Measurement of aerosol size distribution function using Mie scattering—Mathematical considerations

Aerosol Science 38 (2007) 1150 – 1162 www.elsevier.com/locate/jaerosci

Measurement of aerosol size distribution function using Mie scattering—Mathematical considerations Lin Ma∗ Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA Received 11 June 2007; received in revised form 21 August 2007; accepted 22 August 2007

Abstract This paper analyzes the applications of Mie scattering to measure the size distribution function (SDF) of aerosols. Measurement of SDF by Mie scattering usually involves solving the Fredholm integral equations of the first kind based on discrete inputs with uncertainties (e.g., extinction measurements at multiple wavelengths or at multiple angles). A set of inputs which are not mutually independent within the measurement error implies that redundancy exists in the measurements and not all the measurements provide useful information for solving the integral equations. To avoid such redundancy, this paper develops a method to analyze the dependency among the kernel functions associated with Mie scattering. Applications of this method are demonstrated and the results provide valuable insights into the optimization of SDF measurement based on Mie scattering, in terms of minimizing the number of measurements needed and revealing the optimal wavelengths and angles to perform the measurements. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Mie scattering; Size distribution function

1. Introduction Among all the methods developed for the measurement of size distribution function (SDF) of aerosols, methods based on Mie scattering hold a special position and offer unique capabilities for non-intrusive and in situ measurement (Kerker, 1997; McMurry, 2000). These methods can be broadly divided into two categories based on the light source employed. The first category utilizes light sources tunable over a wide spectral range to obtain multi-spectral scattering data and impressive measurements have been demonstrated using these methods (Ramachandran & Leith, 1992; Velazco-Roa & Thennadil, 2007; Wang & Hallett, 1996). However, tuning of the light sources over a wide spectral range limits the time response of these techniques and also complicates the data analysis due to the variation of refractive index over wide ranges of wavelengths. Rapid and continuous measurement is highly desirable in many applications, such as the study of spray combustion (Grassmann & Peters, 2004; Meyer, Roy, Belovich, Corporan, & Gord, 2005; Tishkoff, Hammond, & Chraplyvy, 1982; Winklhofer & Plimon, 1991) and the fast formation/evaporation of aerosols (Vandongen, Smolders, Braun, Snoeijs, & Willems, 1994). Such measurement needs motivate the development of the second category of methods, which relies on scattering data at a single wavelength or a limited number of discrete wavelengths. Variations of methods in this category include the measurement of scattering data at a single wavelength ∗ Tel.: +1 864 656 2336; fax: +1 864 656 4435.

E-mail address: [email protected]. 0021-8502/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2007.08.003

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and multiple angles (Beretta, Cavalieri, & Dalessio, 1984; Swithenbank, Beer, Taylor, Abbot, & McCreath, 1977), measurement of extinction data (i.e., at a fixed angle of 180◦ ) at multiple wavelengths (Ma & Hanson, 2005; Vandongen et al., 1994; Walters, 1980), and measurement of scattering data at multiple wavelengths and multiple angles (Dick, McMurry, & Bottiger, 1994; Dick, Ziemann, & McMurry, 2007; Nagy, Szymanski, Gal, Golczewski, & Czitrovszky, 2007; Szymanski, Nagy, Czitrovszky, & Jani, 2002). Despite their dramatically different experimental arrangements and complexities, all these techniques in both categories involve the same mathematical background, namely solving the Fredholm integral equations of the first kind in the following form: 

Dmax

Dmin

K(Si , D) · g(D) dD = Mi ,

si = s1 , s2 , . . . , sN ,

(1)

where D is the diameter of the aerosols, Dmin and Dmax the lower and upper ranges of the diameter, respectively; K(si , D)’s the kernel functions given by the Mie scattering theory; g(D) the unknown, sought SDF on the interval Dmin D Dmax ; and Mi the measurements performed with uncertainty under condition si (si represents a wavelength or a scattering angle at which the measurement is performed). Fredholm integral equations of the first kind are inherently ill-conditioned problems and extensive literature exists on the algorithms for solving such equations (see Hansen, 1992, and references therein). Assuming a proper technique has been applied to solve Eq. (1), a maximum of N pieces of information about g(D) can be inferred (for example, values of g(D) at N discrete D’s) from the N measurements performed when the kernel functions (K(si , D)’s) are mutually independent. When the kernel functions are not mutually independent, a case that could occur when the wavelengths or angles are not well chosen, some of the measurements performed can be represented by the linear combination of other measurements within the measurement uncertainty and hence do not provide useful information about g(D). In this case, less than N pieces of information about g(D) can be inferred, although a number of N measurements are performed. In practice, such redundancy must be avoided. Furthermore, each measurement (at a certain wavelength and angle) is desired to be as independent to the others as possible to maximize its information content toward the extraction of the SDF. These considerations motivate this work to study the dependency among the kernel functions associated with Mie scattering, and to develop a method to identify the wavelengths and angles to generate mutually independent scattering measurements free from redundancy. The remainder of this paper is organized as follows. Section 2 introduces the mathematical model for the analysis of the kernel functions. Section 3 describes the application of the model to the analysis of SDF measurement methods based on Mie extinction, and Section 4 discusses the effect of refractive index in extinction methods. Section 5 describes the application of the model to the analysis of scattering-based methods. Finally, Section 6 summarizes the paper. 2. Mathematical analysis This section describes the method we developed to quantitatively analyze the dependency among a given set of kernel functions associated with Mie scattering for the measurements of SDF. For clarity, here we develop the model using the example of SDF measurement based on Mie extinction (i.e., measurements performed at an angle of 180◦ ), where extinction measurements at multiple wavelengths (N wavelengths) are used to infer the sought SDF. These extinction measurements are expressed as 

Dmax

Dmin

Q(i , D) ·

D 2 Cn L f (D) dD = i , 4

i = 1 , 2 , . . . , N ,

(2)

where extinction measurement is performed at N discrete wavelengths, i to N ; Q(i , D) is the Mie extinction coefficient at wavelength i and droplet diameter D; Cn the number density of the droplets; L the pathlength; f (D)  Dmax the sought SDF defined such that Dmin f (D) dD = 1 and f (D) dD represents the probability that a droplet has a diameter between D and D + dD; i the extinction measurement performed at wavelength i ; and other notations have the same meaning as in Eq. (1). In Eq. (2), when D 2 Cn L/4 and f (D) are grouped and regarded as the sought function equivalent to g(D) in Eq. (1), Q(i , D)’s are the kernel functions equivalent to K(si , D)’s in Eq. (1). Then

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the notations in Eq. (2) can be simplified into the following form:  Dmax Q1 · g(D) dD = 1 , Dmin

.. . 

Dmax

Dmin

.. . 

Dmax

Dmin

Qi · g(D) dD = i ,

QN · g(D) dD = N ,

(3)

where g(D) represents the product of D 2 Cn L/4 and f (D). Mathematically, a kernel function, Qi , is linearly dependent on the other kernel functions if Qi can be represented by the linear combination of the other kernel functions. In this case, the ith equation in the set of N equations shown in Eq. (3) can be written as the linear combination of the other (N −1) equations. Thus the ith equation, and consequently the ith measurement (i ), does not provide useful information about g(D). Experimentally, due to the existence of measurement uncertainty, a kernel function, Qi , is regarded as “experimentally dependent” on the other kernel functions if Qi can be represented by the linear combination of the other kernel functions within the experimental accuracy. Therefore, we first need to quantify the concept of “experimental dependence”. We define the difference (Qi ) between a kernel function, Qi , and its approximation by the linear combination of other kernel functions as follows:  Qi =

Dmax

Dmin

[Qi − (w1 Q1 + · · · + wi−1 Qi−1 + wi+1 Qi+1 + · · · + wN QN )]2 dD,

(4)

where the w’s are the weights used in the linear combination. In order to compare the relative magnitude between Qmin and the kernel functions, the kernel functions are assumed to be normalized such that i 

Dmax Dmin

Q2i dD = 1,

i = 1, 2, . . . , N.

(5)

Obviously this normalization does not change the essence of Eq. (3). The magnitude of Qi varies with the w’s used in the combination weights used and will be minimized at a certain set of w’s, which can be sought via optimizing min provides a quantitative measure of the accuracy Eq. (4). If we denote this minimum of Qi as Qmin i , then Qi of approximating Qi by the linear combination of the other kernel functions. A small Qmin implies that Qi can be i approximated accurately by the linear combination of other Q’s, and vice versa. Therefore, here we define experimental min is greater than a certain level e, Q dependence of Qi relative to other kernel functions based on Qmin i i . If Qi is defined to be “experimentally independent” relative to the other kernel functions; otherwise, Qi is defined to be “experimentally dependent” relative to other kernel functions and the ith at i does not provide new information toward g(D) within√the measurement uncertainty corresponding to e. Under certain assumptions, it can be derived that e corresponds to ε/ N (Twomey, 1977), where ε is the relative measurement uncertainty in all the i ’s and N the number of measurements performed. Furthermore, if Qmin > Qmin i j (1 i, j N ), it is implied that Qj can be approximated by the linear combination of other Q’s more accurately than Qi , or, in other words, Qj is more experimentally dependent on other Q’s than Qi . Therefore, within a certain level of measurement uncertainty, measurement performed at i provides more useful information than measurement performed at j . Based on the above understanding, a method can be developed to select the most independent kernel functions from a given set of N kernel functions, Q1 to QN . First, Eq. (4) is minimized for each Qi to obtain the corresponding Qmin i (1i N). Second, if all the Qmin ’s obtained in the first step are greater than a preset accuracy level (e) determined from the measurement uncertainty ε, then all the kernel functions are already experimentally independent by definition

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and they all yield useful information toward g(D). Otherwise, the kernel function corresponding to the minimum of the Qmin ’s in the first step is removed from the kernel function set because it is the most experimentally dependent one in the given set. Third, the first and second steps are iterated to remove the kernel functions one by one until all the remaining Qmin ’s are greater than the preset e. Each iteration removes one kernel function, which is the most experimentally dependent one among the kernel functions left. Therefore, at the end of the iterations, all the kernel functions left are experimentally independent for a given experimental accuracy and they are the most independent ones among the given set. Meantime, this selection process also reveals the wavelengths at which the most independent measurements are obtained. It is straightforward to extend the above discussion to other techniques of SDF measurement using Mie scattering, such as techniques involving measurements at multiple angles at a single wavelength or techniques involving measurements at multiple angles and multiple wavelengths. The next section discusses the application of our method to techniques using Mie extinction, and Section 5 discusses the application to techniques using forward Mie scattering. 3. Analysis applied to extinction method Among all the optical methods developed for measuring SDF thus far, methods based on Mie extinction enjoy relative simple implementation and offer good perspective for rapid and continuous measurement capability. These advantages have attracted a considerable amount of research interests, both experimentally and analytically (Cai, Wang, Wei, & Zheng, 1992; Cai, Zheng, & Wang, 1995; Dellago & Horvath, 1993; Horvath & Dellago, 1993; Ma & Hanson, 2005; Tishkoff et al., 1982; Twomey & Howell, 1967; Vandongen et al., 1994). Therefore, here we first apply the method developed in Section 2 for the analysis of SDF measurements based on Mie extinction. In light of the continuing development of laser sources and related wavelength-multiplexing technologies in recent years, it now becomes feasible to perform extinction measurements using laser sources spanning a wide range of wavelengths. Hence, in this work, we analyzed the kernel functions consisting of extinction coefficients at 40 wavelengths equally spaced between 0.25 and 10 m (the spacing is 0.25 m), and examined the potential advantages brought about by the extended spectral range. These kernel functions are denoted as Q1 to Q40 , with Q1 representing the kernel function corresponding to a wavelength of 0.25 m and Q40 that corresponding to a wavelength of 10 m. Using water aerosols as an illustration, here we analyze the dependency of these 40 kernel functions using the method developed in Section 2. First, the extinction coefficients are computed for water aerosols with diameter ranging from 0 to 20 m using the refractive index of water at 22 ◦ C as measured by Hale and Querry (1973). Extension of the analysis to other aerosols with different refractive indexes than water is discussed in Section 4. Second, to examine the dependency between these kernel functions in different diameter ranges, the droplet diameter range is divided into four sub-ranges, 0–5, 5–10, 10–15, and 15.20 m. The iterative process described in Section 2 is then applied to select the most independent kernel functions from Q1 to Q40 for each diameter sub-range. Obviously, similar analyses can be performed based on different divisions of the diameter range in different applications. Fig. 1 shows the history of Qmin during the selection process for the diameter range from 0 to 5 m after 30 kernel functions among Q1 to Q40 have been removed. From Fig. 1 we can see that six kernel functions are experimentally independent if e is set to 5%, corresponding to a relative measurement uncertainty of 12%; and that 10 kernel functions are independent when e is set to 1%, corresponding to a relative measurement uncertainty of 3.2%. As mentioned before, this selection process also indicates the wavelengths corresponding to these kernel functions. The six kernel functions that are most independent are Q1 .Q4 , Q7 and Q20 ; and the corresponding wavelengths are 0.25, 0.50, 0.75, 1.0, 1.75, and 5.0 m. These kernel functions are shown in Fig. 2. As discussed in Section 2, any other kernel functions among Q1 .Q40 can be approximated by the linear combination of these six functions with an e less than 5%. Part (a) of Fig. 3 displays such an approximation, where Q5 , the kernel function at a wavelength of 1.25 m, is approximated by the linear combination of the most independent six kernel functions. The weights are shown in the caption. As can be seen, overall, Q5 can be closely approximated by the linear combination of the six kernel functions shown in Fig. 2. Furthermore, according to the selection process, among the 34 kernel functions removed, Q5 is the most experimentally independent one relative to the six shown in Fig. 2. This suggests that other kernel functions among the 34 removed ones can be approximated by the linear combination of the six most independent ones more closely than Q5 . Therefore, the use of wavelengths other than 1.25 m in addition to the six mentioned above to gain more information about the SDF requires higher experimental accuracy. Therefore, when measurements are accurate enough to resolve the difference shown in part (a) of Fig. 3, seven independent Mie extinction measurements can be performed

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Fig. 1. Qmin during the selection process on the diameter range from 0 to 5 m after 30 kernel functions among Q1 .Q40 have been removed.

Fig. 2. The most independent kernel functions selected from Q1 to Q40 on the diameter range from 0 to 5 m when e is set to 5% and the corresponding wavelengths.

(at the wavelengths shown in Fig. 2 and 1.25 m) in this diameter to allow the extraction of seven inferences of the SDF. Otherwise, measurements at 1.25 m (or other wavelengths) will not provide new information for the determination of the SDF. The comparison between Q5 and its best approximation shown here elucidates the dependency among the kernel functions. In contrast, part (b) of Fig. 3 shows a larger discrepancy between a kernel function and its best approximation by linear combination of other kernel functions. If the removal process shown in Fig. 1 continues, Q3 will be the next kernel function to be removed; therefore, Q3 is the most dependent one relative to the others among the six kernel functions shown in Fig. 2. From Fig. 1, it can be seen that after Q3 is removed, Qmin increases to an e of ∼ 24%. Part (b) of Fig. 3 shows the comparison between Q3 and its best linear approximation by the other five kernel functions left. This comparison suggests a significant difference between Q3 and its best linear approximation which can be safely resolved experimentally (again the weights used in the approximation are shown in the caption of Fig. 3). Therefore, six independent extinction measurements can be reliably performed at the six wavelengths shown in Fig. 1 in this diameter range to obtain six pieces of information about the SDF, even when large experimental uncertainty exists in the measurements. Also note that in Fig. 2, the kernel function corresponding to the longest wavelength ( = 5 m) contains relatively less structures compared with the other five in this diameter range. Therefore, replacement of this

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Fig. 3. (a) Comparison between Q5 and its best approximation by the linear combination of the kernel functions shown in Fig. 2 (corresponding weights used in the approximation: w1 = 0.00932, w2 = 0.0199, w3 = −0.168, w4 = 0.611, w7 = 0.932, and w20 = −0.467). (b) Comparison between Q3 and its best approximation by the linear combination of Q1 , Q2 , Q4 , Q7 , and Q20 (corresponding weights used in the approximation: w1 = 0.0199, w2 = 0.446, w4 = 0.556, w7 = −0.0911, and w20 = 0.0785).

Fig. 4. The number of independent extinction measurements allowed in diameter sub-ranges spanning 0–20 m for water droplets at 22 ◦ C.

kernel function by another one exhibiting a similar profile in this diameter range (e.g., Q16 at  = 4 m or Q24 at  = 6 m) will not cause substantial change to the relative independence among these kernel functions. Results from the selections of kernel functions among Q1 .Q40 for different diameter sub-ranges are summarized in Fig. 4. An error level (e) of 5% was taken as the accuracy limit for Qmin during the selection processes. Fig. 4 shows the number of independent extinction measurements allowed in the diameter sub-ranges spanning from 0 to 20 m (the corresponding wavelengths to perform these measurements are also provided by the removal process). We see from Fig. 4 that six independent extinction measurements can be performed in the diameter range from 0 to 5 and 5 to 10 m to extract six pieces of information about the SDF for each diameter sub-range. The number of independent

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measurements decreases to four and three in the diameter sub-ranges from 10 to 15 and 15 to 20 m, respectively. The reason for this decrease is that the kernel functions (Q1 .Q40 ) become less structured in larger diameter ranges. For example, the kernel function at a wavelength of 0.25 m in the diameter sub-range of 10.15 m is dominated by small ripple structures centered around a constant, as contrasted to the oscillatory structures shown in Fig. 2 in the diameter range of 0.5 m. In the limit of very large diameter, according to the Mie scattering theory, all the kernel functions approach a constant asymptotically and therefore, no longer contain information about the diameter. To validate these results, this analysis is compared with the analysis performed in Ma and Hanson (2005), where the optimal wavelengths for measuring SDF based on Mie extinction are selected by a different approach. The results obtained here, in terms of the number of measurements and the corresponding wavelengths, agree well with the results in Ma and Hanson (2005). Furthermore, these results, when compared with previous studies (Dellago & Horvath, 1993; Horvath & Dellago, 1993; Twomey & Howell, 1967), clearly show the advantage of using wavelengths in a wide spectral range for extracting more information about the SDF. 4. Influence of refractive index on extinction method For some practical applications, it is useful to extend the analysis in the above section performed for water aerosols at 22 ◦ C to other systems. Such extensions essentially involve considering the changes of the kernel functions caused by the refractive index of different aerosol systems. Thus, in this section, we study the influence of refractive index on the extinction technique using the method we developed. We first investigate the influence of varied refractive indices on the kernel functions (the extinction coefficients) to gain some insights into this extension. The extinction coefficients exhibit a weak dependence on the imaginary part of the refractive index. Fig. 5 illustrates this weak influence by comparing the extinction coefficients of water droplets at 22 ◦ C at a wavelength of 2 m with those when the imaginary part of the refractive index is varied. The refractive index of water droplet at this temperature and wavelength is taken to be m = 1.306.0.0011 · i from Hale and Querry (1973). Fig. 5 demonstrates a negligible change in the extinction coefficient when the imaginary part is increased by 25% and a relatively small change when the imaginary part is increased by one order of magnitude. Note that the difference between the calculation at m = n − k · i and that at m = n − 1.25k · i is too small such that the corresponding curves virtually overlap in Fig. 5. Comparisons at other wavelengths display similar behavior. Further observations suggest that changes in the imaginary part mainly cause changes in the magnitudes of the kernel functions but not the oscillatory structures, as indicated in Fig. 5. Because it is the oscillatory structure, not the magnitude, that determine the relative dependence of the kernel functions, the

Fig. 5. Comparison of extinction coefficients at different refractive indices at a wavelength of 2 m. Note that the difference between the calculation at m = n − k · i (n = 1.306 and k = 0.0011) and that at m = n − 1.25k · i is too small to be distinguished and the curves corresponding to these calculations virtually overlap in this figure.

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Fig. 6. Comparison of Qmin at different refractive indices during the selection process after 34 kernel functions among Q1 .Q40 have been removed on a diameter range from 0 to 5 m. The difference between the calculation at m = n − k · i and that at m = n − 1.25k · i is so small that the curves corresponding to these calculations overlap on this figure and are not distinguishable.

imaginary part of the refractive index has a negligible influence on the analysis performed in Section 3, as demonstrated in Fig. 6. In Fig. 6, we calculated the history of Qmin for the wavelength selection process described in Section 3, but based on a kernel function set with the imaginary part of the refractive indices increased by 25% over those used in Section 3. The history of Qmin with the modified refractive index virtually overlaps with that shown in Fig. 1. Consequently, the number of independent extinction measurements and the selected wavelengths remain uninfluenced by the changes in the imaginary part of the refractive index. The extinction coefficients, however, exhibit a strong dependence on the real part of the refractive index. A small increase in the real part compresses the extinction coefficient curve appreciatively and results in more oscillatory features within a fixed diameter range; and a decrease stretches the curve and results in less oscillatory features. Fig. 5 illustrates this effect by calculating the extinction coefficient with the real part of the refractive index varied by 5%. When the kernel functions contain more oscillatory structures within a given diameter range, they are more independent, and vice versa. Hence, an increase in the real part allows more independent extinction measurements in a certain diameter range, and vice versa. Fig. 6 demonstrates this by comparing the history of Qmin with the real part increased by 5% to that shown in Fig. 1. The substantial increase in Qmin shown in Fig. 6 implies that the approximation of kernel functions by linear combination of other kernel functions becomes less accurate and therefore allowing more independent extinction measurements on a certain diameter range. With the above understanding, the analysis in Section 3 was repeated over kernel functions (Q1 .Q40 ) generated based on refractive indices with (1) the imaginary part increased and decreased by 25% and (2) the real part increased and decreased by 5%. The analysis for case (1) yields the same results as those obtained in Section 3 as expected; and the results from analysis for case (2) are summarized in Fig. 7. It can been seen from Fig. 7 that a 5% increase in the real part of the refractive indices enables more independent extinction measurements except in the diameter sub-range from 0 to 5 m. Therefore, for droplets with the real part of the refractive index larger than that of water droplets at 22 ◦ C (e.g., some hydrocarbon fuel droplets), more independent extinction measurements than those shown in Fig. 4 are allowed to infer more information about the SDF, or the measurements require lower experimental accuracy for a fixed number of independent measurements. By contrast, Fig. 7 also shows that a 5% decrease in the real part of the refractive index leads to fewer independent measurements allowed. 5. Analysis applied to scattering method This section applies the method developed to analyze SDF measurement using Mie scattering, another category of techniques extensively used in aerosol research (Kerker, 1997; McMurry, 2000; Swithenbank et al., 1977). We first

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Fig. 7. Comparison of the number of independent extinction measurements allowed at varied refractive indices assuming an e level of 5%.

modify Eq. (3) to the following form:  Dmax S1 · f (D) dD = s1 , Dmin

.. . 

Dmax

Dmin

.. . 

Dmax

Dmin

Si · f (D) dD = si ,

SN · f (D) dD = sN ,

(6)

where the subscripts now refer to angles at which scattering measurements are performed; Si , the ith kernel function, is the total phase function at a scattering angle i at a fixed wavelength in the diameter range from Dmin to Dmax ; f (D) is a function equivalent to the SDF as discussed in Section 2; and si the scattering measurement at i (1 i N ). According to the Mie scattering theory (Bohren & Huffman, 1983), Si = i1 (i ) + i2 (i ),

(7)

where i1 and i2 are intensity functions at a scattering angle i for scattered light polarized perpendicular and parallel to the scattering plane, respectively. Eqs. (4) and (5) are modified accordingly to  Si = 

Dmax

Dmin

Dmax

Dmin

[Si − (w1 S1 + · · · + wi−1 Si−1 + wi+1 Si+1 + · · · + wN SN )]2 dD,

Si2 dD = 1,

i = 1, 2, . . . , N.

(8)

(9)

After these modifications, minimization of Eq. (8) will provide a quantitative measure of the level of accuracy to approximate Si by the linear combination of other kernel functions, and yield the weights used in such linear approximation. The iterative process described in Section 2 is then applied to analyze the scattering method, in terms of determining the maximum number of independent measurements and the angles to perform these measurements.

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Fig. 8. S min during the iteration process after 13 kernel functions have been removed from S1 to S21 on the diameter range from 0 to 5 m.

Fig. 9. The most independent kernel functions selected from S1 to S21 on the diameter range from 0 to 5 m when e is set to 4.5% and the corresponding scattering angles.

As an example, here we consider the measurement of SDF using forward scattering at a wavelength of 0.7 m. Other SDF measurement techniques, such as Mie scattering measurements at right or large angles for achieving spatial resolution (Beretta et al., 1984), can be analyzed similarly. Twenty-one scattering angles (’s) equally spaced between 1 and 11◦ (the spacing is 0.5◦ ) are considered in this analysis. A set of kernel functions consisting of the normalized total phase functions at these angles are calculated for water aerosols at a temperature of 22 ◦ C. These kernel functions are denoted as S1 to S21 , with S1 representing the kernel function at  = 1◦ and S21 at  = 11◦ . These functions are analyzed using the removal method described, and Fig. 8 shows the history of S min on the diameter range from 0 to 5 m after 13 kernel functions have been removed. Fig. 8 suggests that three kernel functions are experimentally independent at an e level of 5%. These three kernel functions (S2 , S14 , and S20 ) are shown in Fig. 9 and they correspond to the normalized total phase functions at 1.5, 7.5, and 10.5◦ . Any other kernel functions among S1 .S21 can be approximated by the

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Fig. 10. (a) Comparison between S19 and its best approximation by the linear combination of the three selected kernel functions shown in Fig. 9 (corresponding weights used in the approximation: w2 = 0.204, w14 = 0.415, w20 = 0.774). (b) Comparison between S14 and its best approximation by the linear combination of S2 and S20 (corresponding weights used in the approximation: w2 = 0.376, w20 = 0.590).

linear combination of these three within an e of 5%. According to the iteration process, S19 is the most independent one relative to the selected three among the 13 kernel functions removed. Part (a) of Fig. 10 shows the comparison between S19 and its best approximation by the linear combination of the three selected ones with the weights listed in the caption. If the iteration process proceeds, S14 will be removed and Fig. 8 shows that the removal of S14 causes S min to increase to about 40%. Part (b) of Fig. 10 compares S14 and its best approximation by S2 and S20 and suggests a difference that can be safely distinguished in experiments. Hence, three independent scattering measurements between 1 and 11◦ at a wavelength of 0.7 m can be performed to reliably extract three pieces of information about the SDF in the diameter range of 0.5 m. Similar analysis was performed among the kernel functions (S1 .S21 ) on different diameter sub-ranges and the results are summarized in Fig. 11. An error level (e) of 5% was taken as the limit of S min during the selection process. Fig. 11 shows that three and five independent scattering measurements are allowed in the diameter sub-ranges of 0–5 and 5.10 m, respectively, and eight in the diameter sub-range of 10–15 and 15.20 m. In comparison with the results for extinction measurements shown in Fig. 4, we see that in the diameter range of 0.10 m, more independent extinction measurements are allowed than scattering measurements. Previously, scattering methods were regarded superior (at least mathematically) to extinction methods in terms of the number of independent measurements allowed (Twomey & Howell, 1967). The analysis here shows that the use of wavelengths over a wider spectral range increases the number of inferences of the SDF by the extinction method to a level comparable to that by the scattering method. However, in the larger diameter range from 10 to 20 m, the scattering method enables more independent measurements than the extinction method. This advantage is due to the fact that the total phase functions, unlike the extinction coefficients, continue to vary significantly with diameter over a larger range, though the scattering signal level often spans a few decades and this large variation poses significant challenges for practical implementation. Application of this method to scattering techniques in a different range of scattering angles or at different wavelengths is straightforward. The consideration of the influence of the refractive index can be treated in a similar manner as in Section 4. Finally, the extension of our method to SDF measurements using Mie scattering at multiple angles and multiple wavelengths, such as in the dual wavelength optical particle spectrometer (Dick et al., 1994; Nagy et al., 2007), can be conducted by analyzing the associated kernel functions generated at the wavelengths used over the range of angles under consideration.

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Fig. 11. The number of independent scattering measurements allowed in diameter sub-ranges spanning 0.20 m for water droplets at 22 ◦ C.

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