Abstract
This paper, second of two parts, presents a parametric study of the forwardscattering corrections for experimentally measured optical extinction coefficients in polydisperse particulate media, since some forward scattered light invariably enters, along with the direct beam, into the finite aperture of the detector. Forwardscattering corrections are computed by two methods: (1) using the exact Mie theory, and (2) the approximate Rayleigh diffraction formula for spherical particles. A parametric study of the dependence of the corrections on mode radii, real and imaginary parts of the complex refractive index, and half-angle of the detector's view cone has been carried out for three different size distribution functions of the modified Gamma type. In addition, a study has been carried out to investigate the range of these parameters in which the approximate formulation is valid. The agreement is especially good for small-view cone angles and large particles, which improves significantly for slightly absorbing aerosol particles. Also discussed is the dependence of these corrections on the experimental design of the transmissometer systems.
Introduction
In Part 11 of two papers, results were presented of a parametric study of the forwardscattering correction factor R and the complementary error factor E for monodispersions. In this paper the results of a similar study carried out for spherical polydispersions of size distribution n(r) (cm- 3 Am-'), r being the radius in ,im, will be presented. For the sake of clarity, only the results obtained with the use of simple unimodal size distributions of the modified Gamma type, such as Dermendjian models 2 Haze M, Haze H, and Cloud C3 (referred to as Haze C in this paper), are presented here. Results for other real size distributions can easily be obtained in a similar manner. The behavior of both R and E, averaged over each of the three size distributions, will be discussed here as functions of each of the following parameters: the mode radius rm; the polydispersity or the spread of the size distribution; and the real (i') and imaginary (m") parts of the complex refractive index m = ' -". The computations have been carried out with both the exact Mie theory solutions in explicit closed form and the Rayleigh diffraction theory approximation, as explained in Part 1. The results of such a study are extremely useful in obtaining the optimum experimental design parameters for the measurement of extinction coefficients in particulate media. 3 
II. Transmission Law (Bouguer's Law)
The transmission law for an electromagnetic plane wave passing through a homogeneous polydisperse aerosol medium ( 
OIext(X) = OIscat(W) + I0abs().
(2) (3) As in Part 1, for the sake of clarity, the molecular contributions to Oext will be ignored here. Then for polydisperse aerosols of size distribution n(r), cm-3 gm-1 , the coefficients are defined by j(W = r 2 7rr 2 Qj(x,m)n(r)drj = ext, scat, and abs. (4) All the quantities in Eq. (4) are the same as defined in Part 1, except that the factor Np (r) for monodisperse particles has been replaced here by the operator
Sr 2 n(r)dr,
where r and r 2 are lower and upper limits of radii.
Sr 2 n(r)dr,
Aerosol size distributions are discussed in a later section.
Ill. Forwardscattering Corrections

A. Exact Mie Theory Formulation
Because of the fact that forwardscattered light invariably enters the detector view cone, Bouguer's law, as defined in Eq. (1) , cannot be used to obtain the true optical depth r in a transmission experiment. But instead, one obtains the apparent optical depth T' related to the apparent volume extinction (ext) and scattering (ncat) coefficients, which are distinguished from the true quantities fext and O'cat, respectively; here, Iabs is clearly not affected by the forwardscattering. Thus, 'xt = lscat + ilabs (apparent). (5) The Mie efficiency factor Qscat is based on the total amount of light lost from the beam by scattering by a particle in all directions; and Q'scat on the amount of light lost by scattering in all directions except within a cone of half-angle 0 in the forward direction, so that their difference is given by 
A. Exact Mie Theory Formulation
where E (and R) are defined in Eqs. (10) and (9) of Part 1. Then a correction factor R and an error factor E, averaged over the particle size distribution between the limits r, and r 2 , may be defined by where Jo and J 1 are Bessel functions of the first kind and of orders zero and one, respectively.
IV. Corrected Transmission Law
By inserting the correction factor H into the transmission equation, one can account for both the direct and forwardscattered radiation. Thus
IV. Corrected Transmission Law
where
IV. Corrected Transmission Law
From the discussion in Part 1, it is clear that for parallel beam transmission systems with an open detector, the half-cone angle 0 = 0(I), so that
IV. Corrected Transmission Law
where
IV. Corrected Transmission Law
L And for a lens-pinhole system 0 = constant, so that R =R (x,0), independent of 1. For the discussion of the experimental design considerations of the two detector systems, see Part 1.
V. Aerosol Size Distributions
Several analytic representations of aerosol size distributions appear in the literature. In this paper, the behavior of R and E is investigated as a function of the mode radius rm, the spread of the size distribution and the complex refractive index m(= m' -im"), the upper and lower limits of radii being 10-2 and 20 gim. For the sake of simplicity, three of the Deirmendjian models, 5 namely, Haze M, and Cloud C3 (referred to here as Haze C), were selected for representing different polydispersities of aerosol size distribution n(r). Since rm is varied between the radii limits 10-1 and 10 gm, the three models are used here in a more general way than was their original intent. It is in that sense that model Cloud C3 is referred to as Haze C. The limits of integration over r are 10-2 and 20 gim.
V. Aerosol Size Distributions
The expressions for the n (r) models and their corresponding mode radii rm are given as follows: terms in the theory of distribution functions to express the same quantity. Perhaps the major disadvantage of the Deirmendjian models is that their falloff behavior for large radii is too sharp: many experimentally measured distributions show a power law behavior, at least in the optically ac-
,.0
In:
,.0
ODE RADIUS rm ,gm
,.0
.8
,.0
.6
,.0
.5 10 tive region. For this reason, we have also considered a power law distribution, given by n(r) = rv1, 10-2 /Am < r < 15gum
,.0
and allowed v to vary between 2 and 4. Note that one cannot talk about either mode radius or polydispersity for a power law haze, only slope v.
VI. Computational Considerations
A parametric study of the correction factor R and the corresponding error factor E was carried out as functions of different combinations of rm, 0, m' and m", whose values occur within the ranges 0.1 < rm < 10 gim, 00 < 0 < 100, 1.33 < m' < 1.65, and 0.0 < m" < 0.1.
VI. Computational Considerations
The computations of H (and E) in Eq. (8) [and Eq.
VI. Computational Considerations
(9)] are made by using the closed form relations given in Part 1 for I(xpmn0). The computations of R for the approximate method are made by using Eq. (10).
VIl. Discussions and Conclusions
The parametric study of H was carried out as a It can be seen from the plots presented here that, in general, for given values of 0 and rm, the higher the polydispersity the higher the error E. This is due to the predominant forwardscattering of the large particle component, implied by the increased polydispersity. We should note, however, that the saturation values of E and R (both 0.5) are not affected by polydispersity, but that these values are reached sooner for a more polydisperse haze than for a less polydisperse haze. A saturation value of E 0.5 implies a Q't value of 1.0.
VIl. Discussions and Conclusions
An explanation for this is as follows. For the same value of 0, the amount of scattered radiation collected at the detector increases as the large particle component increases, which, in effect, reduces Q from a value of about 2.0 (assumed for large particles) to a minimum value of 1.0. The saturation value for E tends to be about 50% for 0 = 100 and 45% for 0 = 4 for Haze M, a difference of about 5%. This difference, however, tends to decrease as aerosol size distributions become less polydisperse, implying a smaller number of large particles, so that the saturation value of E approaches 50%. Figures 11-13 show that, for nonabsorbing aerosols, the agreement between the Mie and the Rayleigh approximation is not as good for small rm as it is for larger rm for all three models. This is to be expected since the Rayleigh diffraction formula Eq. (13) is only valid for large particles. On the other hand, the agreement for both large and small r appears to be better for Haze M (more polydisperse aerosol) than for Haze C, indicating that for more polydisperse size dispersions the agreement between the Rayleigh formulation and the exact formulation tends to improve. 
E. Experimental Design Considerations
Equation (11) shows how the forwardscattering correction factor is included in the transmission law to obtain the measured (or apparent) optical depth a-'. Comments about the two aforementioned experimental geometries, discussed in Paper 1, also apply to the case of polydisperse aerosols, in addition to the following comments regarding the effects of the polydispersity on the experimental design.
E. Experimental Design Considerations
In Paper 1 (Fig. 13) , results were plotted for 1i as a function of xO for the two aforementioned experimental geometries. For the polydisperse case, however, R (x0) has to be further averaged over the volume extinction coefficient to yield R, the path-averaged correction oD factor for polydisperse aerosols, defined by the equation
E. Experimental Design Considerations
The symbols cap(') and bar(-) denote averaging over 2 '
E. Experimental Design Considerations
\ " the pathlength and particle size distribution, respec- for models Haze M, Haze H, and Haze C.
E. Experimental Design Considerations
where z = r/rm,zi = rl/rm.z2 = r2/rm.
E. Experimental Design Considerations
(20)
E. Experimental Design Considerations
~. R, obtained by using Eq. (19), can easily be plotted as
E. Experimental Design Considerations
a function of y (= krm0) for the three size distributions In order to make accurate transmission measurements, it is important that the experimental design be based on those values of y for which either R -1 or R 0.5. But to be able to do so, some reasonable prior knowledge of both the mode radius rm and the polydispersity is required. 
E. Experimental Design Considerations
A more detailed parametric study of the behavior of the forwardscattering effects involved in different transmission measurement experiments designed to measure optical extinction and visibility in the atmosphere will be presented in a separate publication.
G. Results for Power Law Size Distributions
In Fig. 19 These two refractive indices usually produced the largest and smallest errors (respectively) for a given 0 -v combination. Since an increase in v leads to a reduction in large particle content, the shape of these plots is inverted compared with those in Fig. 17 . 
VII. Summary Remarks
From the foregoing discussion, it is obvious that the transmission law as expressed by Eq. (1) cannot be used except in the case of small particles, for which it is merely a good approximation. The effect of not only the particle size distribution, refractive index, and shape, but also the geometry of the optical system on the apparent volume scattering and extinction coefficients must be considered. This study shows that the H(E) decreases (increases) as the size distribution becomes more polydisperse. This result can be explained by the fact that as the polydispersity becomes greater, the relative proportion of larger particles increases, resulting in an increase in forwardscattered radiation within narrower angles. The results are valid for conditions in which multiplescattering effects can be neglected, and only single scattering predominates.
VII. Summary Remarks
The lens-pinhole detector geometry yields the most accurate transmission/extinction results, provided the design conforms to the prescription that the value of y in Fig. 18 should be such that is close to either 1.0 or 0.5. This work was supported by NASA grant NSG-1252 and contract NAS1-15198.

{'paper_id': '1564608', 'title': 'Forwardscattering corrections for optical extinction measurements in aerosol media. 2: Polydispersions.', 's2_url': 'https://api.semanticscholar.org/CorpusID:1564608', 'source': 'S2ORC'}