Atmospheric particulate analysis using angular light scattering
M. Z. Hansen
Using light scattering matrix elements measured by a polar nephelometer, a procedure for estimating the
characteristics of atmospheric particulates was developed. A theoretical library data set of scattering matri-
ces derived from Mie theory was tabulated for a range of values of the size parameter and refractive index
typical of atmospheric particles. Integration over the size parameter yielded the scattering matrix elements
for a variety of hypothesized particulate size distributions. A least squares curve fitting technique was used
to find a best fit for the experimental measurements. This was used as a first guess for a nonlinear iterative
inversion of the size distributions. A real index of 1.50 and an imaginary index of -0.005 are representative
of the smoothed inversion results for the near ground level atmospheric aerosol in Tucson.
1. Introduction
Angular light scattering data from the atmospherenear ground level were obtained with a polar nephe-lometer designed and constructed at the University ofArizona.1 The angular scattering measurements madewith this instrument were processed to acquire the four
scattering matrix elements due to particulates from aset of four intensity measurements at each of a numberof scattering angles. The best data available from theinstrument were used for analysis of particulate char-acteristics.
To analyze the experimental results, Mie theory,which assumes homogeneous spheres, was applied todevelop a theoretical particulate scattering represen-tation for comparison. Mie theory accurately describesthe scatter from the small, typically irregular particlesfound in the atmosphere, however, larger particles (inthe Mie regime) are not particularly well represented. 2
Papers have appeared recently on developing a theoryto correct for the discrepancies34; however, furtherdevelopment is necessary before these methods can be
applied to the extent needed for this study. Therefore,due to the difficulties involved in nonspherical analysis
and the method of data acquisition,1 Mie theory wasnecessarily used for the calculations in this paper.
The majority of this work was done while the author was at Uni-
versity of Arizona, Institute of Atmospheric Physics, Tucson, Arizona
85721; he is now with NASA Goddard Space Flight Center, Labora-
tory for Atmospheric Sciences, Greenbelt, Maryland 20771.
An inversion of the light scattering data was necessaryto obtain the best possible representation for the par-ticulates. The application of inversion methods forremote sensing has been a region of expanding activitysince computer development made the necessary ma-nipulations tractable. Early techniques included thePhillips-Twomey linear inversion5 6 that was appliedby Twomey7 to extract vertical temperature profiles inthe atmosphere. This linear method has been usedextensively with success in the analysis of atmosphericparticulates from multiwavelength extinction data.8 9
The inversion of atmospheric aerosol angular scat-tering data to obtain particulate information has typi-cally met with only marginal success. Westwater andCohen10 felt that the Backus-Gilbert inversion couldretrieve size distributions with angular scattering datafrom their theoretical study with multiwavelengthscattering. Post1 applied this method to multipleangle scattering measurements from water droplets buthad poor results at sizes below 10 Aim in radius. BothPost and Westwater and Cohen used narrow size dis-tributions and still had a deterioration of results at smallsizes. Some success has been achieved in invertingbistatic lidar data of atmospheric particulates12 with thelinear method, but only a very limited data set wasavailable. 13
The reasons for using one inversion scheme over an-other are almost as varied as the investigators, however,the nonlinear algorithm technique of Twomey14 hasshown promise in retrieving particulate size distribu-tions and was chosen on this basis for application toangular scattering measurements in this paper.
II. Data Evaluation Method
An estimate of the size distribution and refractiveindex of an atmospheric sample is made by comparingthe four scattering matrix elements measured at various
angles with the nephelometer with elements producedby theoretical size distributions for various indices ofrefraction. A library data set on magnetic tape wascreated using a subroutine by Dave15 for scattering bya sphere. Matrix elements were recorded for 500 sizeparameters [0.3(0.2)100.1] at every integral angle[00(1°)1800] of scatter for all combinations of a set ofreal indices of refraction (1.40, 1.45, 1.50, 1.54, 1.60) andimaginary indices of refraction (0.0, -0.003, -0.005,-0.01, -0.03). (Size parameter a is 2r times the par-ticle radius divided by the wavelength of incident light,0.5145 gm in this case.) The range of real indices waschosen to encompass a region from near that of water
up past that for silicates. The imaginary index valuesvary from no absorption as for water to a value of -0.03,which is near what King16 has observed.
Subsequently, these data are integrated over the sizeparameters for Junge and two-slope size distributions(Figs. 1-3). The Junge size distributions are calculatedby setting
dN-= Cr(v+l) (1)
where N is the particle number concentration, r is theparticle radius, C is a normalization value set to give 100Ag/m 3 mass loading, and v varies over a typical rangefrom 2.0 to 4.0 in 0.2 steps. The two-slope size distri-butions are calculated by setting
dN C [1 + (r/rB)2]
dr [1 + (r/rA)1']'
where all combinations of rA = 0.04 Alm, rB = 0.4 and 1.0Aim, vi = 2.0, 3.0, and 4.0, and 2 = 0.0, 1.0, and 1.5 areused. These parameters were chosen to give turnovervalues between 0.01 and 0.1 tim. These are above thevalues observed by Twomey,17 but a higher turnoverpoint is necessary if any effect were to be observed onthe scattering data.
Comparisons are made between these size distribu-tions and real data by allowing the mass loading to varyto give a best least squares fit. The quality of the fit isdetermined by the size of H, given by
H = E (bi - di)2, (3)
where is evaluated at idibil3id for a minimum Hand functions as a mass loading adjustment to obtainthe best fit. b is the observed aerosol scattering matrixelement, and d is the corresponding theoretical matrixelement normalized for 100 g/M3. Biasing of the dataaccording to scattering volume is also used. Outside the
Fig. 4. Scattering element S21 integrated over a Junge size distri-
bution for two particle size ranges.
101
100
10-1
E
10-3
10-4
10-50 20 40 60 80 100 120 140 160 180
SCATTERING ANGLE (DEGREES)
Fig. 5. Scattering element M 2 integrated over a Junge size distri-
bution for two particle size ranges.
range of the size parameters on tape, the number con-centrations are inadequate (for any realistic size dis-
tribution and visible wavelengths of light) to affect theobserved scatter and are neglected. The tabulatedcomparisons are evaluated to find the best least squares
fit with reasonable mass loading and to observe anytendencies such as sensitivity to the parameters that arevaried.
Due to the similarity of many of the kernels, littleinformation is gained by using a complete range of an-
gles to obtain a size distribution. The additional time
involved in making excessive measurements can also be
detrimental due to possible changes in the sampledaerosol. Therefore, consideration should be given towhich angles are most critical. Angles where the scat-tered light is minimal have more error and should beavoided. Also, angles where the scattered radiancechanges very quickly are affected more by a positioningerror in the detector. By considering angular scatteringmeasurements made on monodisperse particles,2 onefinds that, for larger nonspherical laboratory aerosols,Mie theory seems to hold best for angles <40°. Smalleraerosols (as they approach the Rayleigh regime) tendto follow Mie theory quite well. This leads one to in-spect where the larger aerosols contribute to the scatter.By looking at Figs. 4 and 5 one observes that the dif-ference due to the large aerosols is limited mainly to theforward few degrees. This not only implies that Mietheory should hold better for a typical aerosol size dis-tribution than for single large aerosol studies, but that,if one desires information content from the largeraerosols, measurements must be made in the forwarddirection, or little information above 1 Am is obtainedfor typical size distributions.
III. Inversion Technique
Continuing with the next step, the inversion methodis considered. A nonlinear algorithm was developed byChahine,18 which essentially assumed delta functionsfor kernels but acquired inherent instabilities due toincreased high frequency content when measurementswere numerous. Chahine's algorithm was modified byTwomey14 to include the entire nonzero region of thekernel. This eliminated the detrimental factor of su-perfluous data and, in fact, caused the inversion to im-prove with additional data due to an effective decreasein measurement error.
The nonlinear inversion has also shown an ability tocope with measurement errors, which greatlystrengthens its position in application to the aerosol sizedistribution problem. The iterative algorithm is
l + ff(r)k(rs)dr I k(s) ]ad r), (4)
where f0(r) is the initial guess size distribution, k(r,s)is the kernel value (the theoretical scattering matrixelement for a single particle), g(s) is the actual measuredscattering matrix element for the collection of particles,and fr) is the modified size distribution. Variable rrefers to the particle radius or size parameter, andvariable s refers to a specific matrix element measure-ment. Examination of the kernels shows that a lot offine structure typically occurs (Figs. 6-9), particularlynear backscatter. Whereas this might be expected tobe beneficial for fine resolution, in practice this struc-ture is too high a frequency to be effective in improvingthe inversion accuracy. Since atmospheric aerosol size
distributions do not seem to have these wild oscillationsand neither do the observed scattering measurements,the fine structure would not seem necessary to resolvethat data even if it were effectively usable. In fact, it
Fig. 9. Scattering element M 2 for single particles.
° 50 100 150 20FREQUENCY
Fig. 11. Power spectrum of D 2 1.
0 250
1N
might be desirable to use smoothing of the kernel toassist in obtaining a stable solution. The power spectraof the kernels also show that the middle frequencies areoften deficient (Fig. 10), and occasionally even lowfrequencies are absent (Fig. 11). This is a strong neg-ative factor in the application of scattering kernels toinversion techniques.
Simple quadrature is used for the integral with thekernels being read from magnetic tape. Each datavalue is successively iterated once through all the par-ticle sizes on tape, modifying the size distribution ac-cording to the kernel's weighting effect. The weightingis scaled to < 1 by dividing by the maximum kernel valuefor a particular angle and matrix element. After eachunknown in the set has been determined from the firstiteration, the process is repeated until the final distri-bution is obtained. Although the inversion lends itselfeasily to programming, care is still required in its ap-plication.
IV. Inversion of Theoretical Data
Initial runs of the inversion program were made ondata generated from Mie theory to establish the accu-racy of the inversion with scattering kernels. Thenephelometer measures the radiance of light scatter,which is a function of the particles' scattering crosssections times their concentrations. It was necessaryto weight the scattering kernels according to an initial,first guess size distribution to obtain reasonable results.Otherwise, there was a strong tendency for the inversionto adjust the large particle concentrations to the pointof instability. The runs were made using only the M2and Ml elements' from five forward angles and twobackward angles. These were chosen to maximize in-formation content with a minimum of data. A specialproblem occurs in applying inversions to the S2 1 and D2 1
elements, as it is possible for the theoretical and mea-sured values to be of opposite sign due to errors in themeasurements or in the first guess size distribution.This would imply a negative particle concentration thatis not allowed. Runs were made with theoretical datafrom Junge distributions. A v of 2, index of 1.54-0.005i,and mass loading of 38 gg/m 3 were used for a first guess,as these values produced a close fit for one of the realaerosol runs. A method of overrelaxation was settledupon as the best technique for applying the algorithm.It has the format
f'(r) = (1 + MR)fO(r), (5)
where[M g~) k(rs)a6x
0(r)k(r,s)dr 'ii k(s)ax (6)
The absolute value of M is raised to the power R, andthat quantity takes the same positive or negative signas M.
Not only did overrelaxation with values of R < 1speed up convergence, but it improved the resultsgreatly (Fig. 12). A too large overrelaxation, however,caused oscillations. A value of 0.7 for R produced thebest stability and convergence although 0.5 gave thefastest convergence. Excessive iterations are not only
15
.100M
cn5
cc
10 15NUMBER OF ITERATIONS
Fig. 12. Convergence of iterative inversion for theoretical data withno error.
102
9n
0
C
1z
lo1
100
l0-'
1-20.1 0.5 1.0 5.0
RADIUS (m)
Fig. 13. Theoretical size distribution inversions for various massloading initial guesses.
costly but tend to produce a more highly structured,atypical size distribution. This is avoided by termi-nating the iterative process after successive iterationswith <0.2% improvement in error.
The inversion program was run with theoretical datato observe the effect of various size distribution firstguesses on the results of the inversion. Large differ-ences between the actual and initial guess mass loadingwere difficult for the inversion to handle if no internalmass loading adjustment is included in the program(Fig. 13). This serves to point out graphically the sizerange of information content of the data and the kindof structure that can occur due to the oscillatory natureof the kernels. The response in the region of informa-tion content is adequate to indicate the proper correc-tion necessary (i.e., higher or lower) in the mass loading.Differences between the actual and initial guess v values(Fig. 14) have much less effect on the inversion. Ex-amination of the region of convergence does show that
7 A' 5, INITIAL GUESSESV= 1.6---1O'- v = 2.4 ...... -
CORRECTIONSOLUTION
100
10°
in-2
I I 111111 X
0.1 0.5 1.0RADIUS (jum)
duce theoretical data, experimental data were ana-lyzed.
A. Curve Fitting
By comparing the experimental data with the theo-retical library data, a best fit was obtained. Typically,about twenty scattering angles with four matrix ele-ments at each angle were used. The fit was weightedby the cosecant of the scattering angle to allow more bias
10'
5.0
Fig. 14. Theoretical size distribution inversions for various Jungeslope initial guesses.
5 10 15NUMBER OF ITERATIONS
E
£1
Fig. 16. M 2 scattering matrix element.20 25
Fig. 15. Convergence of iterative inversion for theoretical data with11% error.
the scattering is mainly sensitive to particles in the 0.2-2,um range, and subsequently this is the region whereresults are applicable.
Random error was added to the theoretical data toobserve its effect on the inversion. Neither 4 nor 11%error had a significant effect on the inverted size dis-tribution. This result is essential to obtaining realisticinversions with experimental data. The convergencelimit on the error (Fig. 15) was, in fact, indicative of thepercentage error in the data; however, additional studyof this point is necessary for verification.
The linear inversion method was also applied to thisproblem initially. However, it could not invert the dataunless the error level was 1% or less. This is an un-realistically low value, especially since Mie theory alonecan account for more than 1% error. Therefore, thelinear method was dropped.
V. Experimental Results
After checking the ability of the inversion to repro-
for larger scattering volumes. After checking the datafit with matrix elements produced by both Junge andtwo-slope size distributions and using truncated datasets that excluded measurements of smaller magnitude,an estimate for the aerosol size distribution was ob-tained. The nephelometer runs for 9 March were av-eraged and gave a Junge best fit with m = 1.50-0.003i,v 2.1, and mass loading = 38 ug/m3 . The two-slopebest fit for the same data was m = 1.50-0.003i, v1 = 2.0,V2 = 0, rA = 0.04 Aum, rB = 1.0 um, and mass loading =
80 Ag/m 3 .The mass loading is not extremely critical as the
largest particles dominate this value while they havemuch less effect on the actual light scatter on which themeasurements are based. Another set of runs on 10March was averaged to yield a Junge best fit with m =1.47-0.005i, v = 2.0, and mass loading = 47 bug/m 3 anda two-slope fit with m = 1.50-0.004i, v 1 = 2.0, v2 = 0.0,rA = 0.04 jum, rB = 1.0 um, and mass loading = 100Atg/m3. Figures 16 and 17 show typical graphs of theexperimentally measured matrix elements plotted incomparison with the theoretical data produced by thecorresponding best fit size distribution. As expected,the curves match closely near the forward direction,which is where Mie theory is believed to hold best andwhere the strongest weighting is placed on the leastsquares fit. The two-slope and Junge distributions thatgave the best fits are of similar form over the size rangeof information content. Therefore, the simpler Jungedistribution was chosen for combining the March datathat gave an overall aerosol characterization of m =1.49-0.004i, v = 2.0, and mass loading = 40 pug/m3.
The strongest sensitivity for the ranges of parametersunder consideration was observed to be the size distri-bution slopes, next in importance was the imaginaryindex, and the least sensitive was the real index.
B. Inversion of Size Distributions
Further improvement in the size distribution esti-mate is attempted by using the library data closest to
10-2
r 1.50 - 005iINITIAL GUESSES
V = 1.6---V = 2.0
= 24.....
,, 10-' , .
E
, 100
eZX
90
60 FF
0
LUInM
30
0 .0I 5 10 15
NUMBER OF ITERATIONS
20 25
Fig. 19. Convergence of iterative inversion for experimental datawith various mass loading initial guesses.
INDICATES 0.2% CONVERGENCE-I ON SUCCESSIVE ITERATIONS
MO INITIAL MASS LOADING GUESSmG 19 ,tg/m3
MO= .38 ,ug/m3 mO= .76 ,tg/m3
-
\ -I… .- 1 _ _
lI I
l-'
the curve fitting results as a first guess for the inversion.Fairly consistent results are obtained by the inversion(as shown by Fig. 18 for the 9 March data) even if theinitial v value is varied above or below the curve fittingresults.
The convergence of the inversion is shown by the rmserror smoothly approaching minimum values as thenumber of iterations increases (Fig. 19). Even when theinitial guess is greatly in error from the data, the rmserror iterates down to the 15% range, which is repre-sentative for all the runs. Attempts were made to im-prove the minimum rms error of the iterated inversionby using different indices of refraction; however, thisexercise just verified the choices of the curve matchingtechnique. Probable causes of this large a convergencelimit (if it is truly indicative of the experimental error)are covered in a preceding paper. 1
Other inverted size distributions with various initialguesses are shown in Figs. 20 and 21. Twelve inversionswere averaged and smoothed to obtain a representativeinverted size distribution (Fig. 22). The results aremost closely modeled by a Junge size distribution withm = 1.50-0.005i, = 1.8, and mass loading of 60Aug/m 3 . The curve is purposely truncated so that onlythe region of sensitivity is shown. A maximum aerosolnumber concentration (or turnover point) of the sizedistribution is not observed since the sensitivity of thekernel drops off sharply below 0.2 Aum, while typicalturnover points occur near 0.01,um for the ground levelTucson aerosol.
VI. Conclusions and Further StudyThe technique developed in this paper has yielded
estimates of atmospheric aerosol characteristics-vis.,size distributions and real and imaginary indices of re-fraction-from measurements of matrix scattering el-ements at various angles. The results are reasonablein comparison with other work19-22 in this field, and thesize distributions match quite well near 0.1 m withnucleopore filter measurements 1 6 that were made at thesame location but are sensitive to particles from 0.1 mon down. To achieve a characterization of the groundlevel Tucson aerosol, measurements should be maderoutinely over an extended period.
Improvement in the stability of the inversion tech-nique might be achieved by smoothing the kernels toremove the higher frequency information. Furthersophistication could be achieved by expanding the in-version routine to include fitting the inversion resultsto a smooth analytic function and using this as a newfirst guess.
A detailed study of the information content would beof special interest. Initial work in this area has shownthat the scattering data used in this study are in theregion of optimum information content. This couldlead to an instrument with a minimum number of fixeddetectors set at carefully chosen angles, which wouldeliminate the need to move the detector and speed themeasurement time.
The author would like to thank Benjamin M. Herman
102
E
-J
C.
0.
Goz.
10'
100
10-'
10-.0.5 1.0 5.0
RADIUS (/m)
Fig. 22. Average of inverted size distributions.
and Walter H. Evans for their assistance on this project.This research was funded by the Office of Naval Re-search under grant N00014-75-C-0208; computer timewas furnished by the National Center for AtmosphericResearch, which is sponsored by the National ScienceFoundation under project 35021004.
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