Retrieval of fractal dimension and size distribution of non-compact soot aggregates from relative intensities of multi-wavelength angular-resolved light scattering

Jun-You Zhang; Hong Qi; Yi-Fei Wang; Bao-Hai Gao; Li-Ming Ruan

doi:10.1364/OE.27.001613

1. Introduction

Soot is the carbonaceous particles produced by the incomplete combustion of hydrocarbons, which widely exists in the flames, exhaust gas, and atmosphere etc [1]. A large number of nano-scale carbonaceous particles can stick together to form soot aggregates with complex fractal structure due to the random Brown motion. Nowadays, soot aggregates have been considered to be the second most important factor (only behind the carbon dioxide) to the climate change due to their direct absorption of solar energy, affection of clouds, and deposition on snows and ices [2]. Airborne soot aggregates are harmful to the lungs, cells, and blood circulation due to its toxicity or as a carrier for other toxic fine particulate matter [3]. Thus, it is very important to measure the morphology and particle size distribution (PSD) of soot aggregates accurately for environment protection. Meanwhile, accurate characterization of the morphology and PSD of soot aggregates is also essential for modeling radiative heat transfer in flame, climate change and environmental quality [4,5]. The significance of the morphology and PSD of soot aggregates has received increasing recognition and considerable attention in recent years .

Generally, the characterization techniques of soot aggregates can be roughly divided into two categories: ex-situ and in situ techniques [6,7]. For the ex-situ methods, which are commonly involved with direct sampling, soot particles can be collected by the physical probe and subsequently analyzed by direct imaging or optical techniques [8]. The analytical techniques include scanning electrical mobility spectrometer (SEMS), electrical aerosol analyzer (EAA) or differential mobility spectrometer (DMS), transmission electron microscopy (TEM), X-ray diffraction (XRD), Raman spectroscopy, to name a few [9]. In terms of the characterization of soot morphology, thermophoretic sampling and analysis by TEM (TS-TEM) is a standard method for accurate estimation of soot morphology and PSD (i.e. the primary particle diameter, fractal properties and size distribution) [10]. However, in order to obtain an accurate result by TS-TEM, thousands of aggregates need to be sampled and analyzed [11]. Meanwhile, the ex-situ methods will inevitably cause perturbation and even modification of soot structure during the sampling process. In addition, these sampling-based methods are only suitable for off-line and probes used in the sampling may be destroyed by the hostile measurement conditions, such as high temperature flames. In a word, applications of sampling-based ex-situ methods are limited by time-consuming operations, inevitable perturbation, off-line soot sampling and restricted measurement conditions [12].

In contrast, the in situ optical diagnostic do not have those shortcomings mentioned above due to its non-intrusive and real-time features, which plays a very important role in investigating the soot formation, growth, aggregation, and oxidation in flames and characterizing the morphology of nanoparticles such as soot, diesel particulate matter, and carbon black. The well-known optical diagnostic techniques contain laser induced incandescence (LII) [13–15], light extinction, elastic light scattering (ELS) and their combinations [16]. In terms of the characterization of the properties of soot aggregates, LII and light extinction are more suitable techniques for determining the soot volume fraction [17,18]. The time-resolved LII signals can also be applied to retrieve the primary particle size [19], size distribution of soot aggregate [20]. However, they are influenced by the dependence of LII on the temperature and primary particle diameter [21,22]. To date, the most common method for determination of soot morphology or PSD is the combination of ELS and the Rayleigh-Debye-Gans polydisperse fractal aggregates (RDG-PFA) scattering theory [23–25], which has been examined by numerical evaluations [23,26] and experimental verifications [27,28]. Given the good agreement between ELS and TS-TEM measurements over the average or distribution of aggregate size, the ELS is considered as an effective method to determine soot aggregate size [25]. A comprehensive review of different types of ELS of particles and their effect on light scattering transfer has been given by Sorensen [29]. In this review, theoretical basis of ELS was summarized for the complete characterization of soot particle system by combining optical structure factor and absolute scattering/extinction measurements, but some vital properties such as complex refractive index, number density of aggregate must be known or independently measured beforehand since the uncertainty of these parameters will lead to obviously inaccurate results. Past years have witnessed sustained efforts aimed at overcoming the defect of ELS. For instance, Köylü and Faeth [30] used the ratio of absolute scattering and extinction to measure primary particle size independent of soot aggregate concentration, but it is still limited by the knowledge of complex refractive index, primary particle size and fractal dimension. Considering relative angular scattering is independent of refractive index and only depends on the PSD, primary particle diameter and fractal properties (i.e. fractal dimension and fractal prefactor), Yang and Köylü [31] used the relative angular scattering to estimate the average aggregate size. Similarly, Oltmann et al. [32] utilized the relative intensity of wide-angle light scattering (WALS) to reconstruct the size distribution of polydisperse soot aggregates. Inspired by their work, if the primary particle size, the PSD and fractal properties of soot aggregates can be determined simultaneously by using relative intensities of angular-resolved scattering. It would be the best solution to overcome the defect of ELS because no additional parameters are required to be known. However, Link et al. [11] have demonstrated that the relative multi-angle scattering approach for a single wavelength cannot retrieve the fractal dimension and the size distribution of polydisperse soot aggregate simultaneously, and the primary particle size and fractal prefactor are still necessary to be known in Link’s study. According to Link’s work, it can be found that there is room for improvement in two aspects: First, the scattering signal can be independent of primary particle size and fractal prefactor by proper transformation. Second, the inversion accuracy can be improved by increasing the wavelength number of incident light.

The motivation of present study is to develop a method using relative angular light scattering to achieve accurate and simultaneous retrieval of fractal dimension and soot aggregate size distribution without knowing other parameters. The present work focuses on two aspects: (i) On the one hand, the distribution of radius of gyration is selected to characterize the size distribution of soot aggregate. Theoretically speaking, it allows the relative multi-angle scattering signals to be independent of the complex refractive index, primary particle size, number density of aggregate and fractal prefactor. Consequently, the remaining fractal dimension and PSD will be retrieved simultaneously with no requirement of prior known parameters. (ii) On the other hand, the multi-wavelength strategy combined with the covariance matrix adaption-evolution strategy (CMA-ES) are applied to improve the inversion accuracy of PSD because the retrieval results are far from satisfactory by using the single wavelength strategy. The reminder of this paper is as follows: First, the forward light scattering theory i.e. RDG-PFA theory is described in Section 2. And then the inverse CMA-ES algorithm and the construction of whole inversion process are shown in Section 3. The results of numerical research on PSD and fractical dimension of soot aggregates are summarized in Section 4. The main conclusions are provided in Section 5.

2. Forward light scattering theory

In principle, the interaction of light with aggregate particles depends on morphological, optical, and concentration properties of aggregate particles. Therefore, the light scattering signals can be used to simultaneously or separately retrieve all the variables involved in light scattering process. In order to retrieve the fractal dimension and PSD of soot aggregates, the first and most essential step in the solution of this complicated problem is to simulate the forward light scattering of soot aggregates efficiently and accurately.

2.1 Fractal-like soot aggregate model

Soot aggregates consist of numbers of nano-scale carbonaceous particles (primary particles) with nearly the same diameter. As shown in Fig. 1(a), due to the Brownian motion, these nano-spheres stick together to form a branching-chain structure which can be well described by the mass-fractal theory. So N, the number of primary particles per aggregate, can be mathematically related with other fractal or basic properties by the following equation [29]

N = k_{f} {(\frac{2 R_{g}}{d_{p}})}^{D_{f}}

where D_f denotes the fractal dimension; dp is the diameter of primary particle and k_f presents the fractal prefactor. As shown in Fig. 1(b), R_g is the radius of gyration, i.e. the root-mean-square radius between primary particles and the centroid of aggregate [29]

Fig. 1 (a) Field-emission scanning electron microscope images of soot particles [33]; (b) Fractal-like soot aggregate.

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2.3 RDG-PFA scattering theory

The RDG-PFA theory is commonly used for quantitative description of light scattering from fractal-like soot aggregates. According to the RDG-PFA theory, the amount of light scattering and extinction are quantified by the cross sections of an aggregate and the number density of the aggregates. The Rayleigh differential scattering cross section of an aggregate is proportional to that of a single primary, and multiplied by the structure factor function and N² to consider the effect of aggregate structure on scattering [29]

\frac{d σ_{sca}^{agg}}{d Ω} = N^{2} \frac{d σ_{sca}^{p}}{d Ω} S (q R_{g}) = N^{2} [k^{4} a^{6} F (m)] S (q R_{g}) where F (m) = {| \frac{m^{2} - 1}{m^{2} + 2} |}^{2}

where a is the radius of primary particles, a = d_p /2, q is the modulus of scattering vector, q = (4π/λ) sin(θ /2).

The total scattering cross-section of an aggregate can be expressed as [29]

\begin{array}{l} σ_{sca}^{agg} = N^{2} σ_{sca}^{p} G (k R_{g}) = N^{2} [\frac{8 π}{3} k^{4} a^{6} F (m)] G (k R_{g}) \\ where G (k R_{g}) = {(1 + \frac{4}{3 D_{f}} k^{2} R_{g}^{2})}^{- D_{f} / 2} \end{array}

where G(kR_g) is a generalization function used to make the total scattering cross section suitable for all structure factors. In this study, four typical structure factor functions are considered as listed in Table 1.

Table 1. Four typical structure factor functions

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The absorption cross section of an aggregate is equal to the absorption cross section of a single primary particle multiplied by the number of primary particles [29]

σ_{abs}^{agg} = N σ_{sca}^{p} = N [4 π k a^{3} E (m)] where E (m) = Im [\frac{m^{2} - 1}{m^{2} + 2}]

Both F (m) and E (m) are functions of complex refractive index m = m_n − m_ki. m_n and m_k are the real and imaginary part of the complex refractive index, respectively. In present study, m assumed to be constant in visible wavelength regions: m = 1.57 - 0.56i [36]. Rigorously speaking, the value of complex refractive index depends on the light wavelength, so a retrieval method independent of complex refractive index can avoid the uncertainty caused by the approximation of complex refractive index.

The total extinction cross section is equal to the sum of absorption and scattering cross section [29]

σ_{ext}^{agg} = σ_{abs}^{agg} + σ_{sca}^{agg}

The size distribution of soot aggregates in the measured volume can be described by some typical PSD functions. For most soot aggregates systems, the PSD approximately follows the Log-Normal (L-N) distribution [35]

p (R_{g}) = \frac{1}{R_{g} \sqrt{2 π} \log σ_{g}} \exp [- \frac{{(\log R_{g} - \log R_{g, geo})}^{2}}{2 {(\log σ_{g})}^{2}}]

where R_{g, geo} and σ_g are the geometric mean and standard deviation of R_g-based distribution. The variation of these two parameters will directly affect the width and peak position of the PSD curve. In this study, the R_g-based distribution is used to characterize the size distribution feature of soot aggregate instead of the more common N-based distribution.

When the incident light intensity is I₀, the absolute intensity of light scattering for polydisperse aggregates measured at the scattering angle θ is [29]

I_{A} (θ) = c_{0} I_{0} n_{agg} \int \frac{d σ_{sca}^{agg}}{d Ω} p (R_{g}) d R_{g} = c_{0} I_{0} n_{agg} \int N^{2} k^{4} a^{6} F (m) S (q (θ) R_{g}) p (R_{g}) d R_{g}

where c₀ is a constant that can be determined by calibration; n_agg denotes the number density of the aggregates, i.e. the number of aggregate per unit volume.

\begin{array}{l} I_{A} (θ) = c_{0} I_{0} n_{agg} k^{4} a^{6 - 2 D_{f}} F (m) k_{f}^{2} \int R_{g}^{2 D_{f}} S (q (θ) R_{g}) p (R_{g}) d R_{g} \\ = C_{0} (n_{agg}, d_{p}, D_{f}, m, k_{f}) \int R_{g}^{2 D_{f}} S (q (θ) R_{g}) p (R_{g}) d R_{g} \end{array}

where C₀ is a constant function involves the complex refractive index, fractal prefactor, primary particle size etc.

In the absolute extinction measurement for polydisperse aggregates, the transmitted light is [29]

I_{T} = I_{0} \exp (- τ_{ext} l) = I_{0} \exp (- n_{agg} σ_{ext}^{agg} l)

where l is the optical path through the measurement volume and τ_ext is the extinction turbidity.

For methods using absolute scattering or absolute extinction, it is crucial to determine some basic properties accurately before inversion, such as m, n_agg, d_p and k_f. These parameters will vary with the ambient conditions, working conditions and other factors. Therefore, to obtain reliable data from existing literatures or databases, these factors should be controlled strictly in the measurement. A more practical approach is to perform the in situ real-time measurement of these basic properties. But the experimental equipment system will become complicated and the measurement accuracy is also a potential challenge. So, it is a desirable choice to reduce the number of parameters required to be determined. With this idea in mind, using relative scattering signal is a good alternative, the mathematical expression of I_R is expressed as

I_{R} = \frac{I_{A} (θ)}{I_{A} (0)} = \frac{C_{0} \int R_{g}^{2 D_{f}} S (q (θ) R_{g}) p (R_{g}) d R_{g}}{C_{0} \int R_{g}^{2 D_{f}} S (q (0) R_{g}) p (R_{g}) d R_{g}} = \frac{\int R_{g}^{2 D_{f}} S (q (θ) R_{g}) p (R_{g}) d R_{g}}{\int R_{g}^{2 D_{f}} p (R_{g}) d R_{g}}

According to Eq. (10), when using the R_g-based distribution, the relative angular scattering is not involved with fractal prefactor and primary particle size. Meanwhile, the multi-angle scattering signals are chosen rather than single-angle signal to overcome the ill-posed nature of the inverse problem. There are some distinguished approaches to realize simultaneous acquisition of the multi-angle light scattering signals experimentally in the existing literatures. For example, multiple detectors are used to make a simultaneous fast measurement at several angles in [37]; or an ellipsoidal mirror is used to image the scattered light over a wide range of angles onto a planer detector in [38]. Furthermore, the multi-wavelength measurement of different scattering signal model can be realized by using a multi-wavelength laser source.

3. Inverse problem

3.1 CMA-ES algorithm

The stochastic heuristic intelligent algorithms such as particle swarm optimization (PSO) algorithm [39], ant colony optimization (ACO) algorithm [40] and genetic algorithm (GA) algorithm [41], are often used to solve the inverse problem of light scattering [42,43]. In present work, the covariance matrix adaption-evolution strategy (CMA-ES) is introduced as the inversion algorithm for inverse problem of soot aggregates, due to its wide application and outstanding performance in machine learning and computer graphics [44].

With the efforts of Hansen, Ostermeier and their colleagues [45–47], the CMA-ES algorithm has been developed to be a well-established algorithm system that contains many variants for different types of problems. In this study, the integral equations of light scattering signals needs to be deconvoluted to reconstruct the size distribution and morphological parameters. Mathematically, it is a typical ill-conditioned multiple dimensional non-separable inverse problem. Given that the good performance of CMA-ES in solving the ill-conditional and non-separable problems [45,46,48], this promising algorithm is suitable to solve the inverse problem of light scattering. What follows is a brief introduction of the theoretical fundamentals of CMA-ES algorithm, heavily based on a tutorial written by Hansen [46], to which the reader can refer for a full and detailed derivation.

In the CMA-ES, if the number of objective parameters to be inverted is n, then the search point is an n-dimensional vector, also called as the objective vector. The basic equation of CMA-ES is to generate the new search points by sampling a multivariate distribution [46]

x_{k}^{(g + 1)} \sim m^{(g)} + σ^{(g)} N (0, C^{(g)}) for k = 1, ..., λ

where

x_{k}^{(g + 1)} \in ℝ^{n}

is the kth offspring individual in the population of λ new search points at generation g + 1; ~denotes the same distribution on the left and right ride;

m^{(g)} \in ℝ^{n}

is the mean value of the search distribution at generation g;

σ^{(g)} \in ℝ^{+}

is the step-size at generation g;

N (0, C^{(g)})

denotes a multivariate normal distribution with zero mean and covariance matrix

C^{(g)}

; and

m^{(g)} + σ^{(g)} N (0, C^{(g)}) ~ N (m^{(g)}, {(σ^{(g)})}^{2} C^{(g)})

;

C^{(g)} \in ℝ^{n \times n}

is the covariance matrix at generation g;

λ \geq 2

is the population size or the number of the offspring individuals. Then, the kernel of CMA-ES is to update distribution parameters

m^{(g)}

,

σ^{(g)}

and

C^{(g)}

. Update of the mean value

m^{(g)}

is achieved by selection and recombination, which means the new mean

m^{(g + 1)}

of the search distribution is a weighted average of μ selected points from samples

x_{1}^{(g + 1)}, x_{2}^{(g + 1)} ..., x_{λ}^{(g + 1)}

[46]

m^{(g + 1)} = \sum_{i = 1}^{μ} w_{i} x_{i : λ}^{(g + 1)}

where μ ≤ λ is the parent population size, i.e. the number of selected points;

w_{i = 1... μ} \in ℝ^{+}

is positive weight coefficients for recombination such that

w_{1} \geq w_{2} \geq ... \geq w_{μ} > 0

and

\sum_{i} w_{i} = 1

;

x_{i : λ}^{(g + 1)}

is the ith best individual out of

x_{1}^{(g + 1)}, x_{2}^{(g + 1)} ..., x_{λ}^{(g + 1)}

and the index i: λ denotes the index of the ith ranked individual, the ‘best’ and ‘ranked’ individuals are both defined by their objective function value. The smaller the individual’s objective function value is, the better the individual is, and the higher its ranking is

f (x_{1 : λ}^{(g + 1)}) \leq f (x_{2 : λ}^{(g + 1)}) \leq \dots \leq f (x_{λ : λ}^{(g + 1)})

where f is the objective function to be minimized.

The update of the covariance matrix is realized by the covariance matrix adaption based on rank-μ update and rank-one update. The aim of rank-μ update is to make the estimation of covariance matrix reliable for the small population. The rank-one update means that there is only one search point per generation for updating the covariance matrix in limited cases. Finally, rank-μ update and rank-one update are combined to generate new covariance matrix [46]

C^{(g + 1)} = (1 + \underset{can be close or equal to 0}{\underset{︸}{c_{1} δ (h_{σ}) - c_{1} - c_{μ} \sum_{i = 1}^{λ} w_{i})}} C^{(g)} + c_{1} \underset{rank-one update}{\underset{︸}{p_{c}^{(g + 1)} {[p_{c}^{(g + 1)}]}^{T}}} + c_{μ} \underset{rank- μ update}{\underset{︸}{\sum_{i = 1}^{λ} w_{i} y_{i : λ}^{(g + 1)} {[y_{i : λ}^{(g + 1)}]}^{T}}}

where

c_{μ} \leq 1 - c_{1}

is the learning rate for the rank-μ update of the covariance matrix update;

c_{1} \leq 1 - c_{μ}

is the learning rate for the rank-one update of the covariance matrix update;

y_{i : λ}^{(g + 1)} = (x_{i : λ}^{(g + 1)} - m^{(g)}) / σ^{(g)}

;

h_{σ}

is the Heaviside function;

δ (h_{σ}) = (1 - h_{σ}) c_{c} (2 - c_{c}) \leq 1

;

p_{c}^{(g + 1)} \in ℝ^{n}

is the evolution path at generation g + 1; The evolution path can be expressed by a sum of consecutive steps which is referred to as cumulation [46]

p_{c}^{(g+1)} = (1 - c_{c}) p_{c}^{(g)} + h_{σ} \sqrt{c_{c} (2 - c_{c}) μ_{e f f}} \frac{m^{(g+1)} - m^{(g)}}{σ^{(g)}}

where

c_{c} \leq 1

is the learning rate for cumulation for the rank-one update of the covariance matrix; μ_eff denotes the variance effective selection mass for the mean. The factor

\sqrt{c_{c} (2 - c_{c}) μ_{e f f}}

is a normalization constant for

p_{c}

.

The evolution path is also utilized to control the step-size. Unlike Eq. (14), applying the same technique in Eq. (15) is to build a conjugate evolution path as [46]

p_{σ}^{(g + 1)} = (1 - c_{σ}) p_{σ}^{(g)} + \sqrt{c_{σ} (2 - c_{σ}) μ_{e f f}} {(C^{(g)})}^{- 0.5} \frac{m^{(g + 1)} - m^{(g)}}{σ^{(g)}}

where

p_{σ}^{(g)} \in ℝ^{n}

is the conjugate evolution path at generation g;

c_{σ} < 1

is the learning rate for the cumulation for the step-size control;

\sqrt{c_{σ} (2 - c_{σ}) μ_{e f f}}

is a normalization constant;

{(C^{(g)})}^{- 0.5} = B^{(g)} {(D^{(g)})}^{- 1} {(B^{(g)})}^{T}

, where

C^{(g)} = B^{(g)} {(D^{(g)})}^{2} {(B^{(g)})}^{T}

is an eigendecomposition of

C^{(g)}

, where

B^{(g)} \in ℝ^{n}

is an orthonormal basis of eigenvectors, and the diagonal elements of the diagonal matrix

D^{(g)} \in ℝ^{n}

are square roots of the corresponding positive eigenvalues.

To update the step-size $σ^{(g)}$ , the matrix norm of conjugate evolution path of $| | p_{σ}^{(g+1)} | |$ is compared with its expected length $E | | N (0, I) | |$ , that is [46]

σ^{(g+1)} = σ^{(g)} \exp [\frac{c_{σ}}{d_{σ}} (\frac{| | p_{σ}^{(g+1)} | |}{E | | N (0, I) | |} - 1)]

where

d_{σ} \approx 1

is the damping parameter for step-size update;

I \in ℝ^{n \times n}

is unity matrix. In present study, the strategy parameters settings of CMA-ES is available in [46].

3.2 Inverse process

Similar to other optimization algorithms, the inverse process of CMA-ES is to search in a finite space within finite searching generations to find the global best point which can minimize the value of objective function (i.e. fitness function).

As shown in Fig. 2, the inversion process can be summarized as follows:

Fig. 2 The flowchart of the inversion process.

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(1) After the initialization, in the generation g, the estimated objective parameters P_est are substituted into the RDG-PFA theory to calculate the estimated optical signals S_est.
(2) The present study is completely based on numerical simulation, so the ideal signals with Gaussian measurement noise are employed to simulate the measured signals S_mea. The ideal signals are obtained by substituting the original objective parameters P_ori into the RDG-PFA theory.
(3) An objective function F_obj is constructed based on the estimated signals S_est and measured signals S_mea.
(4) In the next generation g + 1, CMA-ES algorithm will update P_est based on F_obj.
(5) The above procedure is repeated until either one of the following two conditions is satisfied, i.e. (i) The objective function F_obj is less than the expected accuracy eps; (ii) The generation g reaches the expected maximum generations max.
(6) Finally, the whole process ends and the last generation of P_est is output as the final retrieval results.

The objective function F_obj can be defined as

F_{obj} = \sum_{i = 1}^{N_{λ}} {{\sum_{j = 1}^{N_{θ}} [\frac{S_{est} (θ_{j}, λ_{i}) - S_{mea} (θ_{j}, λ_{i})}{S_{mea} (θ_{j}, λ_{i})}]}^{2}} / (N_{λ} \cdot N_{θ})

where the subscript j denotes the optical signal at jth scattering angle; subscript i denotes the optical signal at ith wavelength; N_θ is the total number of the scattering angles. N_λ is the total number of the incident wavelength. S_est is the estimated optical signals and S_mea is the ‘measured’ (simulated) optical signals, both of which depend on the different scattering signals models.

As shown in Fig. 3, three scattering signal models are investigated in present research: (1) Absolute scattering and transmittance (AS&AT); (2) Absolute scattering (AS); (3) Relative scattering (RS). The prior parameters of these three signals are shown in Table 2. Measurement parameters such as scattering angle or incident wavelength can be determined directly by measurement or calibration. Therefore, they are usually considered to be known. For the method using RS model, the biggest advantage is that no prior parameters are needed.

Fig. 3 The schematic of three different scattering signal models.

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Table 2. Comparison of three scattering signal models

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4. Results and discussions

All the cases are implemented using the Matlab code, and the developed program is executed on an Intel Core i7-3770 PC (3.40GHz) and 15.9 GB RAM. In the test case, the original values of properties are from the measurement results in [10,49], which have been summarized in Table 3.

Table 3. The original values of properties in the test case

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Other parameter settings of inversion are shown in Table 4. It contains the value of experimental parameters which are required in the AS&AT and AS signals model, the iteration settings and searching range of objective parameters.

Table 4. Other parameter settings of inversion

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The search ranges of objective parameters in this study are much larger than their typical value ranges. For instance, the search range of D_f is set to be in the range of 1 to 3 where D_f = 3 is the limit of a completely compact spherical aggregate, while the typical value of D_f is between 1.6 and 1.9 [50]. Although this method can be implemented in a large search space of D_f, this work is only applicable to non-compact soot with D_f ranging from 1.6 to 1.9. Given the stochastic nature of CMA-ES algorithm, the calculation of test case will run 100 times and the average values are selected to reduce the influence of randomness on the retrieval accuracy. The final estimated results of every objective parameter are the average of 100 independent runs. The retrieval results will be evaluated by two measurement and statistical criterions, i.e. relative error and the standard deviation.

The relative error ε_rel is used to evaluate the accuracy of estimated results, which is defined as

ε_{rel} = \frac{| z_{est} - z_{ori} |}{z_{ori}} \times 100 %

where for an objective inverse parameter, z_ori is its original value and z_est is its average of estimated value. Lower relative error means higher retrieval accuracy.

The standard deviation SD, which is used to quantify the dispersed degree of a set of data, is given by

S D = \sqrt{\frac{1}{N u m} \sum_{i = 1}^{N u m} {(x_{i} - a v g)}^{2}}

where for an objective parameter, Num denotes the number of data which equal to the times of runs, x_i is the estimated result of the ith run; avg is the arithmetic mean of

x_{1}, x_{2}, x_{3}, \dots x_{Num}

. Lower standard deviation indicates that the data is distributed closer to their arithmetic mean.

It is emphasized that RDG-PFA theory is the reasonable approximation in the range of validity. x_p = πd_p/λ <0.3 and |m-1| < 1, which have been met simultaneously in our study, are generally considered as the applicable conditions of RDG-PFA. In this circumstance, when compared to more exact theories such as integral equation formulation for scattering (IEFS) or multi-sphere T-Matrix (MSTM), the model error of RDG-PFA on prediction of angular scattering intensity is less than 10% [26] [51].

4.1 Preliminary retrieval results

In the preliminary results, 32 scattering angles, uniformly distributed between 10° and 165°, are chosen as optical signals. Three signal models are combined with three different multi-wavelength strategies to improve the retrieval accuracy. Three different multi-wavelength strategies use different numbers of incident laser wavelength: (i) 550 nm (single-wavelength, or 1-λ); (ii) 432.8 nm, 550 nm and 806.5 nm (3-wavelength, or 3-λ); (iii) 432.8 nm, 550 nm, 660 nm, 770 nm and 806.5 nm (5-wavelength, or 5-λ). According to our previous study, the multi-wavelength strategies can improve the retrieval accuracy [52]. Generally, the more incident laser wavelength is used, the more accurate the retrieval results are obtained, but the laser devices required in the measurement will be more complicated. First, the retrieval process of test case is performed under ideal condition, i.e. without the interference from measurement noise.

Table 5 only shows the retrieval results based on Lin’s [34] structure factors, since the results based on other structure factors are almost the same. In Table 5, it can be found that the retrieval results in a no-noise situation are obtained with high precision, where ε_rel of all three different signal models is less than 0.004%. Simultaneous and accurate retrieval of D_f, N_g and σ_g can be easily achieved in this situation. However, this ideal situation is impossible in a real measurement, the inevitable measurement noise presents a difficult challenge for retrieval process. To make this method more practical, the following results are all obtained under 10% Gaussian noise. The similar trend is embodied in the retrieval results of different structure factors, so the results of Lin’s [34] structure factor is taken as an example in Table 6. It is not surprising that the presence of measurement noise seriously deteriorates the retrieval accuracy. But it is notably that the inversion of D_f is accomplished with surprising precision, especially compared with the retrieval accuracy of size distribution parameters.

Table 5. Retrieval results using single wavelength without measurement noise

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Table 6. Retrieval results with 10% Gaussian measurement noise using different multi-wavelength strategies

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In terms of retrieval accuracy of objective parameters, the AS&AT model exhibits the expected higher precision than the AS model in the inversion of the size distribution parameters. Meanwhile, the estimated D_f of two models are almost same and very accurate (ε_rel < 1%). Due to the improvement of multi-wavelength strategies, the retrieval error of σ_g is within 4%, but even with the multi-wavelength strategies, the retrieval error of another size distribution parameter R_g,geo is still greater than 20%. So the common shortcoming for the schemes using absolute intensity is that the retrieval results of R_g,geo deviates significantly from its original value, resulting in unreliable particle size distribution inversion results. Conversely, in the results of RS model, the difference in retrieval accuracy between three objective parameters is not as great as the results of using absolute scattering. However, ε_rel of all three objective parameters are larger than 6% when using single-wavelength RS signal, the retrieval accuracy of these three objective parameters is not accurate enough. Fortunately, the results of all three objective parameters are effectively improved by using the multi-wavelength strategy. In terms of multi-wavelength strategies, it can be found that for all three signal models, their results are significantly improved after using the multi-wavelength strategies. And the trend SD is positively related to the trend of ε_rel. This reflects the essential reason for the improvement of multi-wavelength strategies, i.e. multi-wavelength strategies can effectively constrain the distribution of retrieval results for each run. Certainly, the constraint effect varies depending on the signal models. For AS&AT and AS models, the retrieval errors of objective parameters are reduced by approximately 50% from single wavelength to three wavelengths. But from five wavelengths to three wavelength, error reduction is only about 1% for AS&AT, and error even increases for AS. And for RS signal model, the retrieval errors of all three objective parameters are steadily decreasing with the increasing of wavelength number especially for R_g,geo. When using the 5-wavelength strategy, the ε_rel of R_g,geo and D_f are 0.31% and 1.92%, respectively, which are both accurate enough, and only the ε_rel of σ_g is far more than 2%, which is 7.55%.

A direct comparison between original PSD curve and the curves based on preliminary results is more suitable for objectively evaluating the retrieval accuracy of R_g,geo and σ_g. So the PSD curves based on the preliminary results of Table 6 are plotted in Fig. 4, where (b) are the enlarge view of peak portion of (a). It can be seen that as long as the multi-wavelength strategy is employed, the recovered PSD curve is closer to the original one. Although using more incident wavelength can make the recovered PSD curves closer to the original one, this improvement is limited. So the recovered PSD curves basically depends on the signal models. The recovered PSD curve through RS signal model is closest to the original one. The PSD curves recovered by 5-wavelength with RS model are substantially coincident with the original PSD, although there is a deviation between the recovered PSD and original one at the peak portion. This reveals that the RS model is a better measurement signal strategy for recovering PSD than AS&AT and AS model. These preliminary results have been proven that retrieval method based on RS signals is feasible and competitive.

Fig. 4 PSD curve based on preliminary results.

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To explain the constraint effect brought by multi-wavelength strategies, 100 independent runs obtained by RS signals for different wavelength on their (R_g,geo & σ_g) fitness value contour surface (FVCS) are plotted in Fig. 5. FVCS are based on the based-10 logarithm of Fitness (i.e. objective function) Value. The multi-wavelength strategies effectively constrain 100 runs to a much smaller range, and as the number of wavelengths increases, the distribution of 100 runs becomes much denser. It contributes to the expected trend that the average value of 100 independent runs becomes more closer to the original value.

Fig. 5 100 independent runs obtained by RS signals for different wavelength plotted on (R_g,geo & σ_g)-FVCS.

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The following study is the optimization of the scattering angle contains optimizing the angle combination and finding the minimum number of angles, which has been studied by some previous works. For instance, and He et. al. [53] improved scattering angle combination by the PCA approach, and Burr used design-of-experiment theory to derive an optimal set of angles, but the improvements from the optimal set of angles are negated by the regularization error and the perturbation of noise [54]. Therefore, we focus on determining the least number of scattering angle.

4.2 Simplification strategy of the scattering angle number in RS model

The retrieval method based on 5-wavelength RS signals has been validated in the above section, but it is based on 32 angle scattering signals. (32 scattering angles are uniformly distributed from 10° to 165°) So this method requires 32 detectors in the real measurement, it is difficult to realize the simultaneous measurement at 32 angle by multiple detectors approach. Although the ellipsoidal mirror can be used to simultaneously measure 32 or more angle signals, it will limit the target size and usage scenario. Thus, if the scattering angle number can be reduced to be no more than 10 or even less, the corresponding measurement device will become simple and easy to construct.

It seems that the two scattering angles is an ideal ultimate solution, but because of the ill-posed nature of the inverse problem, it is impossible to realize the retrieval of objective parameters using only two scattering angle signal even without the measurement noise. To explain that, we compare no-noise (R_g,geo&σ_g)-FVCS obtained by using 2-angle and 5-angle RS signal in Fig. 6. Like the (R_g,geo&σ_g)-FVCS of 32-angle, there is only one global optimal point in the (R_g,geo&σ_g)-FVCS of 5-angle RS signal. However, in the (R_g,geo&σ_g)-FVCS of two-angle RS signal, there are large number of local optimal points whose fitness value are less than 10⁻⁶. The search point may be trapped by any of these point, which means that any of these points can be the final retrieval result. This inevitably leads to inaccurate results, not to mention the inversion results are even worse if they are disturbed by measurement noise. Therefore, more angle signals are needed to ensure the accuracy of the retrieval method.

Fig. 6 The no-noise (R_g_,geo&σ_g)-FVCS obtained by using (a) 2-angle and (b) 5-angle RS signal for single wavelength.

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After multiple attempts of numerous numerical experiments, it was found that the RS signals required at least 8 angles to obtained retrieval results stably under 10% Gaussian measurement noise. These 8 angles are uniformly distributed between 10° and 165°. Table 7 compares the retrieval results of 8-scattering-angle RS signals using different multi-wavelength strategies. With the increasing of incident wavelength number, the ε_rel of R_g,geo decreases but that of σ_g increases. Due to this particular trend, it is difficult to determine which of these two cases yields the most accurate inversion of PSD parameters. The PSD results based on 8-angle and 32-angle under different wavelengths are directly compared in Fig. 7.

Table 7. Retrieval results based on 8-angle RS signals under different multi-wavelength strategies

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Fig. 7 The retrieval PSD based on 8-angle and 32-angle RS signals under different multi-wavelength strategies.

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The PSD curves obtained from the 32 scattering angles are significantly closer to the original one than the PSD curves obtained from the 8 scattering angles. The PSD curve recovered by using 8-angle under 5 wavelength is similar to the original PSD in the overall trend, and the difference between them is concentrated at the peak portion of the curve. Although the maximum deviation at the peak portion of the curve is nearly 20%, if the focus of the inversion is on the reconstruction of the overall trend, this result is acceptable. But this may lead to a more significant deviation when the method is used with real measurement data.

4.3 Efficiency of CMA-ES algorithm

To elucidate the fast and efficient performance, the retrieval results using CMA-EA and IQPSO algorithm are compared. The improved quantum particle swarm optimization (IQPSO) algorithm has proven efficient and stable in realizing multi-parameters inversion [55–57]. The comparison between CMA-ES and IQPSO is on premising of the same signal model, i.e. 5-wavelength 32-angle RS signals as shown in Table 8. The expected maximum generations of the two algorithms are both 1000 generations.

Table 8. Retrieval results using CMA-ES and IQPSO algorithm based on 5-wavelength 32-angle RS signals

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In terms of retrieval accuracy, the results of IQPSO are inaccurate especially in the retrieval of size distribution parameter. And there is a huge difference in the average computing time between the two algorithms. The average computing time of CMA-ES is only 4.58 seconds, while the average computing time of IQPSO is 37.8 seconds, which is more than 8 times that of CMA-ES. The calculation speed of CMA-ES is impressive and mostly attribute to the core of CMA-ES algorithm is based on direct matrix operations such as eigenvalue decomposition. Actually, there is room for improvement in the computing time of CMA-ES algorithm. Converge process curves of 50 runs is depicted in Fig. 8, which are obtained in the case using 5-wavelength 32-angle RS signals. In order to clearly display the results, only 50 of 100 curves are plotted, but the trend is same when 100 curves are plotted.

Fig. 8 Converge curves of 50 runs obtained by 32-angle RS signals under 5 wavelength in (a) no-noise and (b) 10% Gaussian noise.

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In the Fig. 8 (a) without measurement noise, the objective function value of all curves quickly decrease below eps (10⁻¹⁰) in no more than 700 generations, and most of them (41 out of 50) are in no more than 350 generations. In the Fig. 8 (b) with 10% Gaussian measurement noise, the objective function value of all curves still rapidly decrease at the beginning, but soon encounter the low-fitness-value area and keep between 10⁻² and 10⁻³ until its generation reaches the expected maximum one. The effective convergence has already finished before entering the minimum objective value area, so if the expected maximum generations change from 1000 to 500, the whole retrieval process can be finished in 2.29 seconds which proves the CMA-ES is more efficient.

The past few years witnessed increasing focus on using Bayesian inference strategy to achieve a confidence interval for the inverted parameters when compared to obtaining a unique solution [58]. It provides a different kind of solution for our study. Since the Bayesian inference strategy have been used in many studies [59-63], it is expected to achieve the combination of CMA-ES and Bayesian inference strategy in our next stage work. In addition, a few parameterization techniques [64-68] will be adopted to dispose the model error of RDG-PFA in the future work.

5. Conclusions

In the present work, the proposed method is accurately and stably to realize the retrieval of fractal dimension and size distribution of polydisperse soot aggregate by using the relative intensity of multi-angle scattering for multiple wavelength. The outstanding advantage of this method is that the relative scattering intensity can theoretically avoid the independent measurements of non-objective parameters before inversion, and no additional parameters are required to be known. A proof-of-concept numerical demonstration is presented from different perspective. The following conclusions can be drawn:

(1) Compared with the methods using absolute scattering intensity, the method based on relative scattering intensity was proved to be more accurate under 10% Gaussian measurement noise.
(2) The multi-wavelength strategy significantly improves the retrieval accuracy of size distribution parameters because of its excellent constraint effect on the results.
(3) An accurate reconstruction of fractal dimension and PSD curve can be obtained by 32 angle of RS signals under 5 wavelength. The number of scattering angles can be reduced to 8 but the reconstructed PSD will have a deviation of nearly 20% at the peak of the curve.
- (5) The proposed method uses the CMA-ES algorithm, which ensures high retrieval precision and computational speed compared to the IQPSO algorithm.

In conclusion, all numerical results show that the proposed method based on relative scattering intensity is feasible to simultaneously retrieve the PSD of aggregate particles which is a new promising prediction tool. Further research will be conducted to prove the feasibility of this method experimentally.

Funding

National Natural Science Foundation of China (No. 51576053, 51806047).

Acknowledgments

The authors thank Prof. Kuanfang Ren from University of Rouen for useful discussion and exchange of information. A very special acknowledgment is also made to the editors and referees who make important comments to improve this paper.

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Researchers	Notation	Structure factor functions $S (q R_{g})$
Lin et al. [34]	Lin	$\begin{array}{l} S (q R_{g}) = {(1 + \sum_{m = 1}^{M} C_{m} {(q R_{g})}^{2 m})}^{- D_{f} / 2 M} \\ C_{1} = 2 M / 3 D_{f}, C_{2} = 2.5, C_{3} = - 1.52, C_{4} = 1.02 \end{array}$
Dobbins and Megaridis [23]*	D&M	$\begin{array}{l} S (q R_{g}) = \exp [- {(q R_{g})}^{2} / 3] small q R_{g} (q R_{g} < \sqrt{3 D_{f} / 2}) \\ S (q R_{g}) = {(3 D_{f} / 2 e)}^{D_{f} / 2} {(q R_{g})}^{- D_{f}} large q R_{g} (q R_{g} \geq \sqrt{3 D_{f} / 2}) \end{array}$
Köylü and Faeth [35]	K&F	$\begin{array}{l} S (q R_{g}) = \exp [- {(q R_{g})}^{2} / 3] small q R_{g} (q R_{g} < \sqrt{3 D_{f} / 2}) \\ S (q R_{g}) = {(q R_{g})}^{- D_{f}} large q R_{g} (q R_{g} \geq \sqrt{3 D_{f} / 2}) \end{array}$
Yang and Köylü [31]	Y&K	$\begin{array}{l} S (q R_{g}) = {(1 + \sum_{m = 1}^{M} C_{m} {(q R_{g})}^{2 m})}^{- D_{f} / 2 M} \\ C_{1} = 2 M / 3 D_{f}, C_{2} = C_{3} = 0, C_{4} = 1 \end{array}$

Three signals models		Parameters need to be known before inversion
Abbr.	Basic formulas	Measurement parameters	Basic property parameters
AS&AT	Equation (8) & Eq. (9)	$λ, θ, c_{0}, I_{0}, l$	$n_{agg}, m, d_{p}, k_{f}$
AS	Equation (8)	$λ, θ, c_{0}, I_{0}$	$n_{agg}, m, d_{p}, k_{f}$
RS	Equation (10)	$λ, θ$	–

Signal schemes	AS&AT			AS			RS
Criterions	avg	ε_rel[%]	SD	avg	ε_rel[%]	SD	avg	ε_rel[%]	SD
D_f = 1.70	1.700	0.00	5.8 × 10⁻⁶	1.700	0.00	5.9 × 10⁻⁶	1.700	0.00	5.9 × 10⁻⁶
R_g_,geo = 117	117.0	0.00	1.7 × 10⁻³	117.0	0.00	1.7 × 10⁻³	117.0	0.00	1.7 × 10⁻³
σ_g = 2.10	2.100	0.00	1.1 × 10⁻⁴	2.100	0.00	1.1 × 10⁻⁴	2.100	0.00	1.1 × 10⁻⁴
Fitness/Time[s]	3.00 × 10⁻¹¹ / 1.452			2.75 × 10⁻¹¹ / 1.082			2.75 × 10⁻¹¹ / 0.962

AS&AT	Single-wavelength			3-wavelength			5-wavelength
Criterions	Avg	ε_rel[%]	SD	avg	ε_rel[%]	SD	avg	ε_rel[%]	SD
D_f = 1.70	1.713	0.74	0.134	1.704	0.24	0.097	1.698	0.14	0.090
R_g_,geo = 117	164.1	40.3	118.9	145.4	24.3	101.2	145.3	24.2	101.0
σ_g = 2.10	1.967	6.36	0.591	2.044	2.68	0.455	2.047	2.54	0.420
Fitness/Time[s]	9.74 × 10⁻³ / 4.79			1.05 × 10⁻² / 6.04			1.07 × 10⁻² / 6.05
AS	Single-wavelength			3-wavelength			5-wavelength
Criterions	Avg	ε_rel[%]	SD	avg	ε_rel[%]	SD	avg	ε_rel[%]	SD
D_f = 1.70	1.712	0.70	0.135	1.703	0.18	0.100	1.697	0.19	0.103
R_g_,geo = 117	168.9	44.4	127.3	147.9	26.4	107.8	151.9	29.9	117.0
σ_g = 2.10	1.965	6.41	0.625	2.033	3.19	0.484	2.027	3.46	0.456
Fitness/Time[s]	9.52 × 10⁻³ / 3.91			1.04 × 10⁻² / 4.43			1.09 × 10⁻² / 5.03
RS	Single-wavelength			3-wavelength			5-wavelength
Criterions	Avg	ε_rel[%]	SD	avg	ε_rel[%]	SD	avg	ε_rel[%]	SD
D_f = 1.70	1.805	6.19	0.243	1.766	3.87	0.145	1.733	1.92	0.126
R_g_,geo = 117	130.5	11.6	132.0	107.3	8.29	86.29	116.6	0.31	84.73
σ_g = 2.10	2.456	16.9	1.003	2.356	12.2	0.602	2.258	7.55	0.485
Fitness/Time[s]	9.86 × 10⁻³ / 3.88			1.05 × 10⁻² / 4.37			1.05 × 10⁻² / 4.58

	RS (3-wavelength, 8-angle)			RS (5-wavelength, 8-angle)
Criterions	avg	ε_rel[%]	SD	avg	ε_rel[%]	SD
D_f = 1.70	1.707	0.42	0.201	1.769	4.04	0.195
R_g_,geo = 117	174.0	48.7	126.2	129.6	10.8	113.6
σ_g = 2.10	2.052	2.26	0.692	2.270	8.08	0.674
Fitness/Time[s]	9.33 × 10⁻³ / 3.89			9.28 × 10⁻³ / 3.94

Retrieval of fractal dimension and size distribution of non-compact soot aggregates from relative intensities of multi-wavelength angular-resolved light scattering

Abstract

1. Introduction

2. Forward light scattering theory

2.1 Fractal-like soot aggregate model

2.3 RDG-PFA scattering theory

3. Inverse problem

3.1 CMA-ES algorithm

3.2 Inverse process

4. Results and discussions

4.1 Preliminary retrieval results

4.2 Simplification strategy of the scattering angle number in RS model

4.3 Efficiency of CMA-ES algorithm

5. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (8)

Tables (8)

Equations (19)

Optics Express

Experiment		Iteration settings		Search range of objective parameters
c₀	l [cm]	eps	max	D_f	R_{g, geo}	σ_g
0.8	2	10⁻¹⁰	1000	(1, 3)	(1, 1000)	(1, 10)