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SCATTERED SUNLIGHT IN THE ATMOSPHERE, FROM THE MIDDLE ULTRAVIOLET THROUGH THE NEAR INFRARED By RICHARD DOUGLAS MCPETERS A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1975 ACKNOWLEDGEMENTS 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 Sutherland. Finally, it should be noted that the National Science Foundation provided financial support for much of this work through grant number NSFGA4479, A.E.S. Green, princi pal investigator. TABLE OF CONTENTS Acknowledgements Abstract Chapter 1 INTRODUCTION 2 AEROSOLS 2.1 Aerosol Size Distribution 2.2 Physical Properties of Aerosols 2.3 Aerosol Altitude Distribution 3 SOLAR RADIATION IN THE ATMOSPHERE 3.1 Units 3.2 The Extraterrestrial Solar Flux 3.3 Geometry of the Atmosphere 4 SOLAR RADIOMETRY 4.1 The Attenuating Components 4.1.1 Rayleigh Scattering 4.1.2 Mie Scattering 4.1.3 Ozone Absorption 4.2 Standard BouguerLangley Analysis 4.3 Ultraviolet Solar Radiometry 4.4 Instrumentation 4.5 Results 5 THE SOLAR AUREOLE 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 Correction 5.4 Comparison of Measured and Calculated Solar Aureoles 5.5 Chromaticity Diagrams iii 68 68 77 83 85 91 92 95 100 120 6 PHOTOGRAPHIC ANALYSIS 123 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 Appendix LISTING OF SINGLE SCATTERING COMPUTER PROGRAM 147 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 SCATTERED SUNLIGHT IN THE ATMOSPHERE, FROM THE MIDDLE ULTRAVIOLET THROUGH THE NEAR INFRARED By 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 BouguerLangley 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 violet. vii CHAPTER 1 INTRODUCTION 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 halfintegral 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 nonnegligible 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, nonscattering, 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 bution. CHAPTER 2 AEROSOLS 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) distribution. 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. CHAPTER 3 SOLAR RADIATION IN THE ATMOSPHERE 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 perunit 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) 0 Intensity is a term that is often misused and over worked. The radiant intensity is defined as the radiant TABLE 3.1 Radiometric Units Name (spectral) Radiant flux Radiant intensity Radiance Irradiance Units watts/nm watts/(nm sr) watts/(nm sr cm2) watts/(nm sr cm2 watts/(nm sr cm Photometric Units Luminous flux Luminous intensity Luminance Illuminance lumens candle=lumen/sr candle/cm2 lumen/cm2 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 nonvisible 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 source. Viewed at a distance of one astronomical unit, the average distance from the Earth to the sun, the sun sub tends thirtytwo 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 sunEarth 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 TABLE 3.2 Solar spectral irradiance at selected wavelengths. F9() w/m2 nm 0.068 0.545 1.015 1.692 2.023 1.927 1.849 1.816 1.577 1.495 1.154 0.889 0.741 (smoothed) 0.068 0.560 1.040 1.540 2.060 1.960 1.880 1.790 1.610 1.500 1.160 0.893 0.741 A(nm) 250 300 350 400 450 500 550 600 650 700 800 900 1000 a PerkinElmer 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 N O OQ 00 0 0 H 1I ' C, oa Q4 0 ONl 1 S En $1 S .4 Laa *rI H 0U l l I, %DOa, ,4~rc 14 ** **** 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 theatmosphere 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 seventyfive degrees. As the zenith angle approach es ninty degrees the flat Earth sec(O) approximation goes sun ^ h s (a) Flat Earth approximation. 0 , (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 e1cll 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 TABLE 3.3 Air mass as a function of solar zenith angle. 8 secO Bemporad Rozenberg 1.00 1.15 2.00 2.92 5.76 11.5 28.7 go 1.00 1.15 2.00 2.90 5.60 10.4 19.8 3540 1.00 1.15 2.00 2.92 5.65 10.4 19.4 40 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 seqoz6 1.00 1.15 1.98 2.84 5.14 8.04 10.5 11.2 seq T 1.00 1.15 2.00 2.90 5.60 10.3 18.2 23.6 ~ 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 dh 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) 0 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 phere. 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 ( 1 sina(e)/qi)1/2 Rayleigh Mie ozone. (3.10) Attenuation through the atmosphere is thus accurately given for the three species by I = 10 exp(I Ti seqi(O) ) (3.11) The individual and total air masses for the atmospheric species are compared to secant and Bemporad air mass results in Table 3.3. where CHAPTER 4 SOLAR RADIOMETRY 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 depth. 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 NO X4 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 molecules. 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 0 where e R = Rayleigh cross section (cm2) A = wavelength (cm) m = index of refraction of air NO = molecular number density for a standard atmosphere (cm3) NO is usually given as 2.547x1019 particles per cubic centimeter. 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 verysmallparticle 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 Alt. 50 \ Elt. calc. t*    Rayleigh + . Aerosol 40 o Ozone 30 + \ 20 /\ 10 \ 0 105 10 103 102 101 Attenuation Coefficient (km1) 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(zr) 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 h2 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) 4w 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 sizealtitude 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 r1 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 component. 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(A300)/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 km1 is the ozone altitude distribution. No oz oz is also represented by a sum of generalized distribution functions. Wavelength (nm) 320 340 360 600 700 Wavelength (nm) 800 Ozone absorption coefficients in Huggins band (300360) and Chappius band (440850). Figure 4.2 300 1.0 0.1 0.01 0.001 500 0.2139 exp{(h23.0)/4.44} N (h) = N hoz ( 1 + exp{(h23.0)/4.44} )2 + 0.0096 eh/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 accurate. 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  Mie  Rayleigh *r 04 ozone 0.01 0.001 300 400 500 600 700 800 900 Wavelength (nm) A comparison of the three attenuators. Figure 4.3 TABLE 4.1 Water vapor absorption bands. Designation Wavelength (nm) Centroid (nm) 700 740 790 840 930 980 1095 1165 1319 1498 a 800nm ppCpT 'p 718 810 935 1130 1395 4.2 Standard BouguerLangley Analysis The standard technique for analyzing solar radiometer data is based on the attenuation law attributed variously to Bouguer, Beer, and Lambert. I = 10 eT (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 nonzero 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 sunEarth 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. Jo B l og  ( T + To (4.23) J's 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) An AX 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) e1 mi 'i() dA (4.28) AX 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) AX 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 BouguerLangley (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 44.4 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) 1000 313nm 100 10 1.0 0 0 .12 23 0.1 .4 .69 0.01 1 2 3 4 5 6 sec(6) Theoretical BouguerLangley plot for 313nm channel. Figure 4.4 with respect to m (= sece) and designate it the effective optical depth. r (m) =dJ(Xn) (4.30) J( 0) dm or ( dmi f R(X) H0(A) ( i dm eTimi dX Te(m) = (4.31) T emM) = f R(A) H0() eimi 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: 1.2 1.1 1.0  STM , 0.9 40.92 U rl  0...8  rl 'a o 0.7 " \ 0.46 0.6  0.23 0.5  0.00 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. to InJ(X 0) nJ(A0) = mTM  (lY)tl x { 1 (l+mt1) 1y (4.34) Solving for the aerosol optical depth gives the equivalent to eqn. 4.23 which we were seeking for use in the ultra violet. TM n( J(Ao ) t0{((+mt1) 1} mM Jps ml(1( m Mt (lY) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 330nm Best fits 2 3 4 .5 sec(e) to Te0(m) at two wavelengths. Figure 4.6 313nm 0.0 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  (0.96m0.01m2) (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 reintegration 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 stable. 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 S1 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 S5 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 4J O 00 la k S.* 00 MY SO 0) 4 to $4 14 .4 . 0 <^ r ( 0) M * M . C3 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 63 100 700nm 0.0 10 11/22/74 0.46 o 2/4/74 100 600nm 0.0 10 0.46 1.0 0 1 2 3 4 5 6 sec (0) Data for two dates plotted with computer generated BouguerLangley 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. computer T = 0.0, M generated Bouguer 0.046, 0.115, 0.23, Figure 4.10 100 10 100 1.0 0.1 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) formation. 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 indicated. 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. 100 330nm 10  0.0 1.0  o0 2/4/74 0.1 0.69 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 1000 313nm 100 10  0 o 11/22/74 0 1.0 2/10/74 0.0 0.69 0.01 ' 0 1 2 3 4 5 6 sec (0) Data for two days plotted with computer gen erated BouguerLangley plots for T = 0.0,0.046, 0.115,0.23,0.46, and 0.69. Figure 4.12 CHAPTER 5 THE SOLAR AUREOLE 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 particles. 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 sizealtitude 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) r 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 radiusnumber 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) or 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. TABLE 5.1 Parameters for Deirmendjian's size distributions. HAZE MODEL L PARAMETER 5.333x104 8.9443 1.0 0.5 0.05V 4.976x106 15.1186 2.0 0.5 0.071 4.000x10s 20.0 2.0 1.0 0.10. a b a Y max 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 particles. 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 v1 n(r) = N(O) V r (5.9) a {1 + (E)V }2 a This size distribution also vanishes at r = 0, and for r a v1 n(r) % N(0) v r (5.10) a (r) A N(0) r +) (5.11) a 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  P4 0.1  v=4 3 2  100 Size distributions 9 10  0 $4 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. vl v21 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' 10 3 =2.82 RPL a=0.03p 102 HAZE L 101 i0* 0 103 10 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 BackusGilbert 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 modelcomparative 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 47 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 function P(e) = x3/2/4w (5.19) where 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 Tg (+g (5.21) 27r x1T2 1lg (1+g ) ] which contains the HenyeyGreenstein 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 BornBethe 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 BornBethe 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 2 parameter w = 1g it follows that x = ( 1 / w)2 + 2 / 1w ( lcos8 ) (5.24) a complex relationship which can be simplified by introduc ing the parameter v v = 2( 1 1w ) (5.25) in terms of which the phase function may be written as P(e) = A (5.26) where 2 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) x TABLE 5.2 Phase function parameters v, A, and B as a function of v and a. i 0.01 0.03 0.05 0.10 0.30 0.50 2.0 .0203 .0203 .2440 .0203 .0211 .299 .0202 .0223 .201 .0200 .0260 .0910 .0187 .0366 .332 .0172 .0396 .516 2.5 .0286 .00968 .655 .0286 .0102 .664 .0286 .0112 .670 .0286 .0149 .622 .0269 .0324 .0105 .0239 .0407 .394 3.0 .0437 .00444 .741 .0438 .00475 .768 .0440 .0054 .808 .0445 .00835 .870 .0420 .0292 .243 .0353 .0416 .287 3.5 .0766 .00283 .627 .0768 .00310 .664 .0772 .00368 .722 .0782 .00643 .849 .0697 .0314 .291 .0537 .0463 .297 4.0 .1480 .00304 .477 .1490 .00347 .520 .1480 .00430 .584 .1450 .00791 .731 .1100 .0392 .186 .0783 .0552 .415 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 uniformly. 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 nearhorizon 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 atmosphere. b) Angle relationships. Figure 5.3 87 S1s L A A ' Spherical atmosphere scattering geometry. Figure 5.4 hhi R 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) 1/2 xsec8 cos{ sarcos (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 nonzero 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) R where cosy = !oe + {l_(R+h 2 1/2 cosy = scos8s + ~1R+A 1 sinO8cosC (5.36) and 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 U) 41 0 z 0 0C 01 a) 00 $4 WV) to l 4 20 a), E44 Afc Is (XS) = 1 eTosecS (5.38) s 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 pattern. (0 9) f TP(A)SF(A) Ix(s,9) dA = (5.39) Is(s) f TF(A)SF( )Is( s,8) dA AX 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 reinserted 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 nearsunset 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 'I 0 Scattering geometry for ground reflection contribution. Figure 5.6 da 
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