Improvements to the spectral transparency methodfor determining particle-size distribution
A. Ya. Perelman and K. S. Shifrin
The previously developed spectral transparency method is improved in two ways: (a) A calculation scheme
is developed that enables the consideration of light dispersion within a substance. We abandon the assump-tion that m(P) = const and develop a method of calculating N(r) from y*(v) provided that the function m(v)is specified. (b) A new scheme of processing data on y*(v)is proposed that significantly reduces the oscilla-
tion in the answer and requires less initial experimental data.
1. Introduction
The polydisperse light extinction coefficient y*(v),within a medium containing spherical particles of dif-ferent radii r, is determined by the following relation:
y*(v) = | K(v,r,m)f (r)dr. (1)
The following notations are introduced here: the par-ticle distribution
f(r) = wrr2 N(r), (2)
for the minimum experimental information necessaryto determine N(r) with a preset accuracy. In this senseof substantial importance are the cases when Eq. (1)may be solved analytically. Such a possibility appearsif the factor K(v,r,m) may be approximately representedby the van de Hulst formula2
sinpo 1 - cospK(v,m) Q(p) = 2 - 4 p+4
, ~~~~~p . p2
where
y = p(v) = 47r[m() - l.
p = yr, (3)
the wave number P = X-1, the wavelength X, the relativeparticle's refraction coefficient m = m (v), the extinctionefficiency factor K(v,r,m), and the particle-size distri-bution density N(r). Equations (1) and (2) may beregarded as equal to the integral equation for densityN(r). It may be found, from the measured value of'y*(v) and from the efficiency factor K(v,r,m), to beknown from the theory of light diffraction by asphere.
In the general case, the factor K(v,rm) is determinedby Mie formulas. Its values for different v,rm are listedin the tables in Ref. 1. The use of tabulated values forK(v,r,m) inevitably requires application of numericalmethods of inverting Eq. (1). This leads to bulkyschemes of calculating N(r) and-what is most impor-tant-does not provide a general formulation required
In this approximation Eq. (1) has the form
y(v) = S| Q(yr)f(r)dr. (5)
One should distinguish between the function -y(v)given by formula (5) and y*(v) determined experi-mentally. Further, the function oy(v) will be referredto as the spectral transparency of the system. In sup-position (3) we have
Y(v) y*(v)-i i)
For the solution of the inverse problem, i.e., for tiedetermination of the distribution density N(r) fromintegral Eq. (5), the spectral transparency method(STM) was proposed.34 This method presupposedthat:
(a) The desired distribution density has a dispersion,i.e., there exists a finite limit
K. S. Shifrin is with U.S.S.R. Academy of Sciences, P. P. ShirshovInstitute of Oceanology, 193015-Leningrad, U.S.S.R. A. Ya. Perel-man is with S. M. Kirov Forest Academy.
(b) In the whole range of values v under consideration,the refraction coefficient of a substance may be con-sidered constant, equal to some mean value
m(v) = mO (8)
for all v.The first supposition is physically quite warranted.
It simply means that the disperse system has a finitesurface. The second one is, however, a far-reachingidealization. The fact is that, in order to apply theSTM, the coefficient y* (v) must be measured in a widerange of wave numbers 0 < v < , within which as a rulethe refraction coefficient m varies substantially.
A great number of numerical experiments performedby means of the STM5,6 and also results of experi-ments7 8 reveal that, if the main spectral range (Or) inwhich ty*(v) is measured is sufficiently wide [the firstmaximum of the y*(v) curve should be bypassed], theSTM produces good results both for single-modal andbimodal distributions. However, a number of cases
,show that the determined density N(r) contains falseoscillations, particularly in the range that is far from themode. The oscillation decreases if a, the number ofmeasurements, and the accuracy of measurements areincreased. The requirements for the experimental datain a number of cases proved hard to reach. The analysisshowed that these difficulties are associated with thegreat generality of the previously developed STM. TheSTM calculation scheme referred to any N(r) functions.Actually, the N(r) functions representing distributiondensities both in natural environments (atmosphere,ocean) and in technological problems are approximatelylinearly arranged and have comparatively simple forms.This allows representing the real dispersed system asa linear set of standards that make it simple to invert,y*(v). The false oscillations of the answer therewithdecrease significantly, and the requirements for theexperimental data become simpler.
In this paper the STM is improved in two importantdirections.
(a) A calculation scheme is developed that enablesconsidering the light dispersion within a substance.With the new scheme we abandon the assumption thatm (v) = const and point out a way of calculating N(r)from y*(v), provided that the function m(v) is speci-fied.
(b) A new scheme of processing data on y * (v) is pro-posed that significantly reduces the oscillation in theanswer and requires less initial experimental data.
11. Consideration of Light Dispersion Within aSubstance
The formal application of the Mellin convolution tointegral Eq. (5) gives3 4
f(r) f{-i-f (p + i)r,(p) [ y (v)y-Pdyjr-Pdp
rc(p) = r(p) cos2 (9)
where the internal integration should be accomplishedwithin the band 1 < Rep < 3 with subsequent analytical
continuation of the whole integrand function to theband -2 < Rep < 0 ( = Rep).9 Let us present integral(9) as the sum of two items-the first one correspondingto the main spectral range 0 < v < r and the second itemcorresponding to the short wavelength range v > r. Wehave
f(r) = fl(r) + /2(r), (10)
where
ft(r) = -- i (p+)rc(p)Ij(p)r-Pdp (-2 < a* <0).4ir2 i -i_
(11)
Here
I(p) = fl y(v)'(v)p-P(v)dv (Rep < 3),
I2(P) = y(v)p'(v)p-P(v)dv (Rep> 1).
(12)
(13)
As integral (12) is replaced by a quadrature formula, wefind
In turn the expression for f2(r) is determined by thechoice of extrapolations of the functions y(v) and m(v)in the short wavelength range. On the basis of Eq. (7)for the function y(v), it is natural to use an extrapolationof the form
Y(V) ` Co + C1V-1 + .. . + CKV-Kf (v > r). (17)
The results of measurements should provide satisfac-tory accuracy of representation (17) with certain K >0; specifically, for the application of the STM, a reliableestimation of the parameter c is needed. In turn, theinversion accuracy increases if it proves possible to es-timate the values of some other parameters of (17) aswell, for example, to obtain an extrapolation of theform
Y(V) co + civ'1 + C2V 2 ( > T). (18)
It was assumed3 4 that cl = 0 because, under condition(8) by Eqs. (3) and (5), we have y(-v) = y(v). In in-verting the experimental data this condition is violatedeven in case (8), because actually we have to extrapolatethe function y*(v) from Eq. (1) and not the function-y(v). Let us assume that the refraction coefficient re-mains constant only in the short wavelength range andvaries arbitrarily in the main spectral range, i.e., wesuppose that
m(v) = mO (v > lj. (19)
Obviously, this condition essentially generalizes con-dition (8). According to (18) and (19), Eq. (13) gives
Formulas (4), (16), (21), and (25) have been used here.Let us point out that, in the case of short wavelengthapproximations (8) and (18), we obtain
- nf(r) ~_--2(mo - ) Y A y(vj)vj[47r(mo - )vjr] + cTwoO(y~r)
tj=1
+ cico(y~r) + c2T-1c2(yrr)}- (28)
The function y(v) formally introduced by Eq. (5) cannotbe measured. So into inversion formulas (27) and (28)the values of the real polydisperse extinction coefficienty* (vi) should be substituted instead of numbers y (vj);in turn, the parameters co, c, and c2 should be esti-mated proceeding from the function y*(v) not thefunction y(v), as was the case in representation (18).
The numerical calculations for different aerosolmodels (with not too great mean aerosol sizes) show thatthe accuracy of equality (6) improves substantially ifEqs. (1) and (5) take account of the real relationshipbetween the refraction coefficient and the wavelength.1 0
According to this result it turned out that formula (27)gives a better inversion result than does formula (28) foraerosol with a mean radius of F < 3 Atm. This fact is il-lustrated by Fig. 1 representing the microstructuremodel corresponding to F - distribution (A being thenormalization factor):
f(r) = Ar4 exp(-37'1r), (29)
with F = 1 Itm (curve 1), and the results of inverting theoptical information with and without regard to lightdispersion within substance (curves 2 and 3, respec-tively).
Integral (13) may be calculated in terms of functions(23) and, in the case of y(v), extrapolated according toformula (17) with any K as well, if the data on the re-fraction coefficient admit a short wavelength approxi-mation of the form
m(v) 1 + m11V- + m2v 2 , m2 w 0 (v> r).
(a < 3).(30)
(24)
Specifically, we find
sinacoo(a) = cosa-2-+ 1
a
W1(a) = cosa -1 +f 1 -cosb db
W2(a) = cosa - 1.
Let us point out that
oa(a) = O(a2 )(a 0 0), wa(-a) = w.(a).
(25)
(26)
So if short wavelength approximations (18) and (19) areallowable, the solution of integral Eq. (5) is of theform
A known equality is taken into consideration here:
lim m(v) = 1. (31)
From (17), in compliance with Eqs. (4) and (7), wefind
(Rep < 1), (34)In Ref. 11 Eq. (43) was derived by means of the basicoptical theorem, which states the connection betweenthe extinction efficiency factor and the forward scat-tered field amplitude.21 2
Let us write integral Eq. (5) in the form(35)
Functions Ali (a) are easily calculated from Eq. (24).In particular, we have
3 31 1 3_a) =- sia +1 -2 cosa + + -a a2~~ 2 a
2
W-2(a) = +3 sina + 2 cosa +3
In the case of , = 2, the representation coefficientsare of the form
M2CO - mM2C1 + MIC2720 2
m2
3' Q(yr)f(r)dr = g(y),
where the spectral transparency
g(Y) = Y ()
(44)
(45)
(36) Integral Eq. (44). may be written as'1
3 f/(r) cosyrdr = h(y),
(35) where
h(y) =
(46)
(47)2co - yg'(y) - 2g(y)
4
m2CI - 2mlC2Y2i 2
47rm2C2 2
'Y167r2m2 (37)
Let us note the formula
f2(r) c~_, -2m2 [ - co(fr) + - w-,i(or) + 2 2(fr)
= , (38)
corresponding to representation (35) with K = 2, and
(39)
In conclusion of this item we point out that, for ex-trapolation (17) with K = 0, an arbitrary approximationof the refraction coefficient m(v) is admissible. Wehave
/2(r) - 2 p(-)0wo[k(r)r].27r
(40)
Ill. New Inversion Formula for the STM
A new convenient inversion formula for the STM maybe proposed, if there exists the function
v = (y), (41)
which is the inverse of the function y = p(v) determinedby Eq. (4). In practice this modification of the methodmay be used if, for m(v) > 1, there are no essential vio-lations of the condition
and the constant co is determined by formula (7).From formulas (43) and (44) it follows that
Ig(Y) = Y 3 (1 - t2)z(yt)dt,
where
z(t) = 2 rf(r) sintrdr.
(48)
(49)
Integral Eq. (48) is solved directly, and its solution maybe given in the form
Cy(t)dt yg'(y) + 2g(y)f o tdt 2 (50)
The integration of Eq. (49), in turn, yields
JO z(t)dt = 2 Jo f(r)(1 - cosyr)dr. (51)
Equations (46) and (47) follow from Eqs. (7), (50), and(51). So the solution of integral Eq. (5) can be repre-sented in the form
1 POf (r =-J h(y) cosrydy,
7r O(52)
where the function h(y) is determined by Eqs. (45), (47),and
(57) where P, (x) are Legendre polynomials referring to the[-1,1] interval, are orthogonal with the weight of y(c2
+ y 2 )- 2 in the (0,o) range, and, if
(58)
(59)
shows that Eq. (58) gives a correct solution of the statedproblems (44) and (56).
The experimental data on the polydisperse extinctioncoefficient y* (v) may be inverted directly according toEq.(52). For this purpose the data on -y*(v) should bepreviously approximated by a certain linear combina-tion of functions chosen from an appropriate completesystem of functions. A convenient way of such appli-cation of the new modification of the STM is describedin the following section.
F(y) = L an7r'(y),n=O
then
an = 2(2n + )c2 3 F(y)7r (y)y(C2 + y 2)- 2dy.
We shall consider approximations of the form
(65)
(66)
(67)g(y) g.(y) = al7r(y) (O <y < )1=0
and let the distribution f' (r) calculated from formula(52) correspond to approximation (67).
The spectral transparencies g(y) form a set of evenanalytical functions that satisfy the relations
g(y) = g(n) + Q(y- 2) (y _ )
g(y) = O(y2) (y 0).
(68)
(69)
For polynomials (64) only condition (69) is violated, andso it should additionally be required that
IV. Approximation of the Polydisperse ExtinctionCoefficient by Modified Legendre Polynomials
The results of measurements make it possible to de-termine the polydisperse extinction coefficient value
g*(Y) = .Y*[,P(Y)]. (60)
The spectral transparency g(y) is a theoretical functiondetermined by Eq. (44). In solving integral Eq. (44)instead of g(y), we actually have to use the approxi-mation
g(Y) g*(y), (61)
which by virtue of (6) and (45) often proves satisfactory.The choice of the approximating functions should beaccomplished on the basis of the spectral transparencymodels.
For a wide class of colloidal solutions, polymers,clouds, and fogs, particle distribution (2) may be rep-resented as
f(r) = r 2 exp(-,yr) ( > 0, y > 0). (62)
Equation (62) (accurate to a normalization factor)represents the r distribution of particles. The corre-sponding spectral transparency g(y) is equal to thepolynomial in the degrees of expression 4
(OA<y < ).
(70)n
_ al = 0.1=0
However, it is not obligatory to follow strictly condition(70), as the share of the vicinity y = 0 f the g(y) func-tion into the inversion formula is insignificant, and thiscondition may be treated as one of the criteria for ap-proximation and measurement accuracy.
The accuracy of approximation (67) will be estimatedby the quantity
= max gm = maxg(y). (71)y>O gm y>0
When searching for the best approximation (67), it isconvenient to vary simultaneously both parameters nand c. It turns out that, for typical distributions,quantities (71) should be compared for approximations(67) that correspond to pairs of n,c) values from theset
n = 1(1)10, c = 0.5(0.5)6.0. (72)
We shall call optimum those pairs (n,c) for whichquantity (71) takes on the least value at set (72). Ifthere are several pairs of the kind, one should choose theoptimum pair with the least n.
Assume that an optimum pair (n,c) is chosen andoptimum approximation (67) corresponds to it. On thebasis of Eq. (64) we obtain
(63)
So it is natural to approximate g(y) by means of a sys-tem of orthogonal polynomials that are dependent onargument (63). Let us give some results of such ap-proximations."",1 3
It is easy to check that modified Legendre polyno-mials
n y2-Igcn(y) = '1+ i2
1=0 ~ C (73)
where coefficients b are determined in terms of coef-ficients a,. According to Eqs. (47) and (73) we have
Fig. 2. Curves of spectral transparency: (1) accurate curve; (2) curvecorresponding to the nonoptimum pair (4,1); (3) curve corresponding
to the optimum pair (2,2).
-10.30 - ___'1 0.15 d \ .......... 3
0 1 2 3 4 5aFig. 3. Results of inversion: (1) accurate distribution; (2) distri-bution corresponding to the nonoptimum pair (4,1); (3) distribution
corresponding to the optimum pair (2,2).
The values of spectral transparency (82) (for y = 2,um 1) calculated accurately to 10-2 were used to con-struct approximations (67) at set (72) of pairs (n,c).(The factor N determines the scale on the axis of ordi-nates.) For each of these approximations the maximumdeviation of S' was found from Eqs. (71). It turned outthat, for the example under consideration, there arethree optimum pairs (2;2), (3;2.5), and (4;3), and thecorresponding error is ne 1%. Figure 2 representstransparency (82) and its approximations (67) corre-sponding to the optimum pair (2;2) and the nonopti-mum pair (4;1). Figure 3 shows distribution (81) andthe results of inverting the spectral transparency for itsindicated approximations. It follows from Fig. 3 that,for optimum pairs, the inversion error given by Eq. (80)is approximately equal to the approximation error givenby Eq. (71), provided that we ignore particles with smallradii. For example, for the optimum pairs (2;2) and(4;3), the error c (r) = c = 1%, if r > 0.3 rM(rM beingthe distribution density mode). A similar result is validalso for other optimum pairs. On the other hand, boththese errors increase significantly for nonoptimumpairs. Thus, for the pair (4;1), we have 6c 8% andEc(r) - 10%, if r > 0.3 rM.
In conclusion we point out that when an approxi-mation of type (63) is used, the modification of thespectral transparency method allows the determinationof particle distribution (2) with a higher accuracy thanallowed with the earlier scheme. 3 In particular, the newinversion method gives significantly smaller amplitudesof the false oscillations in the desired distribution. Allthe calculations according to the new scheme, includingthe search for optimum pairs (n,c), are easily done onan electronic computer.
g9/!
(75).0
0.5(76)
Integrals (76) are easily calculated, and as a result wefind that the following distribution
fA(r) = ctn(cr) exp(-cr) (77)
corresponds to approximation (67). Hence t(x) is apolynomial of degree n, dependent on coefficients an.Let us point out that from Eq. (77) the relation
lim fc(r) = 0
follows.The use of approximation (67) leads to the natural
extrapolation of the spectral transparency within theshort wavelength range: formula (67) obtained as aresult of processing experimental data on transparencyg(y) in the main spectral interval is analytically con-tinued into the range X - 0. The inversion scheme3
was essentially reduced to an independent descriptionof the main spectral interval and the short wavelengthrange. In particular, the increasing oscillations of dis-tribution (2) at large particles of radii r are the conse-quence of this artificial junction. According to Eq. (78),the modification of the STM considered here auto-matically provides the physically grounded behavior ofdistribution (2) when r» 1. Further, the new inversionscheme enables us to state the direct connection be-tween the parameters of approximation (67) and cor-responding distribution of particles of radii (77). Thisresult may be written as a system of sequential ap-proximations f' (r) (n = 1,2, .. .) to the exact distribu-tion f(r). For example, we have
5. K. S. Shifrin and A. Ya. Perelman, Opt. Spektrosk. 20,143 (1966)[Opt. Spectrosc. 20, 75 (1966)].
6. K. S. Shifrin and A. Ya. Perelman, Izv. Akad. Nauk SSSR Fiz.Atmos. Okeana 1, 964 (1965).
7. K. S. Shifrin and A. Ya. Perelman, Opt. Spektrosk. 20,692 (1966)[Opt. Spectrosc. 20, 386 (1966)].
8. A. S. Lagunov, L. P. Baivel, V. K. Litvinov, and V. K. Kariazova,Opt. Spektrosk. 43, 157 (1977) [Opt. Spectrosc. 43, 86 (1977)].
9. It is essential that the kernel of the equation depend on theproduct of the variables y and r. All the details that refer to in-verting Eq. (5) are given in Ref. 4.
10. The calculations have been performed by S. Todorova, a proba-tioner from Bulgaria, by means of the tables cited in Ref. 1. Inthese tables the dependence of m(v) is given numerically. If m(v)is known analytically, the calculations will be significantly sim-plified.
11. A. Ya. Perelman and K. S. Shifrin, Izv. Akad. Nauk SSSR Fiz.Atmos. Okeana 15, 66 (1979).
12. K. S. Shifrin, Light Scattering in Turbid-Media (GTTI, Moscow,1951; NASA, Washington, D.C., 1968).
13. K. S. Shifrin, A. Ya. Perelman, and V. M. Volgin, Opt. Spektrosk.47, 1147 (1979) [Opt. Spectrosc. 47, in press (1979)].
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