL plots for various aerosol optical depths

generated by our computer modeling of the instruments.

In Figure 4.9 we show readings from the 700nm and

600nm channels for February 4 and November 22, 1974. It

shows that the optical depth for November 22 was about half

that of February 4, and that while TM at 600nm was as

expected slightly larger than at 700nm for the February

data, for the November data it was slightly larger at

700nm than at 600nm.

Data for the 505nm and 420nm channels shown in Figure

4.10 are given for November only, since the new filters

were only recently installed. These readings followed

straight lines very closely, ideal for Langley extrapolation

to JO. From 420nm to 600nm TM decreases with wavelength

approximately as a=1.5.

Plotted along with the 505nm data is a set of data

taken with the Volz six channel instrument, 500nm channel,

on April 27, 1975 in Cedar Key, Florida, an island off the

west coast of Florida. An optical depth of 0.32 is indicated

for Cedar Key for this day. The point plotted at m=1.05





11/22/74 0.46
o 2/4/74






0 1 2 3 4 5 6
sec (0)

Data for two dates plotted with computer
generated Bouguer-Langley plots for T =
0.0, 0.046, 0.092, 0.138, 0.230,
and 0.46.

Figure 4.9

1 2 3 4 5 6
sec (6)

Data plotted with
Langley plots for
0.46, and 0.69.

T = 0.0,

generated Bouguer-
0.046, 0.115, 0.23,

Figure 4.10






is a measurement made in Gainesville on the same day.

TM for Gainesville was 0.60, almost twice that at the

coast. This is due partly to lower dust levels over the

ocean, but more important is the solar heating of land

leading to thermals and a high rate of cloud (and haze)


Figure 4.11 shows February 4 data for the 330nm

channel. On this particular day the data follow: the

nonlinear calculated BL curves quite well.

In Figure 4.12 data for the 313nm channel for two

days, February 10 and November 22, are shown. The point

at m=4.7 which falls above the TM = line is probably due

to inaccurate zeroing of the instrument, since an error

equal to the smallest scale division of the instrument is


On November 22 the aerosol optical depths for the

700nm, 600nm, 505nm, 420nm, and 313nm channels respective-

ly were 0.030, 0.018, 0.023, 0.035, and 0.12. Except for

the 700nm channel, this follows the expected decrease of

optical depth with wavelength with alpha, the wavelength

exponent defined by eqn. 4.24 in the range 1.2 to 1.6.



10 -


1.0 -
o0 2/4/74


0.01 I i I I I I
01 0 1 2 3 4 5 6

sec (0)
Data plotted with computer generated Bouguer-
Langley plots for T = 0.0, 0.046,0.115,0.230,
0.460, and 0.69.

Figure 4.11




10 -

o 11/22/74
1.0 2/10/74 0.0


0.01 '
0 1 2 3 4 5 6
sec (0)
Data for two days plotted with computer gen-
erated Bouguer-Langley plots for T = 0.0,0.046,
0.115,0.23,0.46, and 0.69.

Figure 4.12



The light that constitutes the daytime sky brightness

is composed partially of light that has been Rayleigh

scattered and partially of light that has been Mie scat-

tered. If one is interested in determining characteristics

of the atmospheric aerosol by studying light that has been

Mie scattered, it is logical to concentrate on that region

of the sky in which Mie scattering is maximum: the solar

aureole. The aureole is the circular area of whitish light

around the sun, ten or so degrees in extent, arising from

the sharply forward peaked scattering pattern from aerosol


5.1 Aerosol Models

The variability of the natural aerosol was discussed

in Chapter 2, and it may be concluded that the size distri-

bution is the important variable in solar aureole study.

In its most general form the size distribution includes

variation with altitude, but using a size-altitude distri-

bution adds greatly to the complexity and cost of an

atmospheric scattering calculation while yielding little

increase in accuracy.

The most prominent feature of the natural aerosol size

distribution is the power law decrease in number density

with increasing particle radius first described by Junge

(10). Here the term size distribution refers to the radius-

number distribution n(r): the number of particles per cubic

centimeter within a unit radius range dr at r, where r is

the particle radius in microns. Another useful function is

the cumulative oversize distribution N(r): the total number

of particles per cubic centimeter that have radii greater

than r. Thus N(O) will be the total number density of

particles. The size distribution is related to the cumula-

tive oversize distribution by

N(r) = f_ n(r) dr (5.1)

or alternatively

n(r) = -dN(r)/dr (5.2)

Junge prefers to use a logarithmic scale because of the

wide range of particle sizes and concentrations. His log

radius-number distribution is defined as

NL(r) = -dN(r)/d(log r) (5.3)

He found that continental aerosols are described by

dN(r) = c r- d(log r) (5.4)


n(r) = 0.434 c r-(v+1) (5.5)

This distribution is frequently used directly in calcu-

lations with an externally imposed lower limit ( about 0.01

microns) to avoid r = 0 problems. The constant c depends

on the number of particles per cubic centimeter and the

exponent v is found to range between 2.5 and 4.0. The

majority of aerosol measurements indicate that v = 3

gives a good fit to the natural aerosol distribution.


Parameters for Deirmendjian's size distributions.
























A very widely used functional form for the size

distribution is that of Deirmendjian (37)

n(r) = a r e-(b r) (5.6)

which vanishes at r = 0 and which he called a modified

gamma function, since it reduces to the gamma function when

y = 1.0. The parameters for three models are listed in

Table 5.1. Haze M was introduced to reproduce a typical

marine aerosol, while L represents a continental aerosol.

Haze H represents the stratospheric aerosol submicron


A number of models were tried by Green, Deepak, and
Lipofsky (36), the most useful being their mathematical

spline model

n(r) = c exp ]/ {b(exp( )-l)+l}2 r+3

x Q(r) (5.7)

where a, b, c, and v are adjustable constants and Q(r)

is the efficiency factor.

In our analysis of light scattering, we use a regular-

ized power law that is expressed simply in terms of the over-

size distribution

N(r) = N(0)/( 1 + ()) (5.8)

which corresponds to the size distribution

n(r) = N(O) V r (5.9)
a {1 + (E)V }2

This size distribution also vanishes at r = 0, and for

r a

n(r) % N(0) v r (5.10)
a (r)

A N(0) r +) (5.11)

which is identical with the Junge distribution. This para-

meter v thus corresponds to that of Junge, and a is approx-

imately the radius at which the number frequency is maximum.

Junge (38) was one of the first to obtain a "complete"

size distribution curve in 1955 using an impactor to deter-

mine the distribution of particles larger than 0.05 microns

and an ion counter to size the particles smaller than 0.05

microns. Figure 5.1 shows our regularized power law

SOversize distributions
o 10 -

0 1.0 -

0.1 -

v=4 3 2

- 100 Size distributions

9- 10 -

S1.0 -

0.1 I I I I I
0.1 1.0 10.0
Particle Radius (p)

Figure 5.1

distribution for three different values of v, and in

Figure 5.2 we compare the data points found by Junge with

Deirmendjian's standard Haze L distribution and our regu-

larized power law distribution adjusted to v = 2.8 and

a = 0.03. Pasceri and Friedlander (39) support this type

of distribution, peaking between 0.01 micron and 0.10 micron

and then decreasing as a power law, with more recent

measurements made in 1965 in Baltimore using a cascade

impactor and a rotating disc sampler.

While the simple regularized power law is adequate

to describe a normal size distribution, an additional term

can be added to describe a bimodal distribution.

v-l v2-1
n(r) = N v r- +N 2 r (5.12)
a (()V 2 a2 (1+ 2 ( )2

where the second term may be negative if required.

We attempt to infer the size distribution based on

the scattering pattern of the solar aureole and the aerosol

optical depth as measured by the solar radiometer, yet

this represents only a limited amount of information. Our

technique is to constrain the aerosol size distribution

models to those that are physically reasonable in light of


10 3 =2.82






10 3

10- I.- I I ,
.01 0.1 1.0
Radius (P)
Size distribution measurements of Junge com-
pared with the regularized power law and Deir-
mendjian's Haze L distribution.

Figure 5.2

the cumulation of prior investigations. Other investigators

handle the problem of inversion to find the size distri-

bution without assuming a size distribution model. The

Backus-Gilbert inversion technique (40) for instance may

be capable of establishing the size distribution with no

a priori assumptions. But it has generally been found that

noisy data present difficulties to strict numerical inver-

sion methods. Ours is a parametric model-comparative test

with data approach which has been used successfully in a

number of fields, particularly in nuclear and particle

physics where scattering measurements have reached a high

level of sophistication.

5.2 Mie Phase Functions

The phase function relates the particle size distri-

bution (in the case of Mie scattering) to the intensity

of light scattered as a function of angle. In the case of

Rayleigh scattering the phase function is analytic

PR() = 1- ( 1 + cos28 ) (5.13)

and satisfies the normalization condition

f21f' P(e) sinO de de = 1 (5.14)
0 0

Mie scattering intensities are calculated in terms

of Bessel functions and are not simply analytic, so the

phase function which is the result of averaging Mie scat-
tering over an arbitrary size distribution certainly is

not analytic. The basic definition for a phase function is

that it is the ratio

P(e) = 0 (5.15)
f ae(0) dw
of the differential scattering cross section to the total

cross section. In terms of the complex amplitudes for the

scattered radiation and the scattering efficiency factor

of Van de Hulst

= ( )2 1S112 + IS21 )(5.16)
M (7 2 wC r2 Q (5.16)
2 r2 QQ

This phase function for a single particle is then averaged

over the size distribution

P (8) = j P (e) n(r) dr (5.17)

If n(r) is our regularized power law distribution, the

phase functions at different wavelengths are related by

PM (,v,a,X) = PM(6,v,ka,kX) (5.18)

so that once the phase function is calculated at one wave-

length for an array of v and a values, it is directly

extensible to a range of wavelengths.

Since we infer the size distribution by varying the

model parameters v and a to find the best correspondence

with experiment, it would be more efficient as a matter of

practical search procedure if we could characterize the

phase function analytically as a function of angle. Then

we could search directly on variations of the phase function

and save the time and expense of calculating new phase

functions for every variation of the size distribution.

Henyey and Greenstein (41) proposed the scattering


P(e) = x-3/2/4w (5.19)


x = 1 + g2 2g cos9 (5.20)

Dr. A.E.S. Green has proposed a generalized Henyey-

Greenstein function

gT g 1 1 -1
P(8) = x+T/2 T-g (+g (5.21)
27r x1T2 1l|-g| (1+g ) ]

which contains the Henyey-Greenstein function as a special

case when T = 1. The interesting domain of sharply forward

peaked phase functions is characterized by values of g

near 1.

Recognizing the similarity of sharply peaked light

scattering functions to the strong forward peaking for

allowed transitions in inelastic scattering of high energy

electrons by atoms, Green developed the Born-Bethe approx-

imation analogy. In this approximation the differential

cross section is given by

do 4 k' f(x) (5.22)
da W k x

where k is the incident electron energy and k' the final.

If W is the excitation energy, x is the square of the

momentum transferred and is given by

x = k2 + k'2 2kk' cosO (5.23)

One of the powerful features of the Born-Bethe approx-

imation is that if one knows the angular distribution at

one energy, it can be found at any other energy, closely

akin to what we are trying to do with regard to wavelength.

Recalling eqn. 5.20 we see that the analogues k=l and

k'=g places the two x's in correspondence. Introducing the
parameter w = 1-g it follows that

x = ( 1 / -w)2 + 2 / 1-w ( l-cos8 ) (5.24)

a complex relationship which can be simplified by introduc-

ing the parameter v

v = 2( 1 1-w ) (5.25)

in terms of which the phase function may be written as

P(e) = A (5.26)


x = 4 + ( 2 v ) ( 1 cos8 ) (5.27)

Better fits to the actual phase functions were obtained

for the scattering range 0 to 200 by adding a constant

P(6) = A + B (5.28)


Phase function parameters v, A, and B as a function
of v and a.











































In Table 5.2 we give the best fit v, A, and B para-

meters for an array of v and a size distribution phase

functions. For an arbitrary v and a combination, interpo-

lation from the table gives the three parameters needed to

generate the analytic phase function. This represents a

substantial saving in computational time, since direct

generation of the phase function involves integrating

the Mie scattering coefficients over the size distribution.

5.3 The Single Scattering Calculation

A theoretical calculation of sky brightness assuming

that light reaching the ground has scattered one time

only produces quite good agreement with observation under

conditions of small aerosol optical depth and aureole

scattering angles. In previous sections we have developed

the basis needed for the single scattering calculation:

the altitude distribution models for air, particulates,

and ozone, the absorptive behavior for ozone and the scat-

tering behavior for aerosols and air, and the models to

be used for the aerosol size distribution. We can now cal-

culate expected aureole scattering using these in a real-

istic model of the atmosphere.

The intensity I of monochromatic light observed in

the direction (6, ), zenith angle and azimuth angle of
line of sight, when the sun is at zenith angle 0s is
given by

I(Qs.Q,) = 0f {OR(h,X)P R() + OM(h,A)PM(M)}

XI0 T1 T2 ds (5.29)

where p(8,O) is the scattering angle, h is the altitude
during integration over path s, and I0 is the solar
intensity. T1 and T2 are the transmission factors from the
scattering volume to the ground and from the top of the
atmosphere to the scattering volume.

T2(h) = exp(-f {BSR(S)+0M()+oz(s))ds ) (5.30)

Tl(h) = exp(-IL {8R(1)+0M(1)+oz(1)}dl ) (5.31)

Here s is the path followed by the sunlight in reaching the
scattering volume, and 1 is the path ( of length L)
followed in reaching the ground. These paths should include
the bending of light due to atmospheric refraction for
solar zenith angles near ninety degrees, but we did not
include it in our program since all calculations were for

zenith angles less than ninety degrees.

The altitude distributions used in our model atmos-

phere were the analytic functional forms described in

Chapter 4. The relative altitude distributions are assumed

invariant; changes in aerosol optical depth and ozone

thickness are made by varying the altitude distribution


5.3.1 Geometry

As noted earlier the atmosphere may be regarded as

plane parallel for zenith angles of up to 75. Thus, in the

small angle form of our program the geometry of Figure

5.3a was used. If the sun is at zenith angle es and the

point P in the sky is at zenith angle 8 and relative

azimuth angle to the sun, sunlight must scatter through

an angle V to be observed in the direction of P.

cos* = cos:8 cos 8s + sin 8 sin 9s cos (5.32)

The geometry for near-horizon light scattering is

diagramed in Figure 5.4. We simplify the problem for now

by restricting the calculation to points along a vertical

cut through the sun (iLe. along a line of constant azimuth).

The computer integrates along the path L from ground level

a) Scattering geometry, plane parallel

b) Angle relationships.

Figure 5.3



L A A '

Spherical atmosphere scattering geometry.

Figure 5.4



to altitude A, where A is expressed in terms of the vertical

height h by

A = (R2 + 2Rh + h'seCZ)1/2 R (5.33)

where R is the radius of the Earth. For each point along

L sunlight must travel along a path s, for which the

altitude A' is given by

A' = ((R+A)2 + (h'-h)2sec2' + 2(R+A)(h'-h)
xsec8 cos{ s-arcos (R+)} R (5.34)

The integration is carried out in eighty equal increments

in h to a maximum altitude A of eighty kilometers. We

found this integration step size to give excellent results

when used with Simpson's rule, since doubling the number

of integration points caused only a 0.01% variation.

Although we did not choose to look at non-zero 4

angles, the generalization goes as follows.

The integration along L is unchanged, but along s

A' = {(R+h)2 + 2(R+h)A + A2sec2y}l/2 (5.35)

cosy = !oe + {l_(R+h 2 1/2
cosy = scos8s + ~1-R+A 1 sinO8cosC (5.36)


T2 = exp(-fA 0(A')secy dh' ) (5.37)

The scattering angle ip will still be given correctly by

eqn. 5.32.

Our objective in performing this calculation is to

compare calculated aureole intensities with aureole inten-

sities obtained by our photographic technique, which is

described in detail in Chapter 6. An example of an aureole

photograph is shown in Figure 5.5. A key feature of the

photographic method is that the solar disc (attenuated by

a neutral density 4.0 filter) is recorded along with the

aureole, and it is more accurate to obtain the ratio

of aureole to solar disc intensities than to attempt to

obtain absolute intensities directly from the photograph.

Once this ratio is accurately known, reference to the

direct solar intensity at ground level measured by the

solar radiometer allows the absolute aureole intensities

to be inferred. The intensity of light for the sun's disc

is calculated by



z 0



to l




Is (XS) = 1 eTosecS (5.38)

where To is the total optical depth at ground level and

A1s is the solid angle subtended by the sun.

5.3.2 Wavelength Integration

The aureole data obtained experimentally are not

monochromatic, since an interference filter is used at

309nm and narrow band absorption filters are used at 430nm,

540nm, and 640nm (Wratten filters #98, #99, and #92).

Initially the complete single scattering calculation was

performed at 10nm intervals (at Inm intervals for the

309nm data) and integrated to obtain the expected aureole


(0 9) f TP(A)SF(A) Ix(s,9) dA
= (5.39)
I-s(s) f TF(A)SF( )Is( s,8) dA

where TF is the filter transmission and SF is the relative

film sensitivity. But aureole scattering does not change

rapidly with wavelength, so we now calculate the scattering

at three wavelengths, the filter minimum, peak, and max-

imum wavelengths, and interpolate the scattered intensity

at each scattering angle to wavelengths in between for

integration. The extraterrestrial solar intensity, which

is irregular with wavelength, is removed before the

quadratic interpolation is performed and then re-inserted

at each wavelength after interpolation. This saves

substantial computation while causing less than 1% change.

5.3.3 Ground Albedo Correction

Since light traveling at a zenith angle close to

ninety degrees is close to the ground over a large part of

its path, we felt that including light reflected from the

ground would increase the accuracy of the near-sunset

aureole calculation. Bauer and Dutton (42) measured ground

albedo in Wisconsin from an airplane for different seasons

and terrain and found that wooded terrain in summer had

an average albedo of 0.15 to 0.20. The flat, sandy soil

of Florida might be expected to have a slightly higher

albedo, so we used a value of 0.25 in our calculation to

correspond with that used by Coulson, Dave, and Sekera (43)

in their Rayleigh scattering calculation. We assume that the

ground is a Lambertian reflector, an assumption supported

by but, as they are careful to point out, not required by

the measurements of Bauer and Dutton.

For the purpose of calculating the albedo contri-

bution, the ground beneath the scattering volume along


Scattering geometry for ground reflection

Figure 5.6


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