PUBLISHED IN THE "ADVANCES IN CHEMISTRY SERIES", Vol 227, 83-104 ACS, Washington. DC, 1990

LATEX PARTICLE SIZE DISTRIBUTION FROM TURBIDIMETRY

USING A COMBINATION OF REGULARIZATION TECHNIQUES

AND GENERALIZED CROSS VALIDATION

Guillermo E. Elicabe and Luis H. Garcia-Rubio*

Chemical Engineering Department

College of Engineering

University of South Florida

Tampa, Florida 33620

..

* to whom the correspon~enceshould be addressed.

ABSTRACT

Particle Size Distributions (PSD's) of latexes are estimated from

turbidimetric measurements. The estimation of the PSDls is accomplished

usin gaR e g u1a r i za t ion Techni que (RT). Reg u1ar i za t ion te chni que s

requiere the selection of a constraining parameter known as the

regularization parameter. In this work the regularization parameter is

calculated using the Generalized Cross Validation (GCV) technique. The

use of these complimentary techniques (RT and GCV) is demonstrated

through the simulated recovery of PSD's of polystyrene latexes. Unimodal

and bimodal PSD's of varying breadth and mean particle diameters have

been investigated. The results demonstrate that the combination of these

techniques yields adequate recoveries of the PSD's in almost every case.

The cases where the techniques fail have been identified and strategies

for subsequent recovery are discussed.

- 2 ­

1. Introducti on

When a suspension of spherical particles is illuminated with light

of different wavelengths, the resulting optical spectral extinction

(turbidity) contains information that, in principle, can be used to

estimate the Particle Size Distribution (PSD) of the suspended

particles. The recovery of the PSD from turbidity measurements falls

within the category of lIinverse problems" to which several techniques

have been applied with varying degrees of success U.-1.). In a recent

paper (6), a regularization technique has been successfuly applied to

the estimation of the PSD of polystyrene latexes. Regularization

techniques require the selection of a constraining parameter r known as

the regularization parameter. The selection of the regularization

parameter is critical for the adequate recovery of the PSD (&).

In this paper some of the available methods for the estimation of r

are briefly introduced. Particular emphasis has been placed on the

General Cross Validation (GCV) technique. This technique, in our

particular application, appears to be the most robust among the

techniques available.

In the next section the equations that relate the particle size and

the turbidity are shown and a discrete model for those equations is

described in detail. In section 3.1 the regularized solution of the

discrete model previously developed is introduced. In section 3.2 a

discussion about some of the techniques available for estimating the .. regUlarization parameter is given. In section 3.2.1 the GCY technique is

- :3 ­

revisited. Finally, in section 4 the results of simulated examples are

shown. In all the examples the regularized solution is used along with

the GCV technique to estimate a broad range of PSD's of polystyrene

latexes.

2. Absorption and Light Scattering of Spherical Suspended Particles

The loss of intensity experienced by a beam of electromagnetic

radiation in passing through a sample of suspended particles, recorded

as a function of the wavelength of the incident radiation, is known as

the turbidity spectrum. The turbidity (t), is related to the intensities

at two points separated a distance! by

1 1° L = 9. 1n (-1-) (1]

where: 1° is the intensity at the point where the electromagnetic

radiation enters the sample and it coincides with the intensity of the

source. I is the intensity at the point where the electromagnetic

radiation leaves the sample and it coincides with the intensity at the

detector. For a suspension of monodisperse isotropic spherical

particles, the turbidity can be related to the wavelength of the

incident radiation, the particle diameter (D) and the optical properties

of the suspension through Mie theory (1)

- 4 ­

[2J

where: N is the total number of particles per unit volume in the sample p

and Qext is the extinction efficiency which is function of: i) the real

and imaginary parts of the particle refractive index (nl and kl

respectively); ii) the refractive index of the suspensionmediumnz;

iii) the wavelength of the incident radiation in vacuo),o ; and iv) the

diameter of the spherical particles. Note that the refractive indexes

are, in general, functions of the wavelength.

Eq. [2J can be readily expressed in terms of the particle

concentration (ie; C = weight of particles per unit volume of

suspension)

3C t(>'o,O) = ZpO Qext(>.o,D) [3J

where p is the density of the particle. (For simplicity, the refractive

indexes have been omitted from the argument of Qext from Eq. [3)

onwards) .

If the sample is a mixture with a distribution of particle

diameters, and the PSD can be represented by a differential

distribution, the turbidity can be rewritten as

[4J ..

- 5 ­

where f(D) is such that

[ 5)

If f(D) is normalized with N , Eq. [4J becomes p

[b]

where now

J: f I dO 1 UJ(0) =

Similarly, the turbidity of a polYdisperse suspension, in terms of

concentration C, can be written as

Notice that, by defining:

11 2K(Ao,D) = ~ Qext(Ao,D) 0 [9J

Eq. [4J can be readily identified as a Fredholm integral equation of the ..

first kind, in which K(Ao,D) is the corresponding kernel.

The numerical sol ution to any of the above equations ([4J. [6] or

[8J) must be based on an appropriate discrete model. The solution to

such a model will result in estimates of the number of particles and of

the shape of the PSD. If the integrand in Eq. [4J is d;scretized into

(n-l) intervals, the integral can be approximated at a given wavelength

AO i with a sum,

n T· = E a.. f. [10J

1 lJ Jj=1

I::. ~where T· = T(>'o.) and f. f(D) .1 1 J

The details of the discretization procedure and the resulting

coefficients a.. are given in the Appendix. lJ

If the turbidity is evaluated at m wavelengths ),0i' = 1, •.. , m, Eq.

[4J can be written in matrix form.

1. = A f [ 11]

where

[ 12J

A = [ a .. } [13JlJ

[ 14]

- 7 ­

c

T indicates the transpose.

Eq. [llJ can be written as an equality if the quadrature error f.

introduced in the discretization is considered,

_l=Af+e:: [l~J - ~

Finally, with the addition of the measurement error e:: the discretet -m

equation for the representation of the experimental values of 1 (iej

Im)' can be written as

_1 = A f + e:: + e:: = A _f + E_ [16Jm - -c -m

3. Particle Size Distribution From Turbidity Measurements

3.1. The solution of Eg. [16] using regularization techniques

The discrete model developed in the previous section (ie; Eq.

[16)). transforms the problem of obtaining the PSD from turbidity

measurements into a linear algebraic problem, where n points of the PSD

can be estimated from m turbidity measurements (m has to be greater or

equal than n). If m = n. estimates of the PSD (fd) can be in principle

obtained by the direct inversion of Eq. [16J

[ 17)

- b ­

Alternatively, if m > n, the least squares solution of an overspecified

system of linear equations yields,

[18J

Although the above solutions appear to be straightforward, it is well

documented in the literature (Q-lf). that small errors (ie; quadrature

and/or experimental errors). result in 1arge errors in f d or f l s' The

amplification of the errors, occurs independently of the fact that the

inverses of A and (ATA) can be calculated exactly, and it is a direct

consequence of the near singularity of the matrix A (if m = n), or more

generally (if m > n) of its near incomplete rank. This behaviour can be

explained by the near linear dependence between the functions K(Ao.,D), from which the matrix A was obtained. Notice that in spite of the fact

that these functions depend on the optical properties of the system

under study, the pre-selected range of wavelengths and diameters, and

the number of points that it is desired to recover, a certain amount of

collinearity between some of the functions will be always present.

Adding the fact that at least a small experimental error is also always

present, it is possible to state that Eqs. [17J and [18j cannot give a

solution to the problem under study. However, by constraining the least

squares solution by means of a penalty function, it is possible to

obtain approximate useful solutions. This can be achieved by using all

of the prior information available regarding the PSD (ie; the lItrue" f

vector). For example, it is known that the values of f must be positive

- 9 ­

or zero, that there is an upper and a lower bound on the particle

diameters and that there exist a certain amount of correlation between

successive points on the distribution. If it is recalled that Eq. [18J

is the solution to the least squares problem

2mi n A f - lorn 1 [19J

f

where I-I indicates the modulus, and 11s has been replaced by f.

It is well known that prior information can be introduced by augmenti ng

Eg. [19J with (Q-~)

2 ... min {I A f - .lm 1 + y q(f) } [20j

f

where q(f) is a scalar function that measures the correlation or

smoothness of i , and y is a nonnegative parameter that can be varied in

order to emphasize more or less one of the terms of the objective

functional given by Eq. [20J. If y is set to 0 Eq. [20] reduces to Eq.

[19J, a solution which generally exhibits large oscillations. On the

other hand when y + w the minimization leads to a perfectly smooth

solution [judged by the measure of q(f)J but totally independent of the

values and therefore, useless. It is apparent that intermediate -m

values of y will produce acceptable solutions to the original problem ...

(ie; Eq. [16J) and that those solutions will have the smoothness or

- 10 ­

1

- - - -- -- - - - - - -

• - - • • •

correlation characteristics imposed by the term q(f) in the functional.

Also, for bounded (fTf ), it has been demonstrated that, although the

estimates of f obtained through the solution of Eq. [20J will be biased,

there exists a value of y > 0 such that (11)

[ 21J

where E[-] indicates expected value.

In other words, the error in the estimation of f associated with the

solution of Eq. [20J will be smaller than that associated with the

solution of Eq. [19]. It is necessary, however, to select an appropriate

form for the function q(f) and an adequate value for the parameter y.

Several functions can be chosen to establish the desired

correlation level or the smoothness of f. An interesting class of

functions are those that can be formulated using a quadratic form of the

vector f because they yield an analytical solution to the minimization

problem of Eq. [20]. For example, if q(f) =- iT i, Eq. [20J can be

readily identified with the well known Ridge Regression (1). A more

interesting example in which q(f) = fTKTKi with

a a U 1 -1 0 • - - 0 a 1 -1 0 0

K = [22J

-0 • • 1 -1 0 .. a • • °• 0 1 -1

- 11 ­

• • • • • • •

gives the following q(f)

n 2A

q(f) = I: (f. f. [23]J J- 1)

j=2

which is a typical measure of smoothness. Another less restrictive q(f)

is given by the sum of the squares of the second differences

n-1 q{.f) = L (2: f. [24J

j=2 J

In this case the matrix H = KTK is given by

1 -2 1 0 • • 0 -2 5 -4 1 0 • 0

1 -4 6 -4 1 • 0 H = [25J

a • 1 -4 6 -4 1 0 • a 1 -4 5 -2 0 • • 0 1 -2 1

It will be shown later, that unimodal and bimodal latex

distributions, can be readily analysed using this last quadratic form

with a slight modification that constrains the values of f and f to1 n

be 0, thus incorporating into the solution additional prior knowledge

that had been imposed during the derivation of the discrete model. This

last constraint can be imp1emented by summing a2, with a » 1, to the

(1,1) and (n,n) elements of the H matrix shown above. In this form the

final quadratic form of the q(.f) function for the examples of the next

sections will be

- 12 ­

f.J- 1 [ 26J

Having arrived to an explicit expression for H, it can be shown

that the solution to the constrained problem of Eq. [20J is given by

(11) :

[27]

The value of f obtained using Eg. [27J will be called the regularized

solution of Eq. [16J. Notice that if we use the matrix H of Eq. [25J

with the modification that permits the constrain f f n = 0, (AlA + y1 =

H) is a positive definite symmetric matrix. Thus, it is possible to use

efficient algorithms to perform the inverse.

3.2. Some results about the selection of y

The regularized solution of Eq. [27J requires the selection of the

regularization parameter y. The existing methods for selecting y can be

roughly divided in two; those stemming from applications in physics and

engineering and those developed in statistics.

Among the first, Towmey's analysis of information content (11) has

been applied mostly in atmospheric sciences. The idea of lowmey1s method

is to detect the number of independent pieces of information available

in a set of experimental measurements. This analysis leads to a

- 13 ­

regularization parameter y that also depends on an estimation of the

root mean square value of the measurement noise. Using a completely

different approach, Provencher (15) proposed a method for selecting y in

the problem of inverting the Fredholm integral equation that arises in

the determination of molecular weight distributions (MWD) of polYmers

using photon correlation spectroscopy. Provencher's method ;s analogous

to the standard procedure of constructing confidence regions for the

sought solution. Although the method is rather arbitrary. the results

obtained for the estimation of the MWD were satisfactory.

The methods for the selection of y based on statistics theory were

developed for the so called Ridge Regression (RR). As it was saia

before, RR is a special case of Eq. [27J in which H is the identity

matrix (I). Notice that by defining

x = A K- 1 [ 28J

and

f' = Kf (29J

then

I' = (XTX + 'YI}-lXTIm (30J

Therefore, the regularized solution of Eq. [27J can be seen as a RR, and

the methods specifically developed for estimating 'Y in Eq. [30) can be

directly applied to Eg. [27J. The statistical methods for the estimation

of'Y may be divided into two; those that use a priori information and .. .'those that only use the measured data. Among the first, the method of

- 14 ­

Hoerl et a1 U...1.l.§.) and the closed form solution for the iterative

method described in (16) given by Hemmerle in (17) can be cited. In

these cases an estimation of the variance of the noise (0 2) and an

initial estimate of the solution are needed. Another method, that uses

2only an estimate of 0 , is known as the "range risk" estimate and it is

bri efl y out1i ned in reference (~). Accardi ng to Go 1ub et a1 (18), the

only methods available for estimating y from the data are Maximum

Likelihood, Ordinary Cross Validation (OCV) and Generalized Cross

Validation (GCV). This last three methods do not require any a priori

information and therefore they can be used to completely automatize the

PSD estimation process. Simulated studies (l§.,19), have shown that the

GCV technique is the most reliable and the best theoretically founded

among those using only the measured data. This assertion justifies, in

principle, its selection as a method for estimating y in the context of

PSD estimation from turbidimetric data.

3.2.1. The generalized cross validation technique

The GCV technique is a rotation-invariant version of OCV. This last A ( k)

technique may be derived as follows: define f'(y) as the estimation of

f' = K fusing Eq. [30J with the kth value of.l omitted. The argumentm~(k) th

is that if y is adequate, then [Xfl(y)]k (the k component of the

~(k) h [Xfl (y)J vector) should be a good predictor of 1 (the kt component of mk

- 15 ­

l ). In order to obtain good predictors for all the measurements (k = 1,III

... ,m), y should be chosen as the minimizer of

1 m - (k) 2 Ph) = - I {[ Xi I (-y) ] k - t m } (31j

m k=l k

This function can be expressed as (18):

1 p(y) = - IB(y) [I - Z(y)]

-lm l

2 l32j

m

where: B(y) is a diagonal matrix with jj entry {l/[l - z h)J} being jj

z (y) the jj entry of Z(y) = X(XTX + YI)-lXT. jj

Although the idea developed above seems to be appealing, it was

pointed out by Golub et al (18) that this method fails when the matrix

Z(y) is diagonal because P(y) does not have a unique minimizer, This

behaviour indicates tHat the oev is not expected to perform successfully

in the near diagonal case either. In order to circumvent this difficulty

the GCY technique was introduced as a rotation-invariant form of DeV

(18), The GCV function of y can be defined as the oev function (Eq.

[32]) applied to the following transformed model

I. wuT .lm = WDVT_f l + WUT_E = X fl + WUT_E [33]=

..where U and V are the result of the singular value decomposition of X

- 16 ­

Tx = U D V [34J

and Wis a complex matrix whose elements are:

1 2iT i j kim w.. = - e j, k = 1, 2, .•• ,m [35JlJ lin

where /1 = -1.

Therefore, using the transformed model in Eq. [32J results

1 P(y) = IB{y) [I - Z(y)} il 2

[36J m

Using the fact that as a result of the transformation, Z(y) is a

circulant matrix and hence constant down the diagonals, the last

equation can be expressed as

1[1 - Z(y)J i!2 = V(y) = m ----------::­ [37J

{Trace [I - Z(y)J]2

It can also be shown that

m L

I[I - Z(Y)J 1. I 2 i=1 mv{y) = m -- = m-------- [38) {Trace [I - Z(y)]} 2 n y 2

[.L A.+ y + m-n]1= 1 1

- 17 ­

where .t.. = [Zl' ... , ZmJT= UT.I. and Ai' i = 1, ... ,n are them eigenvalues of (XTX).

4. Recovery of the Particle Size Distribution for Polystyrene Latexes

In this section, Eqs. [27J and [38J will be used to estimate the

PSD's of polystyrene latexes. Using the measurements and the model, Eq.

[38J will be minimized with respect to y. The value of y that minimizes

Eq. [38J will be then used in Eq. [27J to estimate the PSDls.

The turbidity spectra were simulated using the results of section

2. For the simulated experiments, the refractive index of water was

calculated from (20)

3046 n2 = 1.324 + ---2--­ [39J

). 0

with Ao' given in nm. The real and the imaginary parts of the complex

refractive index for polystyrene were obtained from the data of Inagaki

et al (21). Figure 1 shows the optical properties for polystyrene and

for water as functions of the wavelength. Figure 2 shows the

distributions analyzed. Notice that a broad range of possibi lities is

being considered, including bimodal and very narrow distributions. The

mathematical expressions for those distributions are shown in Table I.

together with the values for the leading parameters that characterize

them.

- 18 ­

The simulated turbidity spectra were calculated using a 51 point

discretization of the distributions shown in Table I. The range of

wavelengths was chosen between 200 and 900 nm with a resolution of 1 nm

that results in a value of m = 701. A 3% maximum value random noise

(relative to the maximum turbidity value) was added to the simulated

spectra in order to represent extreme measurement conditions (the actual

instrument noise is less than 0.01 absorption units). The use of an

exaggerated noise level demonstrates that the technique would be robust

for typical measurement errors. The simulated spectra with the added

noise constitute the experimental data.

The number of recovered points on the distributions was n = 51 in

all cases, and the ranges of diameters varied according to the case

being analyzed (see Figure 2). The value of ~ was chosen as 1000.

In order to draw the most general conclusions, a Monte Carlo type

experiment was carried out. Each spectrum was replicated five times

keeping the statistics of the noise constant.

An optimal objective function was defined in the following manner

[ 40J

The value of y that minimizes Eq. [40J would give the best solution in

the context of the regularization technique utilized in this work.

Unfortunately, this objecti ve function can not be evaluated in a real

•situation because it depends on the unknown value of f. However, it

- 19 ­

permits to examine. in a simulated experiment, the performance of Eq.

[38) as estimator of y.

For each case and replication Eqs. [38J and [40J were minimized.

The values of r that minimize those equations, Y and YGCV opt

respectively. are shown in Table II. The last two columns of that table

show the value of the optimal objective function (Eq. [40J) evaluated at

Y and This permits to jUdge how good is the solution obtainedopt YGCV '

with the value of y that minimizes the GCV objective function (Eq. [38])

with respect to that obtained using r ' opt Figures 3 and 4 show Eqs. [38J and [40J as functions of Y in 109­

log plots for cases Band F. The values of Y range from the 37th to the

51th eigenvalue of the corresponding (ATA) matrix in both cases. For

each case two replications are plotted in order to give an idea of

closeness along the y aXis. This is possible because of the fact that

two replications of the same case should give values of Yopt close

together for Eq. [40J. giving an idea about how far are the YGCV values

provided by Eq. [38J from the optimal values. In these figures the

scales of the ordinate aXis are different for each plot to clearly

compare the locations of the minima attained for each function. (The use

of the same scale does not give any additional information and prevents

a clear comparison).

As it can be seen in Table II. the results are very good in almost

all the cases. Note that the method is able to distinguish between

bimodal and unimodal distributions. For example. when the distribution

is bimodal like in case F, the optimum r's are shifted to the left with

-- 20 ­

respect to the unimodal cases (e.g. case B) in order to allow the

inherent oscillations of a bimodal distribution (see Figures 3 and 4).

There is a case (A) in which the values of y provided by Eq. [38]

do not give correct solutions in any replication. The distribution

corresponding to this case has the characteristics of being very narrow

and having a very small number average particle diameter. In order to

analyze if the poor results are either due to both characteristics or if

they only depend on one of them, a simulation using the same

distribution of case A but shifted to the high diameters zone was

carried out. This simulation corresponds to case G and reveals that

although for this case the best solutions provided by the regularization

technique are not as good as those for case A, the GCV technique gives

values of y very close to the optimal ones. On the other hand, the

results obtained for case C show that even though the corresponding

distribution has a high number of small particles, the results obtained

with the GCV technique are still good. Therefore, the poor behaviour in

case A may be attributed to the conjunction of the following

characteristics: a very narrow distribution in the small diameters zone.

As a conclusion about this point it can be said that the GCV technique

is expected to work poorly when the distributions are very narrow and

have a small number average particle diameter.

A cl earer picture of the resul ts can be seen in Figures 5 to 10 in

which the estimated PSD's for cases B to G are shown. In these fi gures

the true distribution and two replications are plotted for each case. • Although the optimal estimates of the PSDls are not plotted, they were

- 21 ­

very close to those obtained with the Y values. In case D (figure 7)GCV a mild oscillation is present in the solution obtained for replication

3. This result may be expected when replications are carried out. The

possibilities of this kind of results are even higher when as in this

case, high levels of noise are present.

5. Summary and Conclusions

The results presented in this paper verify the potential and

versatility of the regularization technique when it is utilized along

with the GCV technique. A completely different problem from those

analyzed in references (~) and (22) using the same combination has been

solved with few and predictable limitations.

The use of these complimentary techniques has been demonstrated

through the recovery of the PSD of polystyrene latexes. Unimodal and

bimodal PSD's of varying breadth and mean particle diameters have been

investigated. The results were mostly satisfactory.

The GCV technique makes possible to integrate the complete

estimation process in a single step with the purpose of monitoring

and/or controlling a variety of heterogeneous systems, among them

emulsion polymerizations.

- 22 ­

6. Acknowledgements

This research was supported by NSF Grants RII 8S07956 and INT­

8602578. Guillermo Elicabe is with a scholarship from Consejo Nacional

de Investigaciones Cientificas y Tecnicas de la Republica Argentina.

7. Literature Cited

1. Wallach. M. L.; Heller, W.; Stevenson, A. F. J. Chern. Phys. 1961,

34, 1796.

2. Wallach, M. L.; Heller. W. J. Phys. Chern. 1964, 68, 924.

3. Yang, K. C.; Hogg. R. Anal,ytical Chemistry 1919,21, 758.

4. Zollars, R. L. J. Call. Interface Sci. 1980, z...1, 163.

5. Mel i k. D. H.; Fogl er, H. S. J. Coll. Interface Sci. 1983, 92, 161.

6. Elicabe. G. E.; Garcia-Rubio, L. H. To be pub1ished in:

J. Coll. Interface Sci. 1988.

7. Kerker, M. "The Scattering of Light and Other Electromagnetic

Radiation"; Academic Press: New York, 1969.

(j. Phillips, D. L. J. Assoc. Comput. Mach. 1962, i. 84.

9. Twomey, S. J. Assoc. Comput. Mach. 1963, lQ. 97.

10. Turchin, V. F.; Kozlav, V. P.; Malkevich. M. S. Sov. Phys. ­

~ 1971. ~, 681.

11. Twomey, S. "Introduction to the Mathematics of Inversion in Remote

Sensin9 and Indirect Measurements"; Elsevier: New York, 1977.

- 23 ­

12. Bertero, M.; De Mol, C.; Viano, G. A. In "Inverse Scattering

Proolems in Optics ll ; Baltes, H. P., Ed.; Topics in Current Physics;

Springer - Verlag, 1980; p 1b1.

13. Hoerl, A. E.; Kennard, R. W. Chern. Eng. Proqr. 1962, 55, 54.

14. Hoerl, A. E.; Kennard, R. W.; Baldwin, K. F. Comm. in Statis. 1975,

,1, 105.

15. Provencher, S. W. Makromol. Chern. 1979, 180, 2Ul.

lb. Hoerl, A. E.; Kennard, R. W. Camm. in Statis. 1976, A5, 77.

17. Hemmerle, W. J. Technometrics 1975, lZ, 309.

18. Golub, G. H.; Heath, M.; Wahba, G. Technometrics 1979,11, 215.

19. Gi bbons, D. 1. Genera 1 Motors Research Laboratori es, Research

Publication GMR 2659, Warren, Michigan, 1978.

20. Maron, S. H.; Pierce, P. E.; Ulevitch, I. N. J. Colloid Sci. 1963,

~, 470.

21. Inagaki, 1.; Arakawa, E. T,; Hamm, R. N.; Williams, M. W. Physical

Review B 1977, ~, 3243.

22. Merz, P. H. Journal of Computational Physics 1980, 38, 64.

.'

- 24 ­

APPENDIX

For the di screti za ti on of Eq. [4J, it is assumed that for a given

wavelength Ao .• the integrand can be approximated by the product of a 1

linear interpolation between two successive points on f(D).

f. = A. + B. D. [Al]J J J J

= A. + B. Dj +l [A2]f j +l J J

!:J.with f = f(D j ), and Dj +1 D. = LID for all j,j J

and the kernel K(Ao ,0) calculated at ,\ 0 • 1

6. 11 2K(,\ 0 1 •• 0) = Ki(D) = --4-- Qext(Aoi·O) 0 lA3]

Dividing the integral in Eq. [4J into n-l sections, and replacing Eqs.

[AI-A3), it is clear that li can be expressed in the following form:

(A 2 + 82 D) dO + ... +

0'+1 + Jr J K,(O) (A. + B. 0) dO + .•. + ..D. 1 J J

J

- 25 ­

Where it was assumed that f(O) = a for 0 > 0 > D . The values for the1 n

parameters A. and B. can be obtained from Eqs. [Al,A2]J J

[A5]

[A6)

Replacing these last two expressions Eq. [A4] becomes:

o 0 0 0 01 f N-l N- 2 r N-l N f N

+[ 60 J Ki(D) D dO - --ro J Ki(D) dO + [j0 J Ki(O) dO­0N_2 0N-2 DN_1

Therefo re a.. ; s gi yen by:lJ

1 O. D. 1 D.f J J- r J = 60 J K;(D) D dO - 6D J K;(O) dD +aij

D. 1 D. 1J- J­

- 26 ­

Dj +1 rDj +1 1 (D j +1 [A8]+ ~ J . Ki(D) dD - ~ J . Ki(D) D dO

D DJ J

for j = 2, •.. , n-l. and

[A9]

[AlO]

For a small increment in the diameters, the integration of the

functions K.(O) can be calculated using a straight line approximation1

and the same step 6D used with f(O). Thus. the integrals in Eqs. [A8­

AIO] can be written as;

(K; ,k+l- Ki k) Ok

2(Ok+l- Ok)

3 3 ( Ki , k+ 1­ K; k) (D k+1­ Ok)

+ 3(D k+l - Ok)

[A12J

- -27 ­

Repl aci ng Eqs. [All-A12] into Eqs. [A8-AIO]. the appropri ate va 1ues for

can be obtained.aij

..

- 28 ­

Table I: Particle size distributions utilized in the simulated

expe ri ments.

oISTRI BUn ON

f(D) = N(a)( (b) (b)­C1/(C/C 2) LOGN 1 + C/(C 1+C 2) lOGN z J p

D [nm]91 °1 o [nm]

92 °2 C1 C2 Ds[nm]

A 175 0.2 - - 1 0 0

B 600 0.3 - - 1 a 0

c 1000 0.65 - - 1 0 0

0 1300 0.3 - - 1 a 0

E 600 0.2 1000 0.1 1 2 0

F 600 0.2 1500 0.1 2 1 0

G 175 0.2 - - 1 0 2000

(a) Np = 68.49xlO 3[part/em}

1 (b) LOGN,' = exp ( -[In (0 - 0 ) - In 0 J2/ 2 o~ }

12; 0;(0 - Os) s 9i '

- 29 ­

Table II: Results obtained for the five replications of each experiment

(A to G) using Eqs. [38) and [40J.

EXP Repli. ropt rGCV rf(yopt ) r f(YGCV)

A

1 2 3 4 5

-151.32XIO_ 151.32x10_ 15 1. 32xl0-15 2.47xlO_ 1b1. 61xl0

.

-187.51x10_ 175.86xIO_ 201.70x10_ 161.61xlO_ 151.32x10

0.1817 0.1626 0.2216 0.1665 0.1182

1.6838 0.4784 8.3119 0.5952 0.2014

8

1 2 3 4 5

-122.05xlO_ 126.20xlO 1~

- j7.83xl0_ 12 6. 20xl0-12 9.73xl0

-126.20xl0_ 126.20xlO_ 129.73xlO_ 12 9. 73xlO-12 6.20x10

0.0833 0.0575 0.0486 0.0671 0.0602

0.0919 0.0575 0.0805 0.0725 0.0681

C

1 2 3 4 5

-92. SOxlO_9 1. 95x10_ 9 1. 39x10_9 2. 50x10_ 102.0SxlO

-106. 93xlO_ 1O1. 78xlO_ 10l.06x10_ 91. 39xlO_ 91.95x10

0.0780 0.0788 0.0770 0.0/70 0.0727

0.0955 0.1155 0.0937 0.0785 0.0779

D

1 2 3 4 5

-109.05x10_ 109. OSx10_S 1. 94XlO_ 1O3.b8xlO_ 119.27xl0

-91. 42x10-10 9.05xlO 11 3.71xlO: 91. 94x10_ 91. 94x10

0.0621 0.0450 0.0687 0.0412 0.0476

0.0624 0.0450 0.1745 0.0541 0.0593

E

1 2 3 4 5

-12 1. 43x10-12 1. 04xlO 12 1. 43x 10: 12 3. 24xl0_ 111. 38x 10

-135.04x10_ 124. 65xlO-12 6. 06xlO-12 3.24x10_ 124.65x10

0.1974 0.1196 0.1653 0.1117 0.1960

0.2177 0.1535 0.1924 0.1117 0.2062

F

1 2 3 4 5

-13 9. 31xlO-12 2. 69x10-12 2.25x10 r

- .:i9.31xlO_ 124.78x10

-111. 33xlO 12 3.96x10: 12 5. 60xlO-12 3.96xl0_ 122.25xlO

0.2484 0.1736 0.2062 0.1417 0.1886

0.3169 0.1763 0.2174 0.1701 0.2085

- .30 ':"

Table II (continuation).

, . , . , - , - , , . -I -111 1. 41XI0_ 112 I 1. 41xlO_ 113 I 1. 41xl0_ 11G

I4 1. 41XI0_ 129.54xlO5

-11 0.29090.29091. 41XlO_ 0.49090.38293. 94xlO_

911 0.23030.23031. 41xlO_ 11 0.30490.30491.41xIO_ 1l 0.20490.19171. 41xlO

31 ­

LIST OF SYMBOLS

p

A Matrix of coefficients resulting from the discretization of Eq.[4]

C Concetration of Particles in g/ml

o Particle diameter in cm

E Expectation operator

f(O) Number of particles with diameter 0

f Estimated dirscrete particle size distribution

I Intensity of the electromagnetic radiation

k Imaginary part of the complex refractive index

! Path length in cm

n Real part of the complex refractive index

N Number of particles per ml

P(y) Ordinary cross validation function (Eq.(31]

Qext Extinction efficiency A

q(f) Smoothing function (Eg. [23J)

Y Regularization parameter (Eg. [20J)

£: Combined measurement and discretization error

A wavelength in cm

- 32. ­

FIGURE CAPTIONS

~t~,~~e,l: Optical parameters of polystyrene and water.

f1Qu~~,2: PSD's used in the simulated experiments.

,~,i,~,ur~,,3: Eqs. [38] (- - - -) and [40J (,., ) for case B ana

replications 3 and 5 as functions of y .

.~,;,g.ur,~" 4: Eqs. [38J (- - - -) and [40J (. ... , ) for case F and

replications 4 and 5 as functions of y.

~i,gure 5: Case B. True PSD ( - - - -) and estimated PSD's using GCV for

replications 3 (oo , . __ . ) and 5 ( ........ ) oil

,F.!,Qu,r,~, ,6: Case C. True PSD (- - - -) and estimated PSQ' s using GCV for

repl ications 1 (. ',', .. ',' ) and 5 ( ....... ) . ~.i gure 7 : Case D. True PSD (- - - -) and estimated PSDls using GCV for

replications 5 (, , , .. ) and 3 ( ....... ) .,

.f~.Q~.r.~ ..8: Case E. True PSO (- - - -) and estimated PSD1s using GCV for

replications 2 (', , , ~" ) and 5 (........ ) ..' ,

Ci,~u~e., 9: Case F. True PSD (- - - -) and estimated PSD's using GCV for

replications 4 (-.. . . , . . ) and 5 ( ....... ) .

.F.i~,~~e, 10: Case G. True PSO (- - - - ) and estimated PSD's using GCV for , ,

repl ications 1 ( ., .... ) and 5 (........ ) "

..

- 33 ­

· ­

2.22 I 10.87

1.99 0.64

1.76 0.41

n, 1.52f- \k1 0.19

n2 '0~ ~--_.! -- ­' ... .-"'" I

8 ' , 375 1.2 200 A)nm]

FIGtJR.E 1

,. ..

I

-... --~ . .. : . : ~ : .. : . . ..~ .'. .- ~':"','" - -. . . '" ; ..~./.., .

. .. . .

:.r . 'Dmin Dmax

. II I I

A 32550 50 11300A

r\ ~ C I 50 13950

"r­~ 12600

I E 50 11450 G

f (0) I II 1\ I F I 50 2550 G 2050 2325

B /\E, \ I I CII I\

I • __0-\ - ....

\

_\ --.I / I ,

~

50 / 1025 2000 2975 3950F o [nm] ~'

FIGURE 2

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0... d..::J (1) (])~ ....... \- ' ­ 0,..-,

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8 1-1 11.<

----- -----:::-:::::":': :'\,,-... ............

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