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Data inversion for cascade impactors: Fitting sums of log-normal distributions
DATA [NVERS~ON FOR CASCADE IMPACTORS: FITTING SUMS OF LOG-NORMAL DISTRIBUTIONS 3. s.
Pt;TTo&
Atmospheric Environment Service, Environment Canada, 4905 Dufferin Street. Downsview. Ontario. Canada (Firsr receiced 24 September 1979 and in j%ralform 4 February 198 I)
Abstrsct-The processing of data from cascade impactors is discussed and the effects of non-ideal response characteristics on impactor measurements are shown. Since atmospheric particle size dis~~buti~ns may contain two or three modes, the sparse data from an impactor cannot give detailed information about the shape of individual peaks. Thus a shape must be assumed, and it is suggested that a log-normal form is appropriate. A procedure is described which fits sums of log-normal distributions to impactor data, taking account of the detailed random errors, where atmosphe-ic data.
response functions. This procedure has been tested with simulated data including it is found to reproduce the input distributions satisfactorily. and with actual
The cascade impactor has become a standard instrument for the measurement of particle size distributions, particularly in ambient air sampling where multi-elemental analysis of the material collected on each impactor stage can give information on the distribution of various pollutants among the particle sizes. However, there remains the probtem of data inversion, i.e. the inference of the particle size distribution function from the measured stage loadings. This is a problem for two reasons. Firstly, the collection characteristics of existing impactors are far from ideal (Rao and Whitby, 1977, 1978). Secondly, an impactor usually provides too few size separations (nine for the Andersen ambient impactor) to resolve accurately the complex size dist~butions occurring in the atmosphere. The latter di~culty forces us to make assumptions about the shape of the size distributions for the various components of the atmospheric particulate matter. In this paper a computational scheme is described which uses impactor stage collection characteristics to convert stage loadings to size distributions.
2. IMPACfOR CRARACTERISTICS
An ideal impactor stage would collect all particles above a certain size, and none below that size, i.e. defining an efficiency function for stage i, Pi (d), as the proportion of particles of aerodynamic diameter d reaching the stage which are collected by it:
The actual functions Pi have been measured for the l Present address: Thornton Research Centre, P.O. Box I, Chester, England.
‘F
‘5
j.,,.
Andersen ambient impactor by Rao and Whitby (1978). These are shown in Fig. 1. It can be seen that they are significantly different from the ideal step functions. The curves for ‘sticky’ particles on a smooth surface are closest to the ideal, while for solid particles on solid impaction surfaces ‘bounce-off’ changes the curves considerably. With a filter paper surface the slope of the functions is reduced further, presumably owing to a filtration effect in the horizontal flow over the surface. It is useful to derive the total impactor response functions Q,(d) from the Pi’s, where Q,(d) is the proportion of all the particles of aerodynamic diameter d reaching the impactor which are collected by stage i. These are related to the P,‘s by
Q&l = P,(d) i-
1
Q)(d)= Pi(d) X 17 [ I- P,(d)] j=O
= P,(d) x
[l-:
These functions, derived from Rao and Whitby’s data for the case of liquid particies, are plotted in Fig. 2. For the ideal impactor the response functions would be ‘top-hat’ functions, equal to one for dio.i < d
measurements
The non-ideal nature of the characteristics has an important influence on the apparent size distributions measured by the impactor. Distortions are introduced both by the unequal spacing of the P curves and by their finite slope. It is convenient to change to a logarithmic particle size variable I = log,,d/d,, where d, is a scaling length, say 1 cm. The expected stage
1709
Fig. 1. Stage efficiency functions P,(d) for an Anderxn impactor using:--liquid particles; --- solid particles and uncoated stainkss plates; -.-.solid particles and a glass fibre filter collection surface. (From Rao and Whitby. 1978.)
Aerodynamic
dtameter,
pm
Fig. 2. Total impactor response functions Q,(d) for an Andersen impactor using “sticky” particles or collection surfaces. In deriving these, the stage efficienciesfrom Fig. 1, with liquid particles, were used for stages 2-6; stages 0 and I and stage 7 were assumed 10 have similar eficiency functions to stage 2 and stage 6; the wall loss is an approximate representation.
loadings can then be calculated from the impactor characteristics by mi
-=
'MO where
_~ J‘ 3c
Q,(l)f(l)dL
f= kdG. 0
M is the mass size distribution of the sampled particles. and MO is their total mass. Using this, the consequences of the form of the impactor characteristics of Fig. 2 are demonstrated in Fig. 3. This shows the impactor data resulting from a mass size distribution uniform in [(dM/dl = constant, I, < I < le), and from a &function (monodisperse) distribution. The uniform distribution does not give a horizontal straight line in Fig. 3. because the stages have different ‘acceptance widths.’ If only the d,,‘s are known, then an approximate correction for this can be made by dividing each stage loading by
* L_,og,,!+ h,Od 1 d,,. i-
0
0
which gives approximate values for the mass size d&f A better approximation distribution d log (did,)’ would be to divide by the area under the i’th curve in Fig. 2, i.e. divide by Wi =
‘I Q,(l)dl. s -I
This would give a horizontal straight line for the uniform distribution data of Fig. 3. except for the first stage and the backup filter, for which wi is undefined. The latter correction, however, is exact only for a uniform distribution. Further inaccuracy arises from
* The 50’?,,cut-offd~ameterd,,,, , isdefincdasthcdiameter for which P,(d) = 0.5.
Data inversion for cascade impactors: fitting sums of log-normal disrributions
1711
. .
.
. *
.
I I
I
Filter
7
I 6
I 5
,
I 3
4
L
/ 2
I
1
1
0
!
wall iof5
Stage
Fig. 3. Impactor data, in terms of pro~rtion of total mass, for (a) a mass size distribution with d.tf/d(logd) uniform from 0.3 to 30gm, (b) a monodisperse aerosol of aerodynamic diameter 2.3 pm. l sticky particles or surfaces; A solid particles and uncoated surfaces; V solid particles and glass tibre filter collection surfaces.
the error illustrated by the d-function data in Fig. 3. Particles of a given diameter, d, can be collected by several stages, in fact ail stages for which Q,(d) is nonzero. This will make the distribution look wider than it is, and, if bounce-off is severe, it will produce spurious measurements on the small-particle stages. To correct this, it is necessary to take account of the detailed impactor response functions Qi in the inversion calculation.
3.
SOME PROPOSED
DATA REDUCTION
SCfiEMES
Inversion problems similar to that described here occur in many fields. A general theory is described in the book by Twomey (1977). A number of different numerical schemes have been proposed specifically for the inversion of impactor
data. The simplest method is to assume that the stage characteristics are ideal. McCain et ai. (1978) describe a computer program which uses this method, followed by smoothing, for in-stack cascade impactors. However, the inherent weakness of the method appears when two successive stages have identical d,,‘s. In this case Qi for the second of these stages is a small but significant peak and some material is collected, but with the ideal impactor assumption there should be nothing collected and so an infinite value is assigned to f(l) there. McCain er al. avoid this problem by ignoring the data for that stage. Similarly, with strong bounceoff the method cannot explain the large amounts reaching the back-up filter, and this measurement also has to be ignored (McCain et al., 1977). By contrast, SundelGf (1967) suggests deliberately using two identical stages consecutively to obtain Qi with a narrow particle size range. His numerical
scheme uses division b! ii, to give a tirst approximation to the value off’iii for a value oil which is assumed IO coincide with the maximum of Q,. Both ordinate and abscissa of each resulting point on the graph of!‘(i) are successively adjusted until the expected stage loadings. corresponding to a smooth curvej’(i) drawn through these points, are equal to the data. Cooper and Spielman (1976) also take account oi the detailed nature of the stage efficiency functions P,. They describe standard smoothing methods which have been proposed to suppress oscillations in the solution caused by ill-conditioning of the discretised equations used in the numerical quadrature. However, these methods fail and their proposed method uses nonlinear programming with a specific condition excluding negative solutions. The results show a good fit with exact simulated data. However, the method does not seem to work well on data with a simulated measurement error. The one case show-n in the paper has a mean random error of only 5”,,, but the solution shows some oscillatory tendency and compares unfavourably with the simple solution (assuming ideal response). Raabe’s (197R) approach for cascade impactor data involves assuming that the size distribution can be described by a single log-normal function, and using a weighted least squares procedure to find the best values for the three parameters involved. The method has the advantage of providing a z2 value to show how good a fit has been achieved. It appears to work very well when a single log-normal distribution is appropriate, e.g. for particles from a single source.
4.
SIZE DISTRIBL’TIOSS
IX THE ATMOSPHERE
To appreciate what is required of a data inversion scheme for ambient atmospheric particle samples, it is necessary to discuss the nature of particle size distributions in the atmosphere. It is generally accepted that mass size distributions have at least two maxima within the range of standard impactors (approx. 0. i-50 pm), with a smaIl-size component which is an accumulation mode of anthropogenic aerosols, the other component being mainly soii-derived. There is now evidence that the soil-based component has two peaks, leading to trimc !a1 distributions (Patterson and Gillette, 1977). This is confirmed by our own measurements analysed as described below (Puttock and Carrie. 1981). An explanation for the presence of two soil-derived peaks comes from Rahn (1976). who analyses particle size distributions in typical soil and the size selectivity of the suspension process. Given that the atmospheric sample may contain three modes, even the specification of just the peak position (or mean), width (or standard deviation) and amplitude of each mode requires nine parameters. The standard Andersen ambient impactor, commonly used for these measurements. provides just nine numbers: the mass on each of the eight stages and on the backup
hlter. It is clearly impossib!e :o obtain an) ;urthe: information about the shape ~4 the individu:tl mode distributions from such data. For this reason, it is necessary to make .; ~G~v: assumptions about the shape of the dtstribuiions for each component of the aerosol. Little 1s known I;! detatl about the shapes of these distributions. The most convenient forms to assume are log-normal fU~lCIlOIlS of particle diameter. i.e. normal distributions 111i Detailed measurements which support use of such functions for various components of the atmospheric aerosol are reported by Whitby t’r al. (1972). Theorettcai aspects of the use of tog-normal distributions for fitting particle size data have been discussed extensively by Jaenicke and Davies I 19761. One advantage 1s the simplicity of transformanon between volume. surface area and number distributions. In fdcr. if it were not for the problem of non-ideal unpacror characteristics, it would bean easy matter to ti: sums 01 log-normal distributions to the experimental peaks by hand as described by Davies ( !974\.
5. CO~~PL.‘T.ATIO~A~.
SlElHOD
A mathematical algorithm has been derived which uses numerical methods to fit observed impactor data with up to three log-normal distributions, takmg into account the detailed impactor stage response functions Pi (Fig. I). An ad hoc search method IS used. which is analogous to fitting by hand. but allows for the arbitrary relation between the normal distribution and the corresponding stage weights. Central to the computation scheme IS a subroutlnc (IMPACT) which will evaluate. for any proposed size distributjon, the expected resulting impactor data. Thus the subroutine evaluates. mi =
x Q,iO!(l)di i‘ -I
where
uj is the amplitude, aj the (logarithmic) standard deviation and I, the mean of the j’th normal distribution. The stage response functions are input to the program initially; from these are calculated Q,(l) at 90 equally spaced points. The m,‘s are then evaluated by trapezoidal integration wtth the tails integrated analytically.
The fitting procedure uses the fact that the normal distribution to be fitted at any stage has a single peak. It assumes that the functions Q,. while far from ideal, do not change the general character of this shape. Also. superposition of distributions is linear, i.e. the data resulting from the sum of two distributions are the sum of the data corresponding to each separately.
Data inversion for cascade impactors: titting sums of log-normal distributions
bwodynomlc
diameter.
1713
firn
Fig. 4. A size distribution with two normal distributions and the corresponding exact simulated data points. (The stage loadings plotted in this and the following figures are normaiised by w,.) The fitted distribution and stage loadings are indistinguishable from the originals on this plot.
It starts by placing the mean, I,, of a normal distribution at a diameter corresponding to the largest (corrected) stage loading m) = miJwi. Always choosing the amplitude a, to give a correct fit of this point. the standard deviation, G,, of the distribution is varied to find the widest distribution such that no measured value is exceeded (apart from an allowance for measurement error). This ‘widest’ criterion, subject to fitting the middle point, is equivalent to choosing the largest possible amplitude a,, where a, is equal to the area under the curve. Thus it represents an attempt to account for as much of the measured material as possible with the current normal distribution. Since the best mean for the distribution is unlikely to be at the point first tried, the above procedure is repeated for other values of the mean, using a search method with the same criterion of largest a,. If the first fitted normal distribution does not account for all the material collected by the impactor, the corresponding stage loadings are subtracted from the measured values and another distribution is fitted to the remainder. This is repeated for a third distribution, if necessary. It is possible for the first distribution fitted to be too wide but still consistent with the data; this can happen because the other distributions are contributing material on either side of the first which is wrongly interpreted as belonging entirely to the first distribution. A problem then arises when the program tries to fit the subsequent distributions and finds no material left. after subtraction of the first, in the regions of overlap between the peaks. To avoid the ‘squeezing out’ of the second and third distributions in this way an allowance is made for the data to be exceeded in the regions of overlap initially. The distributions are therefore fitted twice more each, cyclically, with the others subtracted from the data,
while the allowance for excess overlap is progressively reduced. As described above, the main fitting criterion is not a least square or a minimax criterion, but one which maximises the amount of collected material accounted for by the normal distribution being fitted. This ensures that the measurements are fitted by as few normal distributions as possible and obviates spurious peaks in the overall size distribution. The root mean square deviation from the measured data is calculated and used as an indicator of the quality of the final fit. It is not dear under what conditions, ifany, it should be possible to fit the nine data points exactly with the nine-parameter size distribution. This would depend on the form of the functions Qi or P,. Certainly some parts of the nine-dimensional space of measurements are excluded, i.e. it is possible to specify inconsistent data. To see this, take the extreme example of a finite measurement in stage 2, say, with zero for the adjacent stages 1 and 3. Even for monodisperse aerosol of diameter corresponding to the peak of Qt(i) (Fig. 2), significant amounts are collected on stages 1 and 3. There is no size of particles which reach stage 2 without a significant proportion of them being collected on stages 1 and 3. It is possible to make a preliminary check in the program for inconsistent data of this type.
6. TRIAL RESULTS 6.1. ~irnu~ared data Simulated data can easily be generated for any given combination of log-normal distributions using the subroutine IMPACT. Such data was input to the program for one, two and three distributions, with zero and 10 y., mean simulated errors. The latter were produced by multiplying each data value by a random
number from a distrtbution uniform betu;un 0.3 and 1.2. The charactertsttcs for a nlrer pavr impaction surface and solid particles were used. For exact data. u~th one normal distrtbutton. the program appeared capable of reproducing the input distribution parameters to any destred degree ot accuracy. given enough iterations. Similarly. with two normal distributions the program had no dtfticulty in reducing the r.m.s. dilference between stage loadings corresponding to the titted distributions and the input stage loadmgs to 0.0003. where the total mass was normahsed to I. The results are plotted in Fig. 1. Sotr: that a good fit 1s Indicated by the closeness of the input and fitted distributions, or the mput and titted data points. not by coincidence of the curve and points. The plotted points are in fast corrected by the method described in Section Ldi~tston by ti,. Without this. the stage loadings would bear less resemblance to the dtstribution curves, and could not strictly be plotted
Wtth three mput distributions the limits are reached. nine data points being fttted by a nine-parameter distribution. Again the fit in the simulation was generally good. provided that the input normal distributions were well spaced. but sometimes one normal distribution would be fitted in place of two if the latter were fairly close together. Specifically. tn terms of lop diameter, using standard deviations ofO.2 ia geometric standard deviation of 1.6). this problem was likely to arise if the difference between the means was less than 0.5 (a diameter ratio of approximately 3). When simulated errors are included, it is likely that the changes in the data will be such that there exist normal distributions, slightly different from the input distributions, which fit the erroneous data better. The program may find such distributions. In this case it is clear that the errors in the distributions found are due
I5
-b’
o
5
IO the errors in the data and better :esuit> .L:::no: ?e recovered irom such data The progran: I\ :hcc behaving satisfactorily ICI: rind, dlsti!butt<)n> ‘.rDIG>iir the data at least a~ well 2s the input cistribu:!\,ns Ii: fast. when simulated errors uere Incluki. fo: ’ !x. IWO tir three ~npur normal distributions. :h~ V~:ogram generally was able 10 find distributions which litted the data better than the exxt Input distributmm Typxai results are shown in FIN. 5. uhlch ~r.cludcs th: exact data. that with error added. ‘rnd stage loadings curresponding to the titted distribution. These results show that I: is important not to place too much reliance on every detatl of the results: closespacing of distributions or e:rors in the data can lead to marked changes in the form oi the solutions. However. the results are at least consistent *ith the actual characteristics of the tmpactor and so better than p!ots of stage mass against mass median diameter The) can strongly suggest the presence of peaks in the particle size distributions; proof of the cytstence of such peaks may require an impactor with man! more stages.
The program has been used to analyse the data from several runs of an Andersen impactor with sticky stages in urban Toronto. followed by multielemental analysis of the deposits. It was able to tit the data with two or three log-normal distributions. except m three cases where the difficulties are believed to be due to error in the chemical analysis. Some results arc shown in Figs 6 and ?. Certain components of the aerosol seemed to be common to many elements, as shown by similar normal distributions being fitted. for instance the small-particle peak tat 0.9~m diameter) for vanadium, manganese and aluminium in Fig. 7. Dofferences in origin could also be made clear: in contrast to the other elements plotted. bromine’s main source is known to be the exhaust from motor vehicles using
r
1 1
31
01
IO
Fig. 5. Fitting simulated data from three normal distributions. ----input distribution; a corresponding exact data; l data with simulated errors; - -- - fitted distribution; o corresponding stage loadings. The upper two normal distributions are too close to be resolved. The fitted distribution IS a better fit IO the data IwIth errors) than the Input distnbution.
Data inversion for cascade impactors: fitting sums of log-normal distributions
Aertiynomic
drometer,
1715
pm
Fig. 6. Vanadium in an atmospheric aerosol sample. e observed stage loadings; ~ fitted distribution; o corresponding stage loadings (where different from observed).
i
i
! i
1 i
Aerodynomlc
dmmeter,
gm
Fig. 7. Fitted mass size distributions for four elements in an atmospheric aerosol sample.
leaded fuel, and it is clear that the size distribution of these particles is different. A problem is caused by the fact that, at the Large diameter end, the size distribution of soil-derived particles continues to sizes larger than can be detected by the impactor. Above 15 pm, wall losses are large and increase with particle size. So it is possible to fit wider and wider distributions which are consistent with the data but most of whose mass is in particle sizes too large to reach stage 0. It is therefore necessary to place
an arbitrary limit on the mean of the largest particle distribution. At the small particle end of the size spectrum, the same di~culty does not arise since the backup filter is a reasonably emcient collector for all particle diameters. A definite upper limit is therefore provided for the mass contained in the lower tail of the distribution. One way to obtain useful results for the largest particle distributions is to perform the fitting procedure for an element which is entirely soil-derived (an
enrichment factor of one’!. usmg the artirary upper limit on normaI distribution mean. The mean and standard deviation obtamed are then assumed to be the same for one of the normal distributions for each of the other elements, and so these are tixed. allowing the program only to find the appropriats amplitude. This provides information on the relative proportions of each element in the soil-derived component provided there is no other source of such large particles. The same trick can be used for an element which only has a small proportion of its mass m a particular component, when the data for that ciement cannot specify accurately the parameters of the appropriate normal distribut!on. The mean and standard deviation can be obtained from the tit of an element which is concentrated in that component; and then these parameters can be specified and again only the amplitude needs to be determined.
7. SCJl>IARY
ASD CONCLUSIONS
Since the collection characteristics of cascade impactors are far from ideal, the measured stage loadings are not related in a simple way to particle mass-size distributions. A numerical technique is necessary IO solve the inversion problem. Also, since impactors used in ambient air sampling provide so few data points to describe thecomplex sizedistributions usually present, assumptions have to be made about the shape of the distributions for individualcomponents oftheaerosol. The use of sums of log-normal distributions has some support from theory and observation, and is practically convenient. The algorithm described has been tested with simulated data using real impactor characteristics. With exact data from one or two normal distributions, the input parameters could be reproduced with high accuracy. For data with realistic simulated errors, the fit was generally as close as the errors would allow. The results reveal some of the limitations inherent in data from an impactor with only eight stages. Using field data obtained by multielemental analysis, the method revealed similarities between the .-___ l The enrichment factor E for an element .A is defined by relation to a referenceelement R which is known to beentirely soil derived:
concentration E= concentration
of
A in air :concentration --___ of R in air i concentratton
of A in soil of R in soil’
distributions obtained for a number oCslements. wh~sh were interpreted as sug~estmg the presenzc of the elements in the same ‘component‘ ot’the ~tmosphsr:~ aerosol. .Atmospheric parrl gcn‘lameters Ibo\e rhr: range erally continue to particl, * C. of an impactor, but the suggested procedure should ~1t least reveal the relative abumiance oivarious elcmentj in the very large particle component. ;I’ the mean and standard deviation of the appropriate normal distrlbution are fixed.
Cooper D. W. and Spielmx L .A. I 13-5) 1ht.c ~n~crs~on using nonlinear programmtcs 561th pS.\JIcaI consrralnrs: aerosol size distribution mcdsurement ok :mp.wori -I!-
mosphrric
EnGvnnwnr
9. -2: -~!O
Davies C. N. (1974) SIX C!xrtbutloc 01 .lt::w,pherlc particles. 1. ilercrsol Ser. I. _‘33 -300. Jaenicke R. and Davies C > I lYl61 The mathrmatlsltl expression ofthe sizedrstrtbur:on o,l‘a:mospheric pJr:tclei J. .-liwsoi Sri. 7, 255 - 260 >lcCain J. D.. Clinard G.. Fell\ L. G ,nd Jahnwn J t 197Y! A data reduction system for cascade Impactors Prow~dinys of lhe S,vnrposiunz on Ad: ..xws ~1 PGr:lc-l,n Surf~pirnq md .I~ra.wrcwcvr. Asheville. Sorth Caro1:r.J. ?ilcCain J. D.. McCormak J. F lnd Harrl,: D. H. ( 137?)SonIdeal brhaviour in cascade impacrorj Paper ---j5.; :,t A.P.C.A. 70th Annual Meetmg. Toronto. Patterson E. M. and Gillette D A. (19771 Common.dl!tes In measured size distributions :jr xrosr)!a hak:ng ;L jotIderived component. J. yw;;;;j. R,T 82. 207-I -2082. Puttock J. S. and Barrie L. .-\ ~13Plt En\lronment Canada. A tmosphcric Environmen: Scr\ IC‘C. report -2 RQT I in preparatibn). Raabc 0. G. (1978) .A gc’rx.~i method ror hrring SIX dlstributmns to multicomponent aero& &ta using \rcl_ghted least-squares. Enrir. S,i Tt&w/ 12. I 162 --I I67 Rahn K. A (1976) Silicon .~nd alumln:um !n atmospheric aerosols: crust-air fraciinn.ltion’l .-1:nw&wr*; I:;,! wrw~iw fO, 597-602. Rao A. K. and Whitby K. T rl37-t SonIdeal zllectton characteristics of single stage Jnci cascade impuctclrc. .4~t. Intl. H~J. Ars. J. 38, I74 - ! -3 Rao A. K. and Whitby K T I 137S1 \onldeal collection lmpactori II Cascade characteristics of inertI impactors. J. .4rrosol Xc: 9. 9- IO0 SundelXL. ( 1967) On the accurJtc ca:cula::on oipar:lcle sw distributions in aerosols i:om Impa;tlon data .Srclub Rriniwir Lz(ir (in English) 27. 22-X Taomry S. ( 1977) Inrro&~rc!w ‘,I ill tl;:itw:~r~i~ i trf 1m(‘I’sion in Rt*mor<~ rrnd /d:~. ’ Ilr.c~>?rr;m<,~i~\ Elsa icr. Amsterdam. Whitby K. T.. Husar R. B. 2nd LIU 3. i. 1-I (i9-2) The aerosol size distribution oi Los .-\ngcles smog. In .-l~wxcw? and .A~nrosplt~~ric Chemrstr: (Edlred b! G \l. Hid) I. pp. 237-264. Academic Press. \CU \I’orl
Pt;TTo&
Atmospheric Environment Service, Environment Canada, 4905 Dufferin Street. Downsview. Ontario. Canada (Firsr receiced 24 September 1979 and in j%ralform 4 February 198 I)
Abstrsct-The processing of data from cascade impactors is discussed and the effects of non-ideal response characteristics on impactor measurements are shown. Since atmospheric particle size dis~~buti~ns may contain two or three modes, the sparse data from an impactor cannot give detailed information about the shape of individual peaks. Thus a shape must be assumed, and it is suggested that a log-normal form is appropriate. A procedure is described which fits sums of log-normal distributions to impactor data, taking account of the detailed random errors, where atmosphe-ic data.
response functions. This procedure has been tested with simulated data including it is found to reproduce the input distributions satisfactorily. and with actual
The cascade impactor has become a standard instrument for the measurement of particle size distributions, particularly in ambient air sampling where multi-elemental analysis of the material collected on each impactor stage can give information on the distribution of various pollutants among the particle sizes. However, there remains the probtem of data inversion, i.e. the inference of the particle size distribution function from the measured stage loadings. This is a problem for two reasons. Firstly, the collection characteristics of existing impactors are far from ideal (Rao and Whitby, 1977, 1978). Secondly, an impactor usually provides too few size separations (nine for the Andersen ambient impactor) to resolve accurately the complex size dist~butions occurring in the atmosphere. The latter di~culty forces us to make assumptions about the shape of the size distributions for the various components of the atmospheric particulate matter. In this paper a computational scheme is described which uses impactor stage collection characteristics to convert stage loadings to size distributions.
2. IMPACfOR CRARACTERISTICS
An ideal impactor stage would collect all particles above a certain size, and none below that size, i.e. defining an efficiency function for stage i, Pi (d), as the proportion of particles of aerodynamic diameter d reaching the stage which are collected by it:
The actual functions Pi have been measured for the l Present address: Thornton Research Centre, P.O. Box I, Chester, England.
‘F
‘5
j.,,.
Andersen ambient impactor by Rao and Whitby (1978). These are shown in Fig. 1. It can be seen that they are significantly different from the ideal step functions. The curves for ‘sticky’ particles on a smooth surface are closest to the ideal, while for solid particles on solid impaction surfaces ‘bounce-off’ changes the curves considerably. With a filter paper surface the slope of the functions is reduced further, presumably owing to a filtration effect in the horizontal flow over the surface. It is useful to derive the total impactor response functions Q,(d) from the Pi’s, where Q,(d) is the proportion of all the particles of aerodynamic diameter d reaching the impactor which are collected by stage i. These are related to the P,‘s by
Q&l = P,(d) i-
1
Q)(d)= Pi(d) X 17 [ I- P,(d)] j=O
= P,(d) x
[l-:
These functions, derived from Rao and Whitby’s data for the case of liquid particies, are plotted in Fig. 2. For the ideal impactor the response functions would be ‘top-hat’ functions, equal to one for dio.i < d
measurements
The non-ideal nature of the characteristics has an important influence on the apparent size distributions measured by the impactor. Distortions are introduced both by the unequal spacing of the P curves and by their finite slope. It is convenient to change to a logarithmic particle size variable I = log,,d/d,, where d, is a scaling length, say 1 cm. The expected stage
1709
Fig. 1. Stage efficiency functions P,(d) for an Anderxn impactor using:--liquid particles; --- solid particles and uncoated stainkss plates; -.-.solid particles and a glass fibre filter collection surface. (From Rao and Whitby. 1978.)
Aerodynamic
dtameter,
pm
Fig. 2. Total impactor response functions Q,(d) for an Andersen impactor using “sticky” particles or collection surfaces. In deriving these, the stage efficienciesfrom Fig. 1, with liquid particles, were used for stages 2-6; stages 0 and I and stage 7 were assumed 10 have similar eficiency functions to stage 2 and stage 6; the wall loss is an approximate representation.
loadings can then be calculated from the impactor characteristics by mi
-=
'MO where
_~ J‘ 3c
Q,(l)f(l)dL
f= kdG. 0
M is the mass size distribution of the sampled particles. and MO is their total mass. Using this, the consequences of the form of the impactor characteristics of Fig. 2 are demonstrated in Fig. 3. This shows the impactor data resulting from a mass size distribution uniform in [(dM/dl = constant, I, < I < le), and from a &function (monodisperse) distribution. The uniform distribution does not give a horizontal straight line in Fig. 3. because the stages have different ‘acceptance widths.’ If only the d,,‘s are known, then an approximate correction for this can be made by dividing each stage loading by
* L_,og,,!+ h,Od 1 d,,. i-
0
0
which gives approximate values for the mass size d&f A better approximation distribution d log (did,)’ would be to divide by the area under the i’th curve in Fig. 2, i.e. divide by Wi =
‘I Q,(l)dl. s -I
This would give a horizontal straight line for the uniform distribution data of Fig. 3. except for the first stage and the backup filter, for which wi is undefined. The latter correction, however, is exact only for a uniform distribution. Further inaccuracy arises from
* The 50’?,,cut-offd~ameterd,,,, , isdefincdasthcdiameter for which P,(d) = 0.5.
Data inversion for cascade impactors: fitting sums of log-normal disrributions
1711
. .
.
. *
.
I I
I
Filter
7
I 6
I 5
,
I 3
4
L
/ 2
I
1
1
0
!
wall iof5
Stage
Fig. 3. Impactor data, in terms of pro~rtion of total mass, for (a) a mass size distribution with d.tf/d(logd) uniform from 0.3 to 30gm, (b) a monodisperse aerosol of aerodynamic diameter 2.3 pm. l sticky particles or surfaces; A solid particles and uncoated surfaces; V solid particles and glass tibre filter collection surfaces.
the error illustrated by the d-function data in Fig. 3. Particles of a given diameter, d, can be collected by several stages, in fact ail stages for which Q,(d) is nonzero. This will make the distribution look wider than it is, and, if bounce-off is severe, it will produce spurious measurements on the small-particle stages. To correct this, it is necessary to take account of the detailed impactor response functions Qi in the inversion calculation.
3.
SOME PROPOSED
DATA REDUCTION
SCfiEMES
Inversion problems similar to that described here occur in many fields. A general theory is described in the book by Twomey (1977). A number of different numerical schemes have been proposed specifically for the inversion of impactor
data. The simplest method is to assume that the stage characteristics are ideal. McCain et ai. (1978) describe a computer program which uses this method, followed by smoothing, for in-stack cascade impactors. However, the inherent weakness of the method appears when two successive stages have identical d,,‘s. In this case Qi for the second of these stages is a small but significant peak and some material is collected, but with the ideal impactor assumption there should be nothing collected and so an infinite value is assigned to f(l) there. McCain er al. avoid this problem by ignoring the data for that stage. Similarly, with strong bounceoff the method cannot explain the large amounts reaching the back-up filter, and this measurement also has to be ignored (McCain et al., 1977). By contrast, SundelGf (1967) suggests deliberately using two identical stages consecutively to obtain Qi with a narrow particle size range. His numerical
scheme uses division b! ii, to give a tirst approximation to the value off’iii for a value oil which is assumed IO coincide with the maximum of Q,. Both ordinate and abscissa of each resulting point on the graph of!‘(i) are successively adjusted until the expected stage loadings. corresponding to a smooth curvej’(i) drawn through these points, are equal to the data. Cooper and Spielman (1976) also take account oi the detailed nature of the stage efficiency functions P,. They describe standard smoothing methods which have been proposed to suppress oscillations in the solution caused by ill-conditioning of the discretised equations used in the numerical quadrature. However, these methods fail and their proposed method uses nonlinear programming with a specific condition excluding negative solutions. The results show a good fit with exact simulated data. However, the method does not seem to work well on data with a simulated measurement error. The one case show-n in the paper has a mean random error of only 5”,,, but the solution shows some oscillatory tendency and compares unfavourably with the simple solution (assuming ideal response). Raabe’s (197R) approach for cascade impactor data involves assuming that the size distribution can be described by a single log-normal function, and using a weighted least squares procedure to find the best values for the three parameters involved. The method has the advantage of providing a z2 value to show how good a fit has been achieved. It appears to work very well when a single log-normal distribution is appropriate, e.g. for particles from a single source.
4.
SIZE DISTRIBL’TIOSS
IX THE ATMOSPHERE
To appreciate what is required of a data inversion scheme for ambient atmospheric particle samples, it is necessary to discuss the nature of particle size distributions in the atmosphere. It is generally accepted that mass size distributions have at least two maxima within the range of standard impactors (approx. 0. i-50 pm), with a smaIl-size component which is an accumulation mode of anthropogenic aerosols, the other component being mainly soii-derived. There is now evidence that the soil-based component has two peaks, leading to trimc !a1 distributions (Patterson and Gillette, 1977). This is confirmed by our own measurements analysed as described below (Puttock and Carrie. 1981). An explanation for the presence of two soil-derived peaks comes from Rahn (1976). who analyses particle size distributions in typical soil and the size selectivity of the suspension process. Given that the atmospheric sample may contain three modes, even the specification of just the peak position (or mean), width (or standard deviation) and amplitude of each mode requires nine parameters. The standard Andersen ambient impactor, commonly used for these measurements. provides just nine numbers: the mass on each of the eight stages and on the backup
hlter. It is clearly impossib!e :o obtain an) ;urthe: information about the shape ~4 the individu:tl mode distributions from such data. For this reason, it is necessary to make .; ~G~v: assumptions about the shape of the dtstribuiions for each component of the aerosol. Little 1s known I;! detatl about the shapes of these distributions. The most convenient forms to assume are log-normal fU~lCIlOIlS of particle diameter. i.e. normal distributions 111i Detailed measurements which support use of such functions for various components of the atmospheric aerosol are reported by Whitby t’r al. (1972). Theorettcai aspects of the use of tog-normal distributions for fitting particle size data have been discussed extensively by Jaenicke and Davies I 19761. One advantage 1s the simplicity of transformanon between volume. surface area and number distributions. In fdcr. if it were not for the problem of non-ideal unpacror characteristics, it would bean easy matter to ti: sums 01 log-normal distributions to the experimental peaks by hand as described by Davies ( !974\.
5. CO~~PL.‘T.ATIO~A~.
SlElHOD
A mathematical algorithm has been derived which uses numerical methods to fit observed impactor data with up to three log-normal distributions, takmg into account the detailed impactor stage response functions Pi (Fig. I). An ad hoc search method IS used. which is analogous to fitting by hand. but allows for the arbitrary relation between the normal distribution and the corresponding stage weights. Central to the computation scheme IS a subroutlnc (IMPACT) which will evaluate. for any proposed size distributjon, the expected resulting impactor data. Thus the subroutine evaluates. mi =
x Q,iO!(l)di i‘ -I
where
uj is the amplitude, aj the (logarithmic) standard deviation and I, the mean of the j’th normal distribution. The stage response functions are input to the program initially; from these are calculated Q,(l) at 90 equally spaced points. The m,‘s are then evaluated by trapezoidal integration wtth the tails integrated analytically.
The fitting procedure uses the fact that the normal distribution to be fitted at any stage has a single peak. It assumes that the functions Q,. while far from ideal, do not change the general character of this shape. Also. superposition of distributions is linear, i.e. the data resulting from the sum of two distributions are the sum of the data corresponding to each separately.
Data inversion for cascade impactors: titting sums of log-normal distributions
bwodynomlc
diameter.
1713
firn
Fig. 4. A size distribution with two normal distributions and the corresponding exact simulated data points. (The stage loadings plotted in this and the following figures are normaiised by w,.) The fitted distribution and stage loadings are indistinguishable from the originals on this plot.
It starts by placing the mean, I,, of a normal distribution at a diameter corresponding to the largest (corrected) stage loading m) = miJwi. Always choosing the amplitude a, to give a correct fit of this point. the standard deviation, G,, of the distribution is varied to find the widest distribution such that no measured value is exceeded (apart from an allowance for measurement error). This ‘widest’ criterion, subject to fitting the middle point, is equivalent to choosing the largest possible amplitude a,, where a, is equal to the area under the curve. Thus it represents an attempt to account for as much of the measured material as possible with the current normal distribution. Since the best mean for the distribution is unlikely to be at the point first tried, the above procedure is repeated for other values of the mean, using a search method with the same criterion of largest a,. If the first fitted normal distribution does not account for all the material collected by the impactor, the corresponding stage loadings are subtracted from the measured values and another distribution is fitted to the remainder. This is repeated for a third distribution, if necessary. It is possible for the first distribution fitted to be too wide but still consistent with the data; this can happen because the other distributions are contributing material on either side of the first which is wrongly interpreted as belonging entirely to the first distribution. A problem then arises when the program tries to fit the subsequent distributions and finds no material left. after subtraction of the first, in the regions of overlap between the peaks. To avoid the ‘squeezing out’ of the second and third distributions in this way an allowance is made for the data to be exceeded in the regions of overlap initially. The distributions are therefore fitted twice more each, cyclically, with the others subtracted from the data,
while the allowance for excess overlap is progressively reduced. As described above, the main fitting criterion is not a least square or a minimax criterion, but one which maximises the amount of collected material accounted for by the normal distribution being fitted. This ensures that the measurements are fitted by as few normal distributions as possible and obviates spurious peaks in the overall size distribution. The root mean square deviation from the measured data is calculated and used as an indicator of the quality of the final fit. It is not dear under what conditions, ifany, it should be possible to fit the nine data points exactly with the nine-parameter size distribution. This would depend on the form of the functions Qi or P,. Certainly some parts of the nine-dimensional space of measurements are excluded, i.e. it is possible to specify inconsistent data. To see this, take the extreme example of a finite measurement in stage 2, say, with zero for the adjacent stages 1 and 3. Even for monodisperse aerosol of diameter corresponding to the peak of Qt(i) (Fig. 2), significant amounts are collected on stages 1 and 3. There is no size of particles which reach stage 2 without a significant proportion of them being collected on stages 1 and 3. It is possible to make a preliminary check in the program for inconsistent data of this type.
6. TRIAL RESULTS 6.1. ~irnu~ared data Simulated data can easily be generated for any given combination of log-normal distributions using the subroutine IMPACT. Such data was input to the program for one, two and three distributions, with zero and 10 y., mean simulated errors. The latter were produced by multiplying each data value by a random
number from a distrtbution uniform betu;un 0.3 and 1.2. The charactertsttcs for a nlrer pavr impaction surface and solid particles were used. For exact data. u~th one normal distrtbutton. the program appeared capable of reproducing the input distribution parameters to any destred degree ot accuracy. given enough iterations. Similarly. with two normal distributions the program had no dtfticulty in reducing the r.m.s. dilference between stage loadings corresponding to the titted distributions and the input stage loadmgs to 0.0003. where the total mass was normahsed to I. The results are plotted in Fig. 1. Sotr: that a good fit 1s Indicated by the closeness of the input and fitted distributions, or the mput and titted data points. not by coincidence of the curve and points. The plotted points are in fast corrected by the method described in Section Ldi~tston by ti,. Without this. the stage loadings would bear less resemblance to the dtstribution curves, and could not strictly be plotted
Wtth three mput distributions the limits are reached. nine data points being fttted by a nine-parameter distribution. Again the fit in the simulation was generally good. provided that the input normal distributions were well spaced. but sometimes one normal distribution would be fitted in place of two if the latter were fairly close together. Specifically. tn terms of lop diameter, using standard deviations ofO.2 ia geometric standard deviation of 1.6). this problem was likely to arise if the difference between the means was less than 0.5 (a diameter ratio of approximately 3). When simulated errors are included, it is likely that the changes in the data will be such that there exist normal distributions, slightly different from the input distributions, which fit the erroneous data better. The program may find such distributions. In this case it is clear that the errors in the distributions found are due
I5
-b’
o
5
IO the errors in the data and better :esuit> .L:::no: ?e recovered irom such data The progran: I\ :hcc behaving satisfactorily ICI: rind, dlsti!butt<)n> ‘.rDIG>iir the data at least a~ well 2s the input cistribu:!\,ns Ii: fast. when simulated errors uere Incluki. fo: ’ !x. IWO tir three ~npur normal distributions. :h~ V~:ogram generally was able 10 find distributions which litted the data better than the exxt Input distributmm Typxai results are shown in FIN. 5. uhlch ~r.cludcs th: exact data. that with error added. ‘rnd stage loadings curresponding to the titted distribution. These results show that I: is important not to place too much reliance on every detatl of the results: closespacing of distributions or e:rors in the data can lead to marked changes in the form oi the solutions. However. the results are at least consistent *ith the actual characteristics of the tmpactor and so better than p!ots of stage mass against mass median diameter The) can strongly suggest the presence of peaks in the particle size distributions; proof of the cytstence of such peaks may require an impactor with man! more stages.
The program has been used to analyse the data from several runs of an Andersen impactor with sticky stages in urban Toronto. followed by multielemental analysis of the deposits. It was able to tit the data with two or three log-normal distributions. except m three cases where the difficulties are believed to be due to error in the chemical analysis. Some results arc shown in Figs 6 and ?. Certain components of the aerosol seemed to be common to many elements, as shown by similar normal distributions being fitted. for instance the small-particle peak tat 0.9~m diameter) for vanadium, manganese and aluminium in Fig. 7. Dofferences in origin could also be made clear: in contrast to the other elements plotted. bromine’s main source is known to be the exhaust from motor vehicles using
r
1 1
31
01
IO
Fig. 5. Fitting simulated data from three normal distributions. ----input distribution; a corresponding exact data; l data with simulated errors; - -- - fitted distribution; o corresponding stage loadings. The upper two normal distributions are too close to be resolved. The fitted distribution IS a better fit IO the data IwIth errors) than the Input distnbution.
Data inversion for cascade impactors: fitting sums of log-normal distributions
Aertiynomic
drometer,
1715
pm
Fig. 6. Vanadium in an atmospheric aerosol sample. e observed stage loadings; ~ fitted distribution; o corresponding stage loadings (where different from observed).
i
i
! i
1 i
Aerodynomlc
dmmeter,
gm
Fig. 7. Fitted mass size distributions for four elements in an atmospheric aerosol sample.
leaded fuel, and it is clear that the size distribution of these particles is different. A problem is caused by the fact that, at the Large diameter end, the size distribution of soil-derived particles continues to sizes larger than can be detected by the impactor. Above 15 pm, wall losses are large and increase with particle size. So it is possible to fit wider and wider distributions which are consistent with the data but most of whose mass is in particle sizes too large to reach stage 0. It is therefore necessary to place
an arbitrary limit on the mean of the largest particle distribution. At the small particle end of the size spectrum, the same di~culty does not arise since the backup filter is a reasonably emcient collector for all particle diameters. A definite upper limit is therefore provided for the mass contained in the lower tail of the distribution. One way to obtain useful results for the largest particle distributions is to perform the fitting procedure for an element which is entirely soil-derived (an
enrichment factor of one’!. usmg the artirary upper limit on normaI distribution mean. The mean and standard deviation obtamed are then assumed to be the same for one of the normal distributions for each of the other elements, and so these are tixed. allowing the program only to find the appropriats amplitude. This provides information on the relative proportions of each element in the soil-derived component provided there is no other source of such large particles. The same trick can be used for an element which only has a small proportion of its mass m a particular component, when the data for that ciement cannot specify accurately the parameters of the appropriate normal distribut!on. The mean and standard deviation can be obtained from the tit of an element which is concentrated in that component; and then these parameters can be specified and again only the amplitude needs to be determined.
7. SCJl>IARY
ASD CONCLUSIONS
Since the collection characteristics of cascade impactors are far from ideal, the measured stage loadings are not related in a simple way to particle mass-size distributions. A numerical technique is necessary IO solve the inversion problem. Also, since impactors used in ambient air sampling provide so few data points to describe thecomplex sizedistributions usually present, assumptions have to be made about the shape of the distributions for individualcomponents oftheaerosol. The use of sums of log-normal distributions has some support from theory and observation, and is practically convenient. The algorithm described has been tested with simulated data using real impactor characteristics. With exact data from one or two normal distributions, the input parameters could be reproduced with high accuracy. For data with realistic simulated errors, the fit was generally as close as the errors would allow. The results reveal some of the limitations inherent in data from an impactor with only eight stages. Using field data obtained by multielemental analysis, the method revealed similarities between the .-___ l The enrichment factor E for an element .A is defined by relation to a referenceelement R which is known to beentirely soil derived:
concentration E= concentration
of
A in air :concentration --___ of R in air i concentratton
of A in soil of R in soil’
distributions obtained for a number oCslements. wh~sh were interpreted as sug~estmg the presenzc of the elements in the same ‘component‘ ot’the ~tmosphsr:~ aerosol. .Atmospheric parrl gcn‘lameters Ibo\e rhr: range erally continue to particl, * C. of an impactor, but the suggested procedure should ~1t least reveal the relative abumiance oivarious elcmentj in the very large particle component. ;I’ the mean and standard deviation of the appropriate normal distrlbution are fixed.
Cooper D. W. and Spielmx L .A. I 13-5) 1ht.c ~n~crs~on using nonlinear programmtcs 561th pS.\JIcaI consrralnrs: aerosol size distribution mcdsurement ok :mp.wori -I!-
mosphrric
EnGvnnwnr
9. -2: -~!O
Davies C. N. (1974) SIX C!xrtbutloc 01 .lt::w,pherlc particles. 1. ilercrsol Ser. I. _‘33 -300. Jaenicke R. and Davies C > I lYl61 The mathrmatlsltl expression ofthe sizedrstrtbur:on o,l‘a:mospheric pJr:tclei J. .-liwsoi Sri. 7, 255 - 260 >lcCain J. D.. Clinard G.. Fell\ L. G ,nd Jahnwn J t 197Y! A data reduction system for cascade Impactors Prow~dinys of lhe S,vnrposiunz on Ad: ..xws ~1 PGr:lc-l,n Surf~pirnq md .I~ra.wrcwcvr. Asheville. Sorth Caro1:r.J. ?ilcCain J. D.. McCormak J. F lnd Harrl,: D. H. ( 137?)SonIdeal brhaviour in cascade impacrorj Paper ---j5.; :,t A.P.C.A. 70th Annual Meetmg. Toronto. Patterson E. M. and Gillette D A. (19771 Common.dl!tes In measured size distributions :jr xrosr)!a hak:ng ;L jotIderived component. J. yw;;;;j. R,T 82. 207-I -2082. Puttock J. S. and Barrie L. .-\ ~13Plt En\lronment Canada. A tmosphcric Environmen: Scr\ IC‘C. report -2 RQT I in preparatibn). Raabc 0. G. (1978) .A gc’rx.~i method ror hrring SIX dlstributmns to multicomponent aero& &ta using \rcl_ghted least-squares. Enrir. S,i Tt&w/ 12. I 162 --I I67 Rahn K. A (1976) Silicon .~nd alumln:um !n atmospheric aerosols: crust-air fraciinn.ltion’l .-1:nw&wr*; I:;,! wrw~iw fO, 597-602. Rao A. K. and Whitby K. T rl37-t SonIdeal zllectton characteristics of single stage Jnci cascade impuctclrc. .4~t. Intl. H~J. Ars. J. 38, I74 - ! -3 Rao A. K. and Whitby K T I 137S1 \onldeal collection lmpactori II Cascade characteristics of inertI impactors. J. .4rrosol Xc: 9. 9- IO0 SundelXL. ( 1967) On the accurJtc ca:cula::on oipar:lcle sw distributions in aerosols i:om Impa;tlon data .Srclub Rriniwir Lz(ir (in English) 27. 22-X Taomry S. ( 1977) Inrro&~rc!w ‘,I ill tl;:itw:~r~i~ i trf 1m(‘I’sion in Rt*mor<~ rrnd /d:~. ’ Ilr.c~>?rr;m<,~i~\ Elsa icr. Amsterdam. Whitby K. T.. Husar R. B. 2nd LIU 3. i. 1-I (i9-2) The aerosol size distribution oi Los .-\ngcles smog. In .-l~wxcw? and .A~nrosplt~~ric Chemrstr: (Edlred b! G \l. Hid) I. pp. 237-264. Academic Press. \CU \I’orl