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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
----------------l~
I I
/ J
I J
I I
I I
I /,-
,-'" /
/
" " ,;' ,;'
.... '"
('1"')------ .... _-- .... ----------~------ ---------..
..-(l)
0X W
o ..10 ...
.. ."
(
LL
0X W·
--------
L-l
I
I I I I
.1, I I
o-10 .,
N ~
10 ..
•• 't
; . " .',
':','
.. '
,T'.( ,
I'
"
...---------~---'-----------'O o
-------- - ... ---.. ...- ~-- ... ~ ..:'"
".~ ...... \
I
o V) 0...
("Y") L() <lJ • •:Jo.o.. ~(1)<1J
1- ~
m 0X
(Y) ,.
-
,..-,
E c ~
o
."...... ·l .. .. " '
." oW LO
I ..
"
(.
( ,"
U
0X W
0 U) 0..
..-LD OJ :J '0.. \- Cl..<lJ ...., ~\-, ·
I
I
I ··· I
I I I
"
0'--'oE°cN J-.-,t
0 lD
~ H r..
o Lf)
..'
." ~ '.'
.~ .' ,-'
(
o o U) N
",-. ' .....o .,............
L!)
N (Y) ~
.,-:'. . .... ." .........
", ._~-....-.. -. .....l!1 M
".'
. 0..'(1)0-..... ill
\-
.. ........ ..... ..-.-. ..'
"-... ....... " ... ...
~
.. III •• l1li' •• 0.. X W
-.,. E/Ct I
" , L--I r-, 0 ~ o H r...U)
D.. <lJ ::J \
+J
I
,I I I I I I
-o
(' I:, .. ~ ...
(/',
_..::~-~"'" ----~-
. "
••· · · ·•·.'· ·•·
<1JNLO ' .
\-0..0.. ~ClJ<1J10.-'-
,.........,
E( c L..-.I
o 00
~2
8o H r....U)
CL
:J
I
•l t ,II'
I
I'"
'.I"
'
•
.....'
LL 0X W
0 tf) 0
'-.:t LO (1) .
0... d..::J (1) (])~ ....... \- ' 0,..-,
I I :
: DE I Me
~\.....-.II
·,•I 0 mI I · r.]I ·• J"
8 1-1 11.<
----- -----:::-:::::":': :'\,,-... ............
-
.'
' . o L()
--.. P'"
\,----- -
..
/
~ ,- . ", 'j.'.
J I I I
I I
J I
--- " ..... .. ....... ......... " , .
<!)
Q.. X W
I: J t I ~
( o ,...1
o
o Lf)
o N
..
f (