## Abstract

For the ill-posed inverse problem of LII-based nanoparticle size measurement, recovered primary particle size distribution (PPSD) is sensitive to the uncertainty of LII model parameters. In the absence of reliable prior knowledge, the thermal accommodation coefficient (TAC) and fractal-dependent shielding factor are often required to be inferred simultaneously with the PPSD. In the simplified LII model for low fluence regime, TAC and fractal-dependent shielding factor are combined to define a new fractal-dependent TAC. The present study theoretically verified the feasibility of inferring PPSD and fractal-dependent TAC from the normalized LII signals. Moreover, the inversion is independent of prior knowledge of most full LII model parameters, which is attributed to low laser fluence, normalized signal, and fractal-dependent TAC.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Primary particle size distribution (PPSD) soot aggregate is an important property for human health and global climate modeling [1,2]. Laser-induced incandescence (LII) has developed as a powerful tool for estimating volume fraction, mean primary particle size, and PPSD of soot aggregate [3,4]. Compared with sampling-based ex-situ methods such as transmission electron microscope (TEM), LII has the advantage of non-invasive measurement and is applicable for fast online in-situ measurement [3].

To date, there have been many studies on the determination of PPSD with time-resolved laser-induced incandescence (TiRe-LII) signals. Roth et al. [5] presented an in situ method for inferring particle size distribution by fitting the intensity of TiRe-LII signals. Filippov et al. [6] applied this method to determine the PPSD of various types of aerosols. Kock et al. [7] demonstrated the capability of this method to obtain almost instantaneous soot PPSD in diesel engine combustion with rapidly varying conditions. Lehre et al. [8] extended this method to simultaneously measure PPSD and gas temperature. Lehre et al. [9,10] used the effective particle temperature derived from the ratio of TiRe-LII signals measured at two wavelengths to obtain PPSD, and proved that it can be used to infer PPSD and flame temperature simultaneously. Liu et al. [11] proposed a simple method to determine the soot PPSD by using the effective temperature in the non-sublimation regime. Danker et al. [12] retrieved the soot PPSD from two different characteristic time intervals of TiRe-LII curve.

The above LII-based PPSD recovery studies are hampered by the ill-posed nature of inverse problem, such that the recovered PPSD is sensitive to the uncertainties in the LII model parameters [13]. Under low fluence conditions, the accurate prior knowledge of the thermal accommodation coefficient (TAC or *α*_{T}) is necessary for a reliable estimate of PPSD. Michelsen [14] and Daun [15–17] used extrapolation from low temperature data using and excitation probability and molecular dynamics, respectively, to predict TAC prior considering LII data. Added factors, such as the influence of soot composition and maturity on the TAC values, may further complicate inference. Uncertainties are often lowered by assuming that the PPSD is log normal, reducing the PPSD inference problem to one determining the geometric mean primary particle diameter, *d*_{p, g}, and the geometric standard deviation, *σ*_{d, g}. Kock et al. [18] demonstrated such a scheme, where ex-situ measurements were used to determine *σ*_{d, g} and the TAC and geometric mean diameter were inferred simultaneously from TiRe-LII signals. Sipkens et al. [13,19] later showed that inference of the TAC and particle size is restricted to the transition or high fluence regimes, as the structure of the problem (wherein the particle size and TAC appear as a product) make it nearly impossible to determine TAC and *d*_{p, g} from a conduction-dominated temperature decay. On the other hand, the above studies are generally limited to cases where the thermal shielding effect caused by the fractal structure of aggregate is neglected. Bauer et al. [20] revealed that the fractal parameters highly correlated with the PPSD parameters amplifying the uncertainties in the recovered PPSD, As such, to determine the influence of fractal-dependent shielding factor, it is necessary to obtain prior knowledge of fractal structure and polydisperse distribution of aggregate size to infer the remaining particle properties. Unfortunately, fractal structure and aggregate size vary with the soot generation conditions or environmental conditions, and it is difficult to obtain their accurate values without additional sampling-based measurements.

In the absence of reliable prior knowledge, TAC and fractal-dependent shielding factor are required to be inferred with PPSD simultaneously, but it will not generally be possible to infer these quantities independently. Instead, we define a new fractal-dependent TAC, which combines the overall impact of TAC and fractal-dependent shielding factor on the simplified LII model for low fluence regime. The purpose of this paper is to theoretically evaluate the capability of inferring the PPSD and fractal-dependent TAC of soot aggregates from normalized TiRe-LII signals. In order to avoid the common inverse crime (see Section 3) in theoretical evaluation, this study uses a full LII model to synthesize LII data for evaluation, while uses a simplified model of the low-fluence regime to perform the inversion process. Since the dependence of LII model on aggregate structure, aggregate size distribution, and thermal accommodation coefficient is completely represented by the new parameter *α*_{T, f} (see Section 3), the inference of PPSD and *α*_{T, f} does not require the prior information of these properties. In addition, the inference of PPSD and *α*_{T, f} does not require the prior information of absorption function at the measurement wavelength, the apparent fraction volume, and the calibration parameter when normalized LII signal is used for inversion analysis.

This paper is structured as follows. Section 2 describes the fractal-based full LII model used for LII data simulation, the simplified LII model in the low-fluence regime for inverse problem analysis, and the definition of fractal-dependent TAC. Section 3 explains the inversion process. Section 4 evaluates the feasibility of inferring PPSD and fractal-dependent TAC from normalized TiRe-LII signals, and analyzes log contour of the objective function. Section 5 summarizes the main conclusions of this paper.

## 2. Time-resolved laser-induced incandescence model

#### 2.1 Full LII model

Laser-induced incandescence (LII) involves the effect of laser heating on the soot structure, please see Ref. [21–26]. These works have provided extensive high-resolution TEM images of the soot aggregates and primary particles under a wide range of combustion conditions with the laser heating. These works provide important experimental support for the current modeling of LII signals. The full LII model (i.e. forward model) typically involves solving energy and mass balance equations, which include the effects of various submodels (cf. Fig. 1) [1]:

The internal energy change rate of any aggregate is assumed to result from a sum over the primary particles making up the aggregates, such that [1]:

*N*

_{p}is the number of primary particle in a single soot aggregate;

*c*

_{s}and

*ρ*

_{s}are respectively the specific heat and density of soot aggregate (taken as temperature-dependent in this study, cf. Reference [27]);

*d*

_{p}and

*T*are the primary particle diameter and mean temperature of a single soot aggregate, respectively; and

*t*is time.

The laser energy absorption of a single soot aggregate is determined by its absorption cross-section *C*_{abs} [27]:

*E*(

*t*) is the temporal irradiance profile of the laser;

*F*is the laser fluence;

*q*(

*t*) is the laser temporal variation function and is a Gaussian distribution with a standard deviation

*σ*

_{Laser}= 3.3 ns [20], and the spatial distribution of the laser beam is set as uniform top hat distribution;

*C*

_{abs}is the absorption cross-section of a single soot aggregate can be approximated by Rayleigh-Debye-Gans fractal aggregate (RDG-PFA) scattering theory [28]: where

*λ*

_{inc}denotes the wavelength of the incident pulse laser;

*E*(

*m*) is the soot absorption function which is determined by spectral complex refractive index

*m*. Here,

*λ*

_{inc}is 532 nm, and the corresponding

*E*(

*m*) is 0.3 [29].

The cooling terms in this model generally are prescribed in Ref. [27]. Conduction plays an important role in this model (and the simplified low-fluence model), such that it receives more attention here. A relatively accurate Fuchs boundary sphere model is applied to calculate heat conduction [30]:

*α*

_{T}= 0.37 is the thermal accommodation coefficient [23];

*P*

_{a}= 1 atm is the pressure of ambient air;

*R*

_{m}= 83.145 g·m

^{3}/(mol·K·s

^{2}) is the universal gas constant in effective mass units; subscript

*δ*is the thickness of the boundary layer in the Fuchs approach;

*T*is the temperature in the limiting sphere;

_{δ}*W*

_{a}= 28.74 g/mol is the molecular weight of air;

*γ** is the average value of the specific heat ratio. For a detailed description of this method, see Ref. [30].

The time-dependent particle temperature *T*(*t*) and the time-dependent primary particle diameter *d*_{p}(*t*) can be obtained simultaneously by solving the system of differential equations. In this study, the coupled system of equations is solved by the fourth-order Runge-Kutta approach.

#### 2.2 Fractal aggregate structure

As shown in Fig. 1(b), primary particles of soot aggregates are not isolated nanospheres but agglomerated in the form of fractal structure. For a soot aggregate containing *N*_{p} primary particles, its fractal structure can be mathematically described by [31]:

*N*

_{p}is the number of primary particles in a single aggregate;

*k*

_{f}is the fractal prefactor;

*D*

_{f}is the fractal dimension;

*R*

_{g}is the gyration radius,

*d*

_{p}is the diameter of the primary particles.

Filippov et al. [32] showed that the shielding effect can cause heat conduction rates from primary particles within a single aggregate to be an order of magnitude lower than that for isolated nanospheres, depending on the particle size, fractal structure, thermal accommodation coefficient, and the current flow regime [20]. To account for the shielding effect of soot aggregates, previous works [30,32] replaced the diameter in the conduction expression above with an effective diameter, *D*_{eff}, of a single soot aggregate as follows [30]:

In their studies, *D*_{eff} was assumed to be the projected-area equivalent diameter for fractal aggregates. Based on Eq. (7), Filippov et al. [32] and Liu et al. [30] both proposed a heuristic fractal-like relationship to parametrically express *D*_{eff}. Their studies have not been further tested, and the physical connection between the effective diameter, and the fractal structure of the soot aggregate is complicated [20]. Given that, this study uses the shielding factor *η* to establish the implicit relationship between the shielding effect and specific geometric structure (including fractal structure and particle size) of the aggregate. The mathematical relationship between *D*_{eff} and *η* is defined as follows [32]:

The value of *η* used to simulate LII signals is derived from the Eq. (9) and the heuristic fractal-like relationship in Ref. [30] and [32]. The two characteristic parameters *k*_{h} and *D*_{h} in this relationship are determined by the quadratic function of *α*_{T} in Ref. [30].

#### 2.3 Simplified LII model and definition of fractal-dependent TAC

While simulation data will be evaluated using the full LII model above, analysis (i.e., the inverse problem) will use a simplified model for the low-fluence regime (this avoids inverse crime [33,34]). In this case, the energy and mass changes caused by the thermal radiation, sublimation, oxidation, and annealing are negligible, such that the system is reduced to

and the mass changes are neglected.For the simplified LII model, substituting Eqs. (3), (4), (8) and (9) into Eq. (11) yields dT/dt:

In Eq. (11), the TAC and fractal-dependent shielding factor appear as an isolated product. According to the findings of Ref. [19], it is generally impossible to simultaneously infer two parameters that appear as a product in an equation. Instead, these two quantities are combined to define a new fractal-dependent TAC as the product,

The influence of aggregate structure on the simplified LII model can be fully quantified by *η*. After lumping *η* with *α*_{T}, the influence of aggregate structure on the simplified LII model is completely represented by *α*_{T, f}. Since *η* and *α*_{T} is related to many factors [3], the prior value of *α*_{T, f} tends to have large uncertainty. If the inversion of PPSD is carried out when *α*_{T, f} is a prior model parameter, it is impossible obtain accurate estimate of PPSD because of the inherent deviation of prior value of *α*_{T, f}. Therefore, *α*_{T, f} and PPSD are simultaneously inferred in this study.

As more and more evidence shows that primary particle size is relatively constant within an aggregate and varies more significantly between aggregates [35,36], this study assumes that *d*_{p} is constant in aggregates but obeys a narrow log-normal distribution between aggregates [37]:

*d*

_{p, g}and

*σ*

_{d, g}are the geometric mean primary particle diameter and the geometric standard deviation for the distribution of

*d*

_{p}, respectively. The integration limits in Eq. (13) are 5 nm and 100 nm with enough refinement, so the integration error is negligible. It should be noted that this study is based on the assumption that primary particle size is basically the same within aggregate. The influence of the relatively large primary particle size distribution within aggregate on this study is beyond the scope of this paper and needs further study.

The aggregate size generally varies between aggregates and follow a log-normal distribution of *N*_{p}:

*N*

_{p, g}and

*σ*

_{N, g}are the geometric mean primary particle number per aggregate and the geometric standard deviation for the distribution of

*N*

_{p}, respectively. The integration limits in Eq. (14) are 1 and 600 with enough refinement, so the integration error is negligible.

The time-resolved laser-induced incandescence (TiRe-LII) signal of a single primary particle in any aggregate can be calculated as follows [29]:

*h*= 6.626×10

^{−34}J·s is the Plank’s constant;

*c*= 2.998×10

^{8}m/s is the speed of light;

*k*

_{B}= 1.381×10

^{−23}J/K is the Boltzmann constant;

*λ*

_{mea}is the measurement wavelength; the mean particle temperature

*T*is related to the fractal-dependent TAC.

For an aggregate with *N*_{p} primary particles, the TiRe-LII signal can be calculated as follows [29]:

The total TiRe-LII signal of *N* soot aggregates in the measurement volume can be calculated as follows [20]:

*C*

_{exp}is a calibration parameter determined by experimental conditions, which is related to the detection geometry and collection efficiency of the detector.

In this study, the relative intensity of (normalized) TiRe-LII signal *RS*_{LII}(*t*) is used as the input for the inverse problem, defined by the ratio of the absolute intensity and maximum value of the TiRe-LII signal [20]:

*t*

_{max}is the time when the

*S*

_{LII}(

*t*) signal reaches its maximum $S_{\textrm{LII}}^{\max }$.

It should be noted that even if *D*_{f}, *k*_{f}, and *α*_{T} are constant for all aggregates, the fractal-dependent TAC, *α*_{T, f}, may be constant within aggregates but varies with the polydispersity of *N*_{p} between aggregates. So, we emphasized the dependence on *α*_{T, f} in Eqs. (16)–(18) for different aggregates. According to Eqs. (10)–(12), if *α*_{T, f} is inferred simultaneously with PPSD, temporal mean particle temperature *T* can be obtained in the simplified LII model without prior knowledge of fractal structure. However, according to Eqs. (15)–(18), prior information of fractal parameter *N*_{p} is still required to simulate normalized TiRe-LII signals. In order to make the inversion process based on the simplified LII model independent of the prior information of the fractal structure, Eqs. (16)–(18) are restated below.

First, *S*_{agg}(*t*) in Eq. (16) is assumed to be the product of an equivalent primary particle number per aggregate, *N*_{p, eff} and the integral of *S*_{p}:

*N*

_{p, eff}is constant for different aggregates.

Considering the definition of *α*_{T, f} in Eq. (12), *α*_{T, f} varies for different aggregates. But in order to obtain a constant parameter as the inverse problem variable, the definition of *α*_{T, f} must be extended for all aggregates within the measurement volume. According to the final definition of *α*_{T, f}, it considers the overall impact of TAC and fractal-dependent shielding factor of all aggregates on the simplified LII model. The target value of *α*_{T, f} can be obtained by minimizing the difference between the LII signals generated by the full LII model and the simplified LII model when the other two variables (*d*_{p, g} and *σ*_{d, g}) are fixed at their target values. In this study, the target value of *α*_{T, f} is 0.26.

Based on Eq. (19), Eq. (17) can be restated as:

Then, Eq. (18) is restated as follows

Finally, substituting Eq. (15) into Eq. (21) yields

It can be seen from Eq. (22) that the inversion of *α*_{T, f} and PPSD does not require the prior information of most full LII model parameters, especially aggregate structure, aggregate size distribution, and TAC, which greatly reduces the uncertainty from the model parameters.

## 3. Inversion process

In the present study, we consider simultaneous determination of PPSD parameters and fractal-dependent TAC, i.e., variables **x **= [*d*_{p, g}, *σ*_{d, g}, *α*_{T, f}], from the relative intensity of TiRe-LII signal using a weighted least-squares approach, with an objective function

**b**

_{mea}and

**b**

_{est}are the measured and the estimated values of the measurement data, taken as the normalized TiRe-LII signal at multiple time points, i.e.

**b**= [

*RS*

_{LII}(

*t*

_{1}),

*RS*

_{LII}(

*t*

_{2}),

*RS*

_{LII}(

*t*

_{3}), …]. In this study, the covariance matrix adaptive evolution strategy (CMA-ES) algorithm is used to minimize objective function value, with the details provided in Refs. [38,39].

The overall inverse method presented in this work is evaluated by simulating data instead of real measurements, using some target value of objective parameters **x**_{tar}. The present work avoids inverse crime [33,34] (wherein identical forward and inverse models are used, resulting in a trivial procedure where the target parameters are returned exactly) in two ways.

First, noise is added to the signals. Two different noise models were considered. The first model follows from our previous studies [39,40], and only considering a signal-independent Gaussian error [41]:

**b**

_{tar}is the target signal vector;

*γ*is the scale factor in Gaussian error;

**n**

^{G}is the standard normal random vector for Gaussian error. The second noise model is the general noise model of Sipkens et al. [41], which considers the Poisson-Gaussian error in single-shot signals and the shot-to-shot error,

*n*is a standard normal random variable;

*τ*,

*θ*, and

*γ*are the characteristic parameters of the general model, which are the scale factors in the shot-to-shot error, Poisson error and Gaussian error, respectively;

**n**

^{P}and

**n**

^{G}are respectively the standard normal random vector for Poisson error and Gaussian error.

Second, we employ a double-model approach (cf. Fig. 2), in which simulated data is generated using the full LII model, Eqs. (1) and (2), and then interpreted with simplified LII model of Section 2.3. Characteristics of the inverse procedure can be evaluated based on the degree to which it can return the target value, with appropriate consideration of uncertainties.

## 4. Results and discussion

#### 4.1 Laser fluence selection

Since this study is based on a low fluence case where particle temperature and mass changes caused by thermal radiation, sublimation, oxidation, and thermionic are expected to be negligible, it is necessary to determine the appropriate laser fluence range. A simple method is provided in Ref. [42], in which the transition fluence (the point on the fluence curve where sublimation effects become significant) is given by

*T*

_{ref}is the transition temperature, which depends on the sublimation model parameters;

*T*

_{g}= 300 K is the ambient gas temperature. This yields an estimated reference fluence of 0.126 J/cm

^{2}. Due to the more complex model, polydispersity, particle size variation, and

*N*

_{p}effects considered in this work, we seek to validate this value and expand this discussion using simulated temperature traces.

In Fig. 3(a), as the laser fluence increases from 0.10 J/cm^{2} to 0.15 J/cm^{2}, the peak portion of the primary particle temperature curve becomes more prominent, leading to higher peak temperatures and longer high-temperature duration, so in Fig. 3(b), the particle mass loss increases with the increase of laser fluence. When the laser fluence exceeds the 0.15 J/cm^{2}, the diameter of the primary particles cannot be regarded as a constant because of the continuous mass loss. Figure 3 is obtained when *N*_{p} = 1 and *d*_{p} = 10 nm. For more *N*_{p} and larger *d*_{p}, the thermal shielding effect will be more significant and the specific surface area will be smaller, resulting in a higher peak temperature and a flatter cooling curve, and then more particle mass loss. Therefore, from the perspective of particle mass loss, a suitable fluence of laser should not exceed 0.15 J/cm^{2}.

On the other hand, the simplified LII model only considers the energy change caused by laser absorption and heat conduction, so the laser fluence should be also selected according to the errors caused by the energy equation approximation. As shown in Fig. 4, when laser fluence exceeds 0.10 J/cm^{2}, the ratio of the energy sum of laser absorption and heat conduction to the total energy is less than 90% at high temperature. In terms of the energy equation error in the simplified LII model, a suitable fluence of laser should be lower than 0.10 J/cm^{2}. But lower laser energy density is not conducive to reducing the overall error, because lower laser energy density will make the signal-to-noise ratio of TiRe-LII signal worse [18]. Therefore, a laser fluence of 0.10 J/cm^{2} is selected in the present study.

#### 4.2 Simultaneously inferring PPSD parameters and fractal-dependent TAC

In this study, the simultaneous inference of PPSD parameters and fractal-dependent TAC are performed in different cases. Table 1 summarizes the basic settings of these cases. To investigate the influence of the measurement wavelength number and measurement noise level on the retrieval precision, two different measurement wavelength schemes were used under 4 different measurement noise levels (8 test cases in total) to evaluate the capability of simultaneously inferring PPSD and fractal-dependent TAC from normalized TiRe-LII. Figure 5 shows a set of noisy LII signal synthesized in this study.

Table 2 shows the retrieval results of three objective parameters in all 8 test cases. For the 4 cases of the general noise model, the noisy signals used for one inversion are synthesized from an average of 100 shots, while for the 4 cases of the Gaussian noise model, the noisy signals used for one inversion are from a single shot. Given the stochastic nature of the noisy signal synthesis, 100 different sets of noisy signals are evaluated in each case. Each case is evaluated in the following aspects: (1) SR: The success rate of the all 100 independent inversions; (2) Avg: The average value of all successful inversions of one objective parameter; (3) Std: The standard deviation of all successful inversions of one objective parameter; (4) *ɛ*_{rel}: Relative error of Avg relative to the target value. Here, successful inversion means that the inversion process can eventually converge to a specific value.

Regarding relative error of retrieval results, whether for Gaussian noise or general noise the *ɛ*_{rel} of *α*_{T, f}, *d*_{p, g}, and *σ*_{d, g} inferred at low noise level do not exceed 5%. For the cases of Gaussian noise, it is found that the *ɛ*_{rel} of *α*_{T, f} and *σ*_{d, g} inferred at high Gaussian noise level exceeds 10%, while the relative error of *d*_{p, g} is thoroughly below 5%. For the cases of general noise, the *ɛ*_{rel} of all three objective parameters basically increase as noise level becomes stronger. When using noisy signals measured at two wavelengths, the *ɛ*_{rel} of all objective parameters do not exceed 5% except for the *ɛ*_{rel} of *d*_{p, g} and *σ*_{d, g} at high general noise. When using noisy signals at three wavelengths, the *ɛ*_{rel} of all objective parameters do not exceed 5% except for the *ɛ*_{rel} of *σ*_{d, g} at high general noise. The *ɛ*_{rel} of *α*_{T, f} is thoroughly below 5% in all the cases of the general noise. In addition, the success rate of 100 inversions drops significantly at high noise level, and as the wavelength number of noisy signals increases, the success rate of 100 inversions increases. Table 2 indicates that the simultaneous inference of *α*_{T, f}, *d*_{p, g}, and *σ*_{d, g} can be accurately achieved at a relatively low noise level, but when the noise reaches a certain level, the resulting *α*_{T, f}, *d*_{p, g}, and *σ*_{d, g} may deviate seriously from their target values.

Regarding the standard deviation of retrieval results, the Std of inferred *d*_{p, g} is close to its value magnitude, while for the inferred *α*_{T, f} and *σ*_{d, g}, the Std is at least one order of magnitude smaller than its value. It indicates that the resulting *d*_{p, g} of 100 evaluations are distributed in a wider range, i.e. the uncertainty of *d*_{p, g} is greater than that of *α*_{T, f} and *σ*_{d, g}. The magnitude of the uncertainties of *d*_{p, g} and *σ*_{d, g} are consistent with the results of Bayesian analysis in Ref. [20]. The uncertainty of the inversion results of all objective parameters increases as the noise becomes stronger. Therefore, in order to obtain more accurate inversion results, it is necessary to average the results over multiple independent inversions, especially for *d*_{p, g}.

According to our previous studies [39,40], the retrieval precision of objective parameters is typically improved when increasing the measurement wavelength number, since it brings more available information for inversion. In this study, for the cases under general noise, the retrieval results using three-wavelength noise signals are significantly better than those obtained from two-wavelength noise signals. It indicates that under general noise, increasing measurement wavelength number can effectively improve the inversion accuracy and reduce the uncertainty. But for all cases under Gaussian noise, the increase of measurement wavelength number has little effect on the relative error and standard deviation of three objective parameters.

In addition, it is not comprehensive to evaluate the inversion of the PPSD only from the retrieval precision of characteristic parameters *d*_{p, g} and *σ*_{d, g}. For a more direct evaluation, Fig. 6 shows the PPSD profiles drawn from the target value of *d*_{p, g} and *σ*_{d, g} together with PPSD profiles drawn from retrieval results of *d*_{p, g} and *σ*_{d, g} in Table 2. It can be seen that the inferred PPSD is sensitive to the noise level. The greater the noise level, the wider the inferred PPSD profiles, and the farther the inferred PPSD profiles deviate from the target profile. For the PPSD profiles inferred under Gaussian noise in Fig. 6(a), there is only a slight difference between PPSD profiles inferred from two-wavelength (2*λ*) noisy signal and three-wavelength (3*λ*) noisy signal. However, for the PPSD profiles inferred under general noise in Fig. 6(b), the PPSD profiles inferred from three-wavelength (3*λ*) noise signals are obviously closer to target PPSD profiles than PPSD profiles inferred from two-wavelength (2*λ*) noise signals at high noise level. It is consistent with the trend of the relative error of *d*_{p, g} and *σ*_{d, g} in Table 2, indicating that using noisy signals measured at more wavelengths can improve retrieval results, especially at high general noise level.

#### 4.3 Log contour analysis of the objective function

Solving inverse problem of inferring three objective parameters is to find the point with the lowest objective function value from the three-dimensional (three-variable) domains. Therefore, the involved inverse problem can be analyzed from the perspective of the log contours of the objective function. Since it is computation-consuming to analyze all log-contours of the objective function from the three-dimensional (three variables) search domain with sufficient resolution, only three typical ones are selected from a large number of log contours of the objective function for analysis. These three log contours of the objective function are obtained by respectively fixing three variables to their target values, such as, the log contours of the objective function on the *α*_{T, f}-*d*_{p, g} plane is obtained when *σ*_{d, g} is fixed at 1.2. Figure 7 shows the log contours of the objective function on (a) *σ*_{d, g}-*d*_{p, g} plane, (b) *α*_{T, f}-*d*_{p, g} plane, and *α*_{T, f}-*σ*_{d, g} plane. The objective functions used in Fig. 7(a), (b) and (c) are based on a set of the normalized LII noise signal that can accurately infer *α*_{T, f}, *d*_{p, g}, and *σ*_{d, g}.

Considering the analysis in Ref. [13], the objective function topography of Fig. 7(a) reveal a robust estimate of *σ*_{d, g} and *d*_{p, g}. According to the objective function topography of Fig. 7(c), the estimate of *α*_{T, f} and *σ*_{d, g} is likely to be robust. In Fig. 7(b), there is no minimum point, but a minimum valley. The objective function topography of Fig. 7(b) is consistent with that in Ref. [19] and [13], arising from the conduction-dominated temperature decay. Previous studies [13,19] concluded that it was impossible to infer *α*_{T} and *d*_{p, g} from conduction-dominated LII. But in this study, reasonable *α*_{T, f} and *d*_{p, g} can be inferred from the conduction-dominated LII. In order to further verify the possibility of inferring *α*_{T, f} and *d*_{p, g} from the LII data, Fig. 8 shows the logarithmic objective function of valley lines of Fig. 7 along the coordinate axes.

Figure 8(a) and Fig. 8(d) are respectively the valley line of Fig. 7(a) along the *d*_{p, g}-axis and *σ*_{d, g}-axis. In Fig. 8(a) and Fig. 8(d), the bottom of the valley line is U-shaped, with steep sides, from which the lowest point can be easily distinguished. The lowest points of *d*_{p, g} and *σ*_{d, g} are respectively at 20 nm and 1.2, which are consistent with their target values. Due to the large slopes on both sides of the lowest point, the inversion process can easily converge to the lowest point, which indicates an accurate estimate of *d*_{p, g} and *σ*_{d, g} when *α*_{T, f}, is perfectly known. Figure 8(b) and Fig. 8(e) are respectively the valley line of Fig. 7(b) along the *d*_{p, g}-axis and *α*_{T, f}-axis. Although the bottom of the valley line is extremely flat, the insets of Fig. 8(b) and (e) prove the existence of the lowest points of *d*_{p, g} and *α*_{T, f}, at about 16.8 nm and 0.22, respectively. Due to the extremely small slopes on both sides of the lowest point, the inversion process is difficult to converge to the lowest point and may require a large number of iterations, which indicates that it is difficult but theoretically feasible to estimate *d*_{p, g} and *α*_{T, f} when *σ*_{d, g} is perfectly known. Figure 8(c) and Fig. 8(f) are respectively the valley line of Fig. 7(c) along the *σ*_{d, g}-axis and *α*_{T, f}-axis. In Fig. 8(c) and (f), the lowest points of *σ*_{d, g} and *α*_{T, f} are respectively at 1.2 and 0.26, which are consistent with their target values. But in Fig. 8(f), the valley line has two U-shaped bottoms with steep sides. The one on the right is the second lowest point of *α*_{T, f}, which is about 0.36. Given that, it can be expected to accurately estimate *σ*_{d, g} and *α*_{T, f} when *d*_{p, g} is completely known, but it may happen that the retrieval process is trapped in the second lowest point where *α*_{T, f} is about 0.36.

Figure 7 and Fig. 8 actually correspond to three relatively simple two-variable inversions. Table 3 shows the retrieval results of these three two-variable inversions using the same noisy signal as Fig. 7 and Fig. 8. Each inversion is repeated 20 times to calculate the average results and the standard deviation. The 20 inversions finally converged to the same result, so the standard deviation of all variables is 0. The average estimated value of all variables is consistent with the minimum value in Fig. 8. the minimum value of *α*_{T, f} and *d*_{p, g} can be obtained stably (SR = 100%) in the current log contour topography of objective function. Regarding the average iteration counter, inferring *α*_{T, f} and *d*_{p, g} takes almost twice as much time as other two two-variable inversions. It can be seen that inferring *α*_{T, f} and *d*_{p, g} is much more difficult. In addition, the estimate of *α*_{T, f} and *σ*_{d, g} has a 70% probability of failure. According to the results, the failed inversions are all be trapped in the area where *α*_{T, f} is about 0.36. All these results verify our analysis of Fig. 8.

## 5. Conclusion

This proof-of-concept study numerically evaluates the feasibility of simultaneously inferring PPSD and fractal-dependent TAC from normalized TiRe-LII signals. All the numerical evaluations are performed in the low-fluence case where the overall impact of TAC and fractal-dependent shielding factor on the simplified LII model can be defined as the fractal-dependent TAC. The full LII model is employed to synthesize normalized TiRe-LII signals for numerical evaluations, while the simplified model is used for the inverse problem analysis, which avoids the inverse crime. Based on normalized TiRe-LII signals and fractal-dependent TAC, the simplified LII model is independent of prior knowledge of most full LII model parameters, including fractal structure, the polydispersity of aggregate size, and absorption function at the measurement wavelength, the apparent fraction volume and calibration parameters.

The possibility of inferring PPSD parameters (*d*_{p, g} and *σ*_{d, g}) and fractal-dependent TAC (*α*_{T, f}) is numerically evaluated in 8 different cases under Gaussian noise or general noise. Regarding the relative error of retrieval results, the inference of *d*_{p, g}, *σ*_{d, g}, and *α*_{T, f} is accurate at low noise level but significantly biased at high noise level. For Gaussian noise, when more measurement wavelengths are used, the retrieval results are only slightly improved. For general noise, when more measurement wavelengths are used, especially for high noise level, the retrieval results are significantly improved. Regarding the uncertainty of retrieval results, to obtain a reliable estimate of *d*_{p, g}, it is necessary to average multiple independent estimates. Regarding the log contour analysis of the objective function, although the inversion process is expected to be difficult due to the flat bottom line, the unique solution existing on the valley line theoretically proves the feasibility of inferring *d*_{p, g} and *α*_{T, f}.

In summary, all numerical results reveal the potential capability of simultaneously inferring PPSD and fractal-dependent TAC from normalized TiRe-LII signals. In this proof-of-concept, the inversion method is evaluated by simulation data instead of real measurements. Considering the essential difference between the simulation data and real measurements, the feasibility and predictive power of the method needs to be validated in controlled experiments with real measurement data. Further research will be conducted to prove the feasibility of this method experimentally. Furthermore, considering the large uncertainty of the inversion results in the current study, a full Bayesian inference will be employed in our future work to obtain reliable uncertainty estimates.

## Funding

National Natural Science Foundation of China (51976044); National Science and Technology Planning Project (2017-V-0016-0069); Foundation for Heilongjiang Touyan Innovation Team Program.

## Acknowledgements

The authors would like to thank Dr. Timothy Sipkens, Prof. Steven Rogak, and Miss Yilin Zhao for their direct input, insightful discussions, and editorial work on this manuscript. This paper also supported by the China Scholarship Council.

## Disclosures

The authors declare no conflicts of interest.

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