# Lidar backscattering model for soot aerosols

Romain Ceolato, Jérôme Yon, Matthew Berg

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Romain Ceolato, Jérôme Yon, Matthew Berg. Lidar backscattering model for soot aerosols. Journal of Quantitative Spectroscopy and Radiative Transfer, 2025, pp.109532. 10.1016/j.jqsrt.2025.109532. hal-05128729

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Submitted on 29 Jul 2025

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# Lidar backscattering model for soot aerosols

Romain Ceolato<sup>a,∗</sup>, Jérôme Yon<sup>b</sup>, Matthew J. Berg<sup>c</sup>

## ARTICLE INFO

### Keywords:

Lidar

Backscattering

Soot

Black carbon

Fractal aggregates

## ABSTRACT

Aerosol lidar remote sensing relies on the interpretation of backscattered light to retrieve particle properties of interest. This study presents a novel analytical backscattering model for soot or black carbon aerosols, based on the Rayleigh-Debye-Gans for Fractal Aggregates (RDG-FA) approximation. This model enables simple analytical calculation of key lidar parameters such as the differential backscattering cross-section, lidar ratio (LR), mass backscattering coefficient (MBC), along with the backscattering color ratio (CR) and Ångström exponent (BAE). The behavior of these lidar parameters is investigated as a function of wavelength, size, and aging. Key findings include a general decrease in LR, along with a clear dependence of CR and BAE on aerosol size as black carbon aggregates grow, for both freshly emitted and aged soot. These findings highlight the ability to model the variations of these parameters throughout the life cycle of black carbon, capturing the evolution from chain-like freshly emitted particles to more spherical aged particles. Furthermore, this study demonstrates the potential of the RDG-FA backscatter model to calculate key lidar parameters and improve the retrieval of soot aerosol products.

## 1. Introduction

Soot, or black carbon (BC), consists of fractal-like aggregates of carbon particles that are commonly produced by the incomplete combustion of hydrocarbon fuels and biomass materials [1]. Soot is a major contributor to climate forcing and the second leading anthropogenic contributor to global warming because of its strong absorption of solar radiation in the atmosphere [2–4]. Soot emissions affect cloud formation [5], present carcinogenic risks to human health, and, due to their small size, can cause lung diseases [6]. These aerosols also degrade urban air quality and visibility [7]. In other contexts, soot particles emitted by aircraft engines act as ice nuclei, potentially forming contrails and cirrus clouds [8–11]. Despite these known effects, the overall climate impact of soot remains uncertain, largely due to variations in soot composition and structure. Therefore, accurate modeling of black carbon aerosols, focusing on the particle number and mass concentration, is critical for a better understanding of their effects on climate and health [12,13].

Modeling the light-backscattering behavior of soot aerosols is crucial for enhancing lidar-based remote-sensing retrieval algorithms [14]. Several light-scattering models can accurately simulate the radiative properties of soot aggregates. For instance, the Discrete Dipole Approximation (DDA) has been applied to soot aggregates with realistic

structures [15]. Yet, the DDA is constrained by the available computational resources and becomes limited when ensemble-averaged results are required [16]. Another method, the T-Matrix [17] or the Multiple Sphere T-Matrix (MSTM) [18], is also well-suited for modeling soot light-scattering [19]. The advantage of the MSTM method is that it analytically accounts for multiple scattering between component particles in a soot aggregate, making it more accurate than the DDA in some cases. Implementing these models for remote-sensing applications remains challenging because of the significant computational time they require. Typically, only a few aggregates can be accommodated, often making these models impractical for interpreting lidar observations of smoke or black carbon aerosols. Therefore, there is value in exploring approximate simplified light-scattering models for black carbon aerosols, including the use of spheroidal shapes [20–22]. If this is done while maintaining a level of physical insight, simplified models could lead to improved lidar inversions and reveal more physical properties of the black carbon aerosols.

The aim of this study is to present an analytical backscattering model for soot aerosols. We assume that the light backscattering cross-sections can be approximated by the Rayleigh-Debye-Gans for Fractal Aggregates (RDG-FA) approximation, which has been widely used in similar studies [23]. While not as rigorous as the DDA or MSTM

<footnote><sup>a</sup> ONERA, The French Aerospace Lab, Université de Toulouse, Toulouse, FR 31055, France</footnote>

<footnote><sup>b</sup> Normandie Université, INSA Rouen, CNRS, CORIA, UMR 6614, Rouen, F-76000, France</footnote>

<footnote><sup>c</sup> Kansas State University, Department of Physics, 1228 N. Martin Luther King Jr. Dr., Manhattan, KS, 66506-2901, USA</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: romain.ceolato@onera.fr (R. Ceolato).</footnote>
methods, the RDG-FA is simple and yields good results for small to medium-sized aggregates [24,25], including the backscattering behavior [26]. The simple analytical relationship in the RDG-FA between the cross-sections and aggregate morphology is a significant advantage, albeit an approximation. As such, one can appreciate the popularity it has gained, particularly in laser diagnostics within combustion science. However, its use in remote-sensing applications remains relatively limited, especially in lidar systems [27], where the backscattering behavior is of paramount importance.

## 2. Fractal aggregate model

The terminology associated with carbonaceous aerosols is diverse and often inconsistent, resulting in uncertainty when defining, modeling, and measuring these particles. Here, the generic term of soot is used to refer to the variety of combustion-derived carbonaceous products, including black carbon. Soot aggregates are formed via the agglomeration of nanometer-sized primary carbon particles, called monomers. Due to their small size, the monomers are subject to Brownian motion, which drives the agglomeration to form irregularly shaped aggregates. The aggregate structure is ideally represented as a cluster of touching, non-overlapping spherical monomers with radius $R_m$. This radius depends on the combustion source and typically ranges from $R_m \sim 10–20$ nm for hydrocarbon fuels [28] to $R_m \sim 30$ nm for biomass [29]. Assuming a repetitive process in which two monomers of equal size merge to form an aggregate, this process becomes self-similar, leading to an exponential increase in aggregate size. This property is revealed by a power-law behavior in the so-called pair-correlation function $A(r)$, which describes the probability of finding a monomer at a distance $r$ from a reference monomer. The pair-correlation function is proportional to the average number of monomers in a shell of radius $r$ and thickness $dr$ centered on a given monomer, and is useful to describe the morphology of any particle or aggregate with diffuse boundaries [30]. Given the spatial dimension $d=3$, the function is given by

$$A(r) = A_0 r^{D_f-d} h(r/\xi), \quad (1)$$

where $D_f$ is the mass fractal dimension, or fractal dimension for short and $A_0$ a constant. This is one of the key parameters for quantifying the overall aggregate morphology.

More compact aggregates have higher values, whereas those with branched or chain-like structures have lower values; all in the range $D_f \in (1,3)$. The fractal dimension is influenced by the physics of the soot formation, source, and aging processes. It is common to regard $D_f$ as a descriptor of the soot-aggregate lifespan. It is generally accepted that freshly-emitted soot, from fuel combustion or small biomass fires, exhibits a structure with $D_f \sim 1.8$, whereas larger values, up to $D_f \sim 2.6$ can be observed [31–33] after aging processes. Such processes include coating by water or other compounds, restructuring due to atmospheric forces, and the emergence of multiple fractal-dimension super-aggregates, i.e., the aggregates-of-aggregates observed in larger forest fires [34]. Since the aggregates are of finite size, there is a cutoff function $h(r/\xi) \neq 1$ in Eq. (1) on a length scale comparable to the aggregate size, given by

$$h(r/\xi) = \exp \left[ - \left( \frac{r}{\xi} \right)^{\gamma} \right]. \quad (2)$$

where, $\gamma$ is the so-called stretching exponent, which quantifies the sharpness of the pair-correlation cut-off and occurs at the length-scale $\xi$, corresponding to the size of the aggregate.

The size of the aggregate, denoted as $\xi$, can be characterized in several ways. A popular choice is the radius of gyration, $R_g$, which appears in the RDG-FA theory developed below, and represents the radius of a sphere with the same moment of inertia as an aggregate

<figure><img src="image_4.png" /><figcaption>Fig. 1. Pair-correlation function $A(r)$ for the canonical DLCA fractal aggregate. The open circles correspond to the calculation of $A(r)$ and the dashed line represents the analytical expression of Eq. (23) below with $N = 90$, $D_f = 1.8$, $k_o = 1.3$, and $R_g = 219$ nm. The aggregate is shown inset.</figcaption></figure>

with $N$ monomers. It is defined as the root-mean-square distance of the monomer centers from the aggregate center of mass, i.e.,

$$R_g = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (r_i - r_{cm})^2} \quad (3)$$

where $r_i$ is the position of the $i$th monomer and $r_{cm}$ is the aggregate’s center of mass. The aggregate mass, as quantified by the number of monomers $N$, follows the classic fractal scaling law [23,35,36],

$$N = k_o \left( \frac{R_g}{R_m} \right)^{D_f}, \quad (4)$$

where $k_o$ denotes the fractal prefactor. The prefactor is anti-correlated with the fractal dimension [37,38] and controls the shape anisotropy and local packing of the aggregate [39,40]. This prefactor can be thought of as the degree of compactness of the aggregate branches. Although often less considered than $D_f$, $k_o$ is relevant as it changes with the presence of coatings, for example [41]. Overall, the morphology of a soot fractal aggregate can be described by the pair of parameters $(D_f, k_o)$.

The Diffusion Limited Cluster Aggregation (DLCA) algorithm can be used to simulate soot aggregate morphologies using the physics of agglomeration, and the results compare well to real soot aggregates [4, 23]. This close agreement with experimental observations makes DLCA particularly valuable across multiple scientific and engineering applications because it accurately captures the fractal nature of real-world soot aggregates, providing reliable predictions of optical properties for combustion modeling, atmospheric radiative forcing calculations, laser-based diagnostics, and climate change studies. Quantities like the monomer pair correlation $A(r)$ can be conveniently calculated from such model morphologies. An example is shown in Fig. 1 along with the model aggregate. The DLCA describes the aggregation process in a dilute regime, where the monomer–monomer distance is significantly greater than the size of the resulting aggregate [36]. Using Eq. (1) and a Gaussian cut-off function [42], a value of $D_f = 1.8$ is obtained by fitting the $A(r)$ to the model data. In the following, this aggregate, where $N = 90$, $D_f = 1.8$, and $k_o = 1.3$ will be referred to as canonical DLCA fractal aggregate, following previous work [26,40,43].
### 3. Backscattering model for soot aggregates

We propose an approximate model for the light-backscattering by fractal aggregates of small absorbing particles, indicative of freshly-emitted soot or black carbon aerosols. Our model is based on concepts used in condensed matter and combustion studies, mainly the RDG-FA and so-called q-space analysis [44].

#### 3.1. Rayleigh–Debye–Gans

The Rayleigh–Debye–Gans for Fractal Aggregates (RDG-FA) approximation refers to a model used to describe the scattering of light from fractal aggregates. It can be derived from the Maxwell equations following two main approximations, as described by Berg et al. [45].

First, a monomer is assumed to scatter light in such a way that the wave-phase shift $\rho_m = 2kR_m|m-1|$ across the monomer diameter is negligible where $m(\lambda) = n+ik$ is the complex-valued specular refractive index and $k = 2\pi/\lambda$ is the wave number. Thus, the electromagnetic field within a monomer is uniform. Small spherical monomers with radii smaller than the wavelength, such as $R_m \ll \lambda$, should scatter incident light with a negligible phase shift, ensuring that $\rho_m$ stays below unity. In other words, the first approximation is that each monomer sees only the incident wave and will scatter the light in the Rayleigh regime. Note, that this approximation depends on $\lambda$ explicitly through $k$ and implicitly through $m(\lambda)$. As the monomer size increases, the phase shift becomes larger, yielding significant errors when applying the RDG-FA. According to the results of Farias et al. [46], $\rho_m$ should remain less than approximately 0.6 for an error of less than 10%.

Secondly, it is assumed that each monomer scatters the incident light independently, implying that monomer–monomer multiple scattering is disregarded. Evaluating this condition is more complex as it relies on several factors, including $N$, $m$, and the detailed morphology of the aggregates. While the first approximation can be verified at the monomer level, the second condition is more difficult to verify at the aggregate level, especially for large aggregates, see [45,47,48]. Precautions are necessary because a significant phase shift may occur across the aggregate, which can be quantified using the aggregate phase shift, defined as $\rho_{agg} = 2kR_g|m-1|$, where it is understood that $m = m(\lambda)$ for simplicity. As recommended by Sorensen et al. [23,25], $\rho_{agg}$ should not be greater than 3 for errors below 10%.

Despite its simplicity, the RDG-FA often agrees well with light-scattering measurements in the case of soot [23], including in the backscattering direction [26]. However, this is only true if the model is averaged over an ensemble of random aggregate-orientations. To summarize, the accuracy of the RDG-FA depends on $R_m$, $D_f$, $\lambda$, and $m(\lambda)$. Further discussion on this point is provided later, especially regarding the approximation of lidar parameters for soot aerosols.

#### 3.2. Backscattering cross-section

The differential scattering cross-section of a fractal aggregate is expressed as the product of $N^2$, the aggregate structure factor or Debye factor $S$ (Section 3.3), and the Rayleigh differential cross-section $dC_{Ray}^{sca}/d\Omega$, i.e.,

$$ \frac{dC^{sca}}{d\Omega}(q) = N^2 S(q, R_g, D_f) \frac{dC_{Ray}^{sca}}{d\Omega}(q), \quad (5) $$

The scattering wave-vector, $q$, is an important parameter for the RDG-FA. It is defined as $q = k^{inc} - k^{sca}$, where $k^{inc}$ and $k^{sca}$ are the incident and scattered wave-vectors, respectively [23], with a magnitude of $q$ expressed as,

$$ q(\theta, \lambda) = 2k \sin \left(\frac{\theta}{2}\right) = \frac{4\pi}{\lambda} \sin \left(\frac{\theta}{2}\right), \quad (6) $$

with units of inverse length. Thus, $q^{-1}$ can be interpreted as a length scale of importance for the light-scattering process in the RDG-FA

approximation; although it may not be obvious what that length scale refers to. In a general sense, one can envision that the electromagnetic waves will not resolve length scales smaller than $q^{-1}$ in this approximate view. Such a concept can be useful to interpret the physics behind certain light-scattering systems, such as for soot aggregates as explained by Sorensen [44]. We apply this view in the following to propose the backscattering model.

Using the Rayleigh–Debye–Gans (RDG), the Rayleigh differential scattering cross-section for a single arbitrary shaped monomer can be expressed as follows,

$$ \frac{dC_{Ray}^{sca}}{d\Omega}(q) = k^4 R_m^6 F(m)G(qR_m)^2 \quad \text{with} \quad F(m) = \left| \frac{m^2 - 1}{m^2 + 2} \right|^2, \quad (7) $$

where $G$ is the monomer form-factor or Gans factor, which describes how a single monomer scatters light. A number of different analytical expressions have been proposed for a variety of particle shapes. In the case of spheres, the expression is as follows [44],

$$ G(qR_m) = \frac{3 \left| \sin(qR_m) - qR_m \cos(qR_m) \right|}{(qR_m)^3}, \quad (8) $$

and reaches unity, i.e., $G \approx 1$, when $qR_m \ll 1$ for small monomers in the Rayleigh scattering regime. Consequently, the light-scattering is isotropic, and the monomers can be considered as point scatterers or Rayleigh particles.

Finally, the differential backscattering cross-section for a soot fractal aggregate is defined in the backward scattering direction, i.e., for $\theta = \pi$, where $\theta$ is the (polar) scattering angle,

$$ dC^{bac} := \left. \frac{dC^{sca}}{d\Omega}(q) \right|_{\theta=\pi} = F_{11}(\pi) \frac{C^{sca}}{4\pi} \quad (9) $$

where $F_{11}(\pi)$ is the backward element of the Stokes scattering matrix and $C^{sca}$ is the scattering cross-section. To simplify, the shorthand notation $dC$ is used here to represent $dC/d\Omega$, Eq. (9). Using Eq. (5) in the backward direction, it becomes:

$$ dC^{bac} = N^2 S(q_\pi, R_g, D_f) \frac{dC_{Ray}^{bac}}{d\Omega}(q_\pi), \quad (10) $$

with $q_\pi = q(\theta = \pi, \lambda) = 2k$ defined as the backscattering wave-vector. In conventional light-scattering studies, the scattering angle is varied to change $q$, and thus, probe different length scales ($q^{-1}$). Note, however, that in elastic backscatter lidar applications, where the angle is fixed at $\theta = \pi$, different length scales can still be probed by varying $\lambda$. For common lidar-wavelengths, $q_\pi = 0.0354 \text{ nm}^{-1}$, $0.0236 \text{ nm}^{-1}$, and $0.0118 \text{ nm}^{-1}$ for $\lambda = 355 \text{ nm}$, $532 \text{ nm}$, and $1064 \text{ nm}$, respectively.

Using Eq. (9), the aggregate's differential backscattering cross-section becomes:

$$ dC^{bac} = N^2 k^4 R_m^6 F(m) S(q_\pi, R_g, D_f). \quad (11) $$

or using the aggregate volume $V = \frac{4}{3}\pi N R_m^3$,

$$ dC^{bac} = 9\pi^2 \frac{V^2}{\lambda^4} F(m)S(q_\pi, R_g, D_f). \quad (12) $$

For polarized light, as used in lidar systems, the differential backscattering cross-section for a sphere simplifies to a straightforward expression,

$$ dC^{bac,vv} = dC^{bac,hh} \cos^2(\pi) = dC^{bac} \quad (13) $$

where $vv$ or $hh$ represent vertical–vertical or horizontal–horizontal configurations. This expression is independent of the polarization configuration in the backward direction ($\theta = \pi$), provided that the emitter and receiver share the same polarization. A depolarization model has been proposed to account for cross-polarization configuration using RDG-FA [49]. However, the use of accurate electromagnetic light-scattering models, such as the Discrete Dipole Approximation (DDA) or the Multiple Sphere T-Matrix Method (MSTM), is recommended to investigate the depolarization properties of soot fractal aggregates [19].
### 3.3. Backscattering structure factor

The structure factor $S$, which appears in Section 3.2 describes the angular structure of the scattered-light intensity in the RDG-FA approximation. It has been used extensively in X-ray scattering studies of condensed-matter systems [50] and to characterize combustion by-products [51,52]. The structure factor is dimensionless and is defined as the square of the Fourier transform of $A(r)$. Thus, $S$ contains information about an aggregate's structure as represented in reciprocal space, in other words, in $q$-space. In light-scattering, $S$ can be partitioned into different regimes according to the product $qR_g$ upon which it depends: (1) a nearly constant forward-scattering regime for $qR_g \ll 1$, (2) a so-called Guinier regime, where a distinctive cross-over occurs between functionalities at $qR_g \sim 1$, and (3) a quasi-constant power-law regime for $qR_g > 1$. Notably, the slope of $S$ versus $qR_g$ in the latter regime is approximately equal to $-D_f$ [23,40].

Let us define the backscattering structure factor, $S(q_\pi)$, as the Fourier transform of the monomer pair-correlation function $A(r)$ into $q$-space in the backscattering direction, i.e., $q = q_\pi$. There are various analytical expressions available for $S(q)$ that can be utilized to represent $S(q_\pi)$. All of these expressions share the common feature of capturing the distinctive characteristics of the Guinier and power-law regimes as previously described. A popular expression is

$$S(q_\pi) = \left[ 1 + \frac{2(q_\pi R_g)^2}{3D_f} \right]^{-D_f/2} \quad (14)$$

Dobbins and Megaridis [53] have proposed another expression,

$$S(q_\pi) = \begin{cases} \exp\left(-\frac{q_\pi^2 R_g^2}{3}\right) & \text{if } q_\pi R_g < \sqrt{\frac{3D_f}{2}}, \\ \left(\frac{3D_f}{2eq_\pi^2 R_g^2}\right)^{-\frac{D_f}{2}} & \text{if } q_\pi R_g > \sqrt{\frac{3D_f}{2}}, \end{cases} \quad (15)$$

where $e = 2.718$. Yet more complicated structure factors have been proposed [40,54], but we find them beyond the scope of the study. Both Eqs. (14) and (15) exhibit the well-established Guinier law regime. The Fisher and Burford expression is advantageous in that it represents a single analytical expression. However, it tends to overestimate the structure factor in the power-law regime. Consequently, we use the Dobbins and Megaridis expression to model soot or black carbon fractal aggregates.

We use the $A(r)$ function from the simulated canonical DLCA aggregate (see Fig. 1) to test the proposed backscattering model. Fig. 2 shows $S(q_\pi)$, via Eq. (15), for the three common lidar wavelengths. Note the different regimes that depend on the values of $R_g$. A major difference with conventional $q$-space analysis of light-scattering by soot is that the angle here is fixed at $\theta = \pi$, i.e., $q = q_\pi$, but the wavelengths vary, and the different regimes are thus observed as a function of $R_g$ now. The intent in using the simulated aggregate here is to investigate the domain of validity of the (fractal) power-law regime and the analytical expression for $S$ in the backscattering direction, Eq. (15). Different $R_g$ values are investigated by varying $R_m$ from 5 nm to 35 nm, which equates to $R_g$ ranging from 50 nm to 360 nm for constant fractal morphology. For aggregates with yet larger values of $R_m$, poorer results are found.

### 4. Analytic expressions of lidar parameters for soot aggregates

We now apply the approximate backscattering model developed in Section 3.2 to simulate the radiative properties relevant for elastic backscatter lidar measurements of soot aerosols. A key advantage here is the derivation of simple analytical expressions for the most important lidar-relevant parameters.

<figure><img src="image_5.png" /><figcaption>Fig. 2. Backscattering structure factor, $S(q_\pi)$ computed as the Fourier transform of the simulated canonical DLCA aggregate pair-correlation function of Fig. 1 for three wavelengths. Here, $R_g$ is varied by scaling $R_m$. Solid lines correspond to the Dobbins and Megaridis expression for $S(q)$, Eq. (15), with $D_f = 1.8$.</figcaption></figure>

#### 4.1. Volume backscatter coefficient

The lidar equation yields an expression for return signals [55–57], and can be derived from the Radiative Transfer Equation (RTE) [14]. In the single-scattering approximation (SSA), the backscattered intensity $I^{\text{bac}}$ is often found in its scalar form [55,56] as, $I^{\text{bac}}(r) \sim U(r)r^{-2}$ where $r$ is the range, or the distance between the aerosol particles and the lidar.

The attenuated backscatter function $U$ is a fundamental quantity measured by lidar and is given by

$$U(r) = \beta(r) \exp \left( -2 \int_{0}^{r} \alpha(r') dr' \right), \quad (16)$$

where $\alpha(r)$ and $\beta(r)$ are the range-dependent volume extinction and backscattering coefficients, respectively. For simplicity, the range dependence $r$ will be omitted in the following.

The volume backscatter coefficient $\beta$ in Eq. (16) has units of [m$^{-1}$ sr$^{-1}$], and is expressed by

$$\beta = \int_{R_{\text{min}}}^{R_{\text{max}}} n(R) \, dC^{\text{bac}}(R) \, dR \quad (17)$$

where $n(R)$ is the particle-size distribution, i.e., the number of particles per unit volume, assuming an isotropic scattering medium of randomly oriented soot particles, each with mean geometric radius $R$, where $R_{\text{min}}$ and $R_{\text{max}}$ are the minimum and maximum geometric mean radii. The differential backscattering cross-section $dC^{\text{bac}}$ can, in general, be obtained from the Stokes phase matrix [58] or from the RDG-FA approximation, i.e., Eq. (9).

For an ensemble of randomly-oriented polydisperse aggregates, the mean volume backscatter coefficient $\beta$ is determined by integrating the monodisperse cross-section over all aggregates, given by Eq. (12), following a log-normal size distribution with a total number concentration of $n_o$. Simple expressions for the mean cross-sections of polydisperse aggregates have been proposed [59] using monodisperse aggregate cross-sections and a polydispersity factor, $C_p$, leading to a simplified expression for $\beta$ as follows,

$$\beta = \int_{R_{g,\text{min}}}^{R_{g,\text{max}}} n(R_g) \, dC^{\text{bac}}(R_g) \, dR_g = n_o C_p \, dC^{\text{bac}}(R_g). \quad (18)$$
The C<sub>p</sub> factors can be calculated for self-preserving systems with a scaling distribution, following Sorensen [23] as,

$$C_p = \exp\left(\eta \ln^2 \sigma\right) \qquad (19)$$

where η = 2 for aggregates in the Guinier Regime or η = 1/2 for aggregates in the power-law regime. For a typical polydisperse log-normal distribution with σ = 2, C<sub>p</sub> reaches values up to 1.27 and 2.61, in the Guinier and power-law regimes, respectively. We note that Sorensen and Wang [60] find that C<sub>p</sub> = 1.57 for DLCA aggregates with D<sub>f</sub> = 1.75. We take the simplification that C<sub>p</sub> = 1 in the following.

## 4.2. Lidar ratio

The lidar problem is inherently under-determined, as demonstrated by Eq. (16), which shows that two variables, α and β, are to be derived from a single measurement. This requires an additional relationship between α and β, namely the extinction-to-backscatter ratio, or Lidar Ratio (LR). This quantity, with units [sr], is defined as [61–63]

$$LR := \frac{\alpha}{\beta} = \frac{\langle C^{\text{ext}} \rangle}{\langle dC^{\text{bac}} \rangle} = \frac{4\pi}{\omega F_{11}}, \qquad (20)$$

where ⟨·⟩ is the average aggregate cross-section and ω the single-scattering albedo. Note that the LR is expressed in units of solid angle and depends significantly on particle properties, including both refractive index and size. Values for LR can be established from extensive lidar campaigns [64,65], measured using advanced lidar techniques [66], or from light-scattering simulations involving irregularly shaped-particles [19,26,67,68].

We now apply the RDG-FA approximation to the LR, starting with the approximate expressions for the absorption and scattering cross-sections:

$$C^{\text{abs}} = \frac{6\pi}{\lambda} V E(m), \qquad (21)$$

and

$$C^{\text{sca}} = \int_{0}^{4\pi} \frac{dC^{\text{sca}}}{d\Omega} \left( \frac{1 + \cos^2 \theta}{2} \right) d\Omega = \frac{24\pi^3}{\lambda^4} g(R_g) V^2 F(m) \qquad (22)$$

where E(m) = Im { (m² - 1) / (m² + 2) } is the soot absorption function. Liu et al. [4] suggest that 0.32 ≤ E(m) ≤ 0.4 and g(R<sub>g</sub>) is an analytical expression defined by Dobbins et al. [53],

$$g(R_g) = \left[ 1 + \frac{4}{3D_f} (k_o R_g)^2 \right]^{-D_f/2} . \qquad (23)$$

Thus, the aggregate extinction cross-section C<sup>ext</sup> can now be approximated as

$$C^{\text{ext}} = C^{\text{abs}} + C^{\text{sca}} = \frac{6\pi}{\lambda} V E(m) + \frac{24\pi^3}{\lambda^4} g(R_g) V^2 F(m). \qquad (24)$$

Although approximate, Eq. (24) aligns closely with the results from rigorous numerical simulations [23,69]. An expression for the LR then results from Eqs. (12) and (24) as

$$LR = \frac{2\lambda^3}{3\pi} \frac{1}{V} \frac{E(m)}{S(q_\pi)} \frac{F(m)}{S(q_\pi)} + \frac{8\pi}{3} \frac{g(R_g)}{S(q_\pi)} \qquad (25)$$

For small and medium-size fractal aggregates in the Guinier regime, g(R<sub>g</sub>) ~ 1, and S(q<sub>π</sub>) ~ 1, we simply have

$$LR = \frac{2\lambda^3}{3\pi} \frac{1}{V} \frac{E(m)}{F(m)} + \frac{8\pi}{3} \sim \frac{E(m)}{F(m)} \left( \frac{\lambda}{R_m} \right)^3 \qquad (26)$$

This expression demonstrates that the lidar ratio increases with wavelength: it is lower in the ultraviolet (UV) range, moderate in the visible range, and higher in the infrared range. Using this analytical equation for the lidar ratio (LR), large values can be obtained for small monomers since the ratio E(m)/F(m) is non-zero for soot. Notably, for non-absorbing aggregates where E(m) = 0, the expression simplifies to the well-known value of 8π/3.

## 4.3. Mass backscattering coefficient

Volume backscattering coefficients can be converted into equivalent-mass concentrations by division with a reference mass backscattering cross-section (MBC). Let us define the MBC, which has units [m² sr⁻¹ kg⁻¹], as the ratio

$$\text{MBC} = \frac{dC^{\text{bac}}}{m} = \frac{dC^{\text{bac}}}{\rho V} = \frac{3dC^{\text{bac}}}{4\pi\rho N R_m^3}, \qquad (27)$$

where m is the aggregate mass, ρ is the monomer particle density with units [kg m⁻³]. Here, we use ρ = 1800 kg m⁻³ since it is the most commonly reported value for black carbon and mature soot.

$$\text{MBC} = \frac{dC^{\text{bac}}}{\rho V} = \frac{9\pi^2 V}{\lambda^4} F(m)S(q_\pi). \qquad (28)$$

Using Eq. (11), this becomes

$$\text{MBC} = \frac{3N}{4\pi\rho} k^4 R_m^3 F(m) S(q_\pi, R_g, D_f). \qquad (29)$$

The MBC is useful for retrieving the mass concentration m<sub>o</sub> of an aerosol of soot aggregates with units of [g m⁻³] from the backscatter coefficient β measured by a lidar instrument, i.e.,

$$m_o = \frac{\beta}{\text{MBC}}. \qquad (30)$$

In addition, there is a relationship between the more commonly used mass-extinction coefficient (MEC) and mass-backscattering coefficient (MBC) using the LR as follows,

$$\text{MBC} = \frac{dC^{\text{bac}}}{\rho V} = \frac{\langle C^{\text{ext}} \rangle}{\rho V LR} = \frac{\text{MEC}}{LR} \qquad (31)$$

The MBC can be regarded as the backscattering counterpart of the mass-absorption coefficient (MAC), which is widely used [4,70]. However, very limited data are available for MBC values in the literature.

## 4.4. Backscattering color ratio

Light-scattering at multiple wavelengths has been proposed to retrieve particle-size information about particles using different coefficients [71]. In particular, multi-wavelength lidar systems typically measure the aerosol backscatter coefficient within a common volume at several spectrally distinct wavelengths, such as 1064, 532, and 355 nm. A common parameter used in multispectral lidar is the backscattering Color Ratio (CR), which serves as an indicator for particle size and composition, and is defined as the ratio of aerosol particle backscattering cross-sections measured at two wavelengths as,

$$\text{CR} = \frac{dC^{\text{bac}}(\lambda_1)}{dC^{\text{bac}}(\lambda_2)}, \qquad (32)$$

where λ₁ > λ₂. This convention has been used in CALIOP retrieval algorithms [72,73] and others [74]. Other definitions are sometimes adopted [75], therefore, comparisons between studies should be made with care.

Backscattering CR is widely used in remote-sensing, including ground-based, airborne, and satellite observations [76–78] and is particularly sensitive to the presence of sub-micrometer particles, making it an essential parameter for studies of fine and ultrafine aerosols [79, 80]. A smaller (larger) CR value is indicative of a smaller (larger) particle size. A commonly used guideline is that a CR value less than 0.6 indicates fine or accumulation-mode aerosols, while larger values indicate coarse-mode aerosols [81]. Naturally, it should be noted that this is not a universal rule, but rather an empirical observation that depends on the specific type of aerosol in question. The computation of backscattering CR predominantly relies on empirical relationships rather than comprehensive theoretical models due to inherent complexities in aerosol microphysics and light-scattering. While the Lorenz–Mie theory provides a solid framework for calculating wavelength-dependent
scattering properties of spherical particles, its direct application to irregularly-shaped particles, such as fractal aggregates, faces significant limitations.

Here, we propose to derive a simple expression for the backscattering CR using the RDG-FA approximation, using Eq. (11) or (12),

$$CR = \frac{k_1^4}{k_2^4} \frac{F(m_1, \lambda_1)}{F(m_2, \lambda_2)} \frac{S(q_{\pi,1}, \lambda_1)}{S(q_{\pi,2}, \lambda_2)} = \frac{\lambda_2^4}{\lambda_1^4} \frac{F(m_1, \lambda_1)}{F(m_2, \lambda_2)} \frac{S(q_{\pi,1}, \lambda_1)}{S(q_{\pi,2}, \lambda_2)}, \quad (33)$$

where $q_{\pi,1}$ indicates calculating $q_{\pi}$ with $\lambda_1$, etc.

We now apply the Dobbins–Megaridis formulation for the structure factor, which is appropriate for mature soot, assuming the refractive index is constant across the visible and near-infrared spectral range, exhibiting no wavelength dependence for $F(m)$ [70]. Three distinct cases can be identified based on the size of the aggregate:

(i) For small aggregates ($q_{\pi} \ll R_g$), the structure factor $S(q_{\pi}) \sim 1$, and the CR remains constant for given values of the refractive index $m$, and wavelengths $\lambda_1$, and $\lambda_2$,

$$CR \approx (\lambda_2/\lambda_1)^4, \quad (34)$$

(ii) For medium-sized aggregates in the Guinier regime ($q_{\pi} \sim R_g$), a simple expression for the CR can be derived as,

$$CR = \left(\frac{\lambda_2}{\lambda_1}\right)^4 \exp\left[\frac{(q_{\pi,2}^2 - q_{\pi,1}^2)R_g^2}{3}\right], \quad (35)$$

(iii) For large aggregates in the power-law regime ($q_{\pi} > R_g$), the CR can be expressed as

$$CR \approx (\lambda_2/\lambda_1)^{4-D_f}, \quad (36)$$

Using the RDA-FA model for soot particles, a range of possible CR values can be obtained, for example, between 0.20 and 0.45 for the wavelengths 355 and 532 nm.

### 4.5. Backscattering Ångström Exponent

The Backscattering Ångström Exponent (BAE) is an alternative quantity to describe the wavelength dependence of the backscattering cross-sections. It is defined as the spectral slope of the cross-sections at two wavelengths when plotted in log scale, and can be linked to the CR as,

$$BAE = - \frac{\ln CR}{\ln[\lambda_1/\lambda_2]} = \alpha_{\lambda} \ln CR \quad \text{where} \quad \alpha_{\lambda} = \ln(\lambda_1/\lambda_2) > 0. \quad (37)$$

The BAE offers several advantages over the CR, including enhanced sensitivity for characterizing ultrafine soot particles and greater robustness to calibration errors. These features have contributed to its growing adoption in inverse methods used for aerosol research and monitoring [82,83]. However, like the CR, its application also relies on empirical relationships.

Using the RDG-FA approximation, and following a similar procedure as for the CR, a straightforward analytical expression for the BAE is obtained as,

$$BAE = 4 - \alpha_{\lambda} \ln \left( \frac{F(m_1, \lambda_1)}{F(m_2, \lambda_2)} \right) - \alpha_{\lambda} \ln \left( \frac{S(q_{\pi,1}, \lambda_1)}{S(q_{\pi,2}, \lambda_2)} \right). \quad (38)$$

Let us now derive simple expressions for the BAE using the Dobbins–Megaridis formulation for mature soot, as previously done for the CR. For small and large aggregates, the BAE is found to be constant, with $BAE \approx 4$ and $BAE \approx 4 - D_f$, respectively. For medium-sized aggregates, a simple expression is derived as follows,

$$BAE = 4 - \alpha_{\lambda} \left[ \frac{(q_{\pi,2}^2 - q_{\pi,1}^2)R_g^2}{3} \right]. \quad (39)$$

This simple expression allows one to define a possible range of BAE values, for instance between 4 and 2 across any wavelength combination, for soot fractal aggregates with $D_f = 2$. This highlights the advantage of using the BAE over the CR for inferring size information, as BAE is not influenced by the selection of wavelengths.

## 5. Application to atmospheric black carbon

We now investigate the use of the approximate expressions developed above to study two types of soot aerosols: freshly-emitted and aged. Freshly-emitted black carbon is expected to be close to the canonical DLCA soot fractal aggregate used above, while aged black carbon typically undergoes slight restructuring and/or acquires coatings of various organic compounds. Note that brown carbon aerosols are beyond the scope of this study.

### 5.1. Freshly-emitted black carbon

For freshly-emitted black carbon, we maintain a constant fractal morphology characterized by a fractal dimension $D_f = 1.8$, a prefactor $k_o = 1.3$, and a monomer radius $R_m = 25$ nm. The selection of a suitable value for the complex refractive index $m$ for soot is still debated due to the numerous intricate and interconnected sources of uncertainty [84]. The value of $m = 1.95 + 0.79i$ (BB06-h), recommended by Bond and Bergstrom [70], is conservatively used here for freshly-emitted black carbon since it is widely accepted in light-scattering studies [85], aerosol remote-sensing [86–88], and climate modeling [89]. Alternative values, as recently proposed in several studies [4,90,91], may be used depending on the application. In many studies, the refractive index is assumed to be wavelength dependent, although here, a constant value is used as a simplification made to facilitate comparisons between the different lidar wavelengths, $\lambda = 355$ nm, $\lambda = 532$ nm, and $\lambda = 1064$ nm.

The backscattering cross-section exhibits a monotonic increase with aggregate size, as illustrated in Fig. 3(a). It approaches a power-law proportional to $R_g$ with a power of $2D_f$ and $D_f$, for small and large aggregates respectively, as seen in Eq. (12). The LR decreases from its higher values in the Rayleigh regime, as shown in Fig. 3(b) and predicted by Eqs. (25) and (26), and then reaches a plateau for larger aggregates in the power-law regime. These findings align with the reference values employed in space lidar retrieval algorithms at 355 nm [92] and with experimental data from short-range lidar [93] at 532 nm. The MBC increases monotonically with aggregate size as shown in Fig. 3(c), reaching a plateau for large aggregates. This parameter has not been extensively studied in the literature. However, several MEC measurements have been reported for different sources around $8.7 \text{ m}^2 \text{ g}^{-1}$ at 532 nm [94]. Using Eq. (31) and this averaged value for the MEC, we estimate a range of possible values for the MBC between 0.029 and $0.058 \text{ m}^2 \text{ g}^{-1} \text{ sr}^{-1}$, for a LR between 150 sr and 300 sr.

As mentioned, both the CR and BAE are used as indicators of particle size, and recall that they are ratios of backscattering cross-sections at two wavelengths. It means that the scattering regimes can differ for the same $R_g$ depending on the choice of wavelength. Fig. 4 illustrates the size-dependent behavior of these ratios. These results are consistent with experimental observations, which indicate that as the aggregate size increases, the CR increases while the BAE decreases. For atmospheric black carbon aerosols, CR values range from 0.20 to 0.25 at $\lambda = 1064$ nm and $\lambda = 532$ nm [95]. In contrast, BAE values typically fall between 2 and 4 for fine mode aerosols, particularly urban pollution and combustion particles [96], and can become negative for large particles, especially non-spherical aerosols [74,97]. For small aggregates, the CR and BAE reach constant values in the Guinier regime, as described by Eqs. (36) and (39). These values can be considered asymptotic in the absence of spectral variability of the Re{$m$}, which is a common approximation for soot particles. Interestingly, the BAE reaches the accepted value of 4, given by Lorenz–Mie theory, and does not depend on the choice of wavelengths. For medium-size aggregates, the CR and BAE vary monotonically with respect to $R_g$ as expressed by Eqs. (36) and (39). For these aggregates, the size could plausibly be inferred from lidar measurements at two wavelengths. For large aggregates, as shown in Fig. 4, the CR and BAE values reach their upper
<figure><img src="image_6.png" /><figcaption>Fig. 3. Backscattering cross-section in (a), lidar ratio in (b), and the mass-backscattering coefficient in (c) as a function of the radius of gyration for freshly-emitted black carbon fractal aggregate with $D_f = 1.8$, $k_o = 1.3$, and $R_m = 25$ nm, as calculated with the approximate RDG-FA model. Shaded in gray is the estimated MBC value from the MEC reported in the literature, for an LR between 150 sr and 300 sr.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Backscattering color ratio (a) and Ångström exponent (b) as a function of the aggregate radius of gyration $R_g$ for freshly-emitted black carbon aggregate with $D_f = 1.8$, $k_o = 1.3$, and $R_m = 25$ nm, as calculated with the RDG-FA model.</figcaption></figure>

and lower limits, respectively. This illustrates a limitation of the RDG-FA, particularly when employing the Dobbins–Megaridis expression, as aggregates become large compared to the wavelength. In this example, the monomer phase-shift $\rho_m$ remains below 0.6 for the three wavelengths, while the aggregate phase-shift $\rho_{agg}$ exceeds the recommended value of 3 only at 355 nm for aggregates larger than $R_g > 90$ nm, according to the criteria in Section 3.1. This indicates that the error at 355 nm will be greater and that a more accurate treatment, such as DDA or MSTM, should be used for large aggregates, especially in the ultraviolet.

## 5.2. Aged soot

The aging process of soot remains poorly understood, despite its importance in atmospheric science and remote-sensing. During aging, freshly-emitted fractal soot particles undergo restructuring and accumulate coatings of secondary species like sulfate, organics or nitrate — a process known as internal mixing [98,99]. As a result, the aggregates become more compact, with fractal dimensions larger than the standard $D_f = 1.8$ [31], and the monomer size increases due to organic coating during aging. This effect is modeled by an effective monomer radius and an effective refractive index, which account for the organic coating around the bare elemental carbon monomers, reflecting the increase in monomer size and the decrease in the imaginary part of $m$ [100–102].

For aged black carbon, the fractal morphology has been adjusted to account for the restructuring, i.e. compaction, of the soot aggregate shape during the atmospheric aging process. The pair of parameters $D_f = 2.6$ and $k_o = 0.8$ is used for aged soot, which are commonly reported values for aged black carbon [31,85]. A core–shell model is

employed to represent the presence of organic coating on elemental carbon particles. Here, a spherical elemental carbon core with a radius of $r_c$ is covered by a spherical organic or water-soluble shell with an outer radius of $r_s = r_c + tr_c$, where $t$ represents the coating percentage. The volume fraction of coating $f = 1 - (r_c/r_s)^3$ is a dimensionless factor commonly used to quantify the amount of coating and is related to aerosol aging. Effective medium theory, such as Maxwell–Garnett and Bruggemann approximations, has been used to model coated monomers for the RDG-FA [103]. For lightly-coated particles, the effective refractive index $m_{eff}$ can be easily calculated using the Maxwell-Garnett model [104] as follows:

$$m_{\text{eff}} = m_c \left( 1 + \frac{3f \left( \frac{m_c^2 - m_s^2}{m_s^2 + 2m_c^2} \right)}{1 - f \left( \frac{m_c^2 - m_s^2}{m_s^2 + 2m_c^2} \right)} \right)^{1/2} \quad (40)$$

where $m_c$ and $m_s$ are the refractive index of the core and shell parts, respectively. Here, a core of elemental carbon with refractive index $m_c = 1.95 + 0.79i$ (BB06-h) is coated by a shell of organic material with refractive index $m_s = 1.55$ as recommended by Horvath [105]. The effective refractive index is calculated using Eq. (40) as $m = 1.86 + i0.63$ for a 20% coating in volume. This value aligns with previously reported results and approaches the lower value (BB06-l) recommended by Bond et al. [70]. Overall, there remains a lack of consensus on the causes and mechanisms of soot aging and its effect on optical properties, making it an ongoing area of research. Note that the values suggested here are purely indicative to demonstrate the capability of the present backscatter model.
<figure><img src="image_8.png" /><figcaption>Fig. 5. Effect of monomer radius $R_m$ on lidar ratio (LR) at 532 nm as a function of radius of gyration $R_g$, as calculated with the RDG-FA backscattering model. In (a) is freshly-emitted black carbon with $m = 1.95 + 0.79i$ while (b) shows aged black carbon with $m = 1.86 + 0.63i$. The latter one accounts for a shell coating of $t = 20\%$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Effect of aging on the color ratio (CR) and backscattering Ångström exponent (BAE) for $\lambda_1 = 532$ nm and $\lambda_2 = 355$ nm, as a function of the radius of gyration, as calculated with the RDG-FA backscattering model, for freshly-emitted and aged black carbon.</figcaption></figure>

Let us now investigate how aging affects the lidar parameters, focusing on the changes in fractal dimension ($D_f$), monomer size ($m$), and mean radius ($R_m$), using the RDG-FA backscattering model. Fig. 5 illustrates the effect of aging on the LR, including fractal restructuring, growth in monomer size, and a decrease of $\text{Im}\{m\}$, resulting from coating and internal mixing as explained above. As $R_m$ increases, the LR decreases for both freshly-emitted and aged black carbon, with a particularly strong decrease for the former. The averaged LR values range from ~150 sr for freshly-emitted soot to ~80–100 sr. This variation in LR is consistent with experimental measurements in urban regions, where freshly-emitted fractal soot particles dominate particulate emissions during peak traffic hours and subsequently undergo aging processes after sunrise [93]. Fig. 6 illustrates the effect of aging on the CR and BAE, with values typical of those reported for fresh smoke [106]. The change in fractal morphology between freshly-emitted and aged black carbon has a significant effect on the CR and BAE for large $R_g$. This effect can be explained by the fact that the CR and BAE are primarily sensitive to the overall aggregate size $R_g$ in the Guinier or power-law regimes.

The analysis presented here provides physical insight into the modeling of lidar-relevant parameters for freshly emitted and aged soot. However, these results should be treated with caution. For instance, for an aggregate with a constant $N$, monomer growth directly affects $R_g$. Furthermore, coating affects both $D_f$ and $k_o$, the latter describing the aggregate anisotropy and the monomer packing density. Recent work has proposed an empirical law to account for the restructuring of soot due by coating [41]. It is important to note that this analysis uses an approximate backscattering model based on RDG-FA theory. Therefore,

the results should be viewed as estimates rather than exact numerical results. Such approximations are often appropriate in remote-sensing, given the substantial uncertainties in the physicochemical properties of aerosols and their evolution over time due to environmental interactions. These findings may ultimately help us to model key lidar parameters for black carbon emissions and to understand how aging affects these parameters.

### 5.3. Limitations

The applicability of RDG-FA is subject to the validity of the fractal model for atmospheric soot morphology [107–109]. Studies have shown that soot can experience significant restructuring, such as compaction, due to organic coatings. Such restructuring is expected to increase the fractal dimension, thereby challenging the validity of analytical expressions for the structure factors. While practical, these empirical structure factors are not without limitations. For instance, the expression proposed by Dobbins and Megaridis, though widely accepted in combustion applications, is generally applicable to chain-like fractal aggregates with fractal dimensions below 2. Moreover, the transition between distinct scattering regimes, i.e., Guinier and power law, may require additional methods to ensure the correct application of regime-specific expressions. Internal multiple scattering may also introduce small deviations in the structure factor slope within the power-law regime, particularly at short wavelengths, such as in the ultraviolet spectrum [47].

Different strategies should be employed to assess the robustness of these empirical structure factor expressions. One approach is to conduct
measurements at multiple wavelengths to verify consistency [110], or to employ complementary techniques, such as electron microscopy, to enhance the reliability of the determined structure parameters. Another method involves deriving the structure factor through numerically exact light-scattering simulations, such as the DDA or MSTM, applied to numerically generated fractal aggregates with well-defined morphologies. Recent work by Argentin et al. (2023) demonstrates that neglecting internal coupling, i.e., multiple-scattering between monomers, can result in an overestimation of the structure factor at large scattering angles, including backscattering. Cross-comparison with light-scattering simulations has been used to strengthen confidence in the use of empirical structure factor expressions, while also proposing corrections using the phasor analysis introduced by Berg et al. [111]. Although such numerical considerations deliver high-fidelity predictions, their substantial computational cost restricts their application mostly to validation rather than routine analytical use. Nevertheless, these numerical comparisons provide important benchmarks for the validation and enhancement of novel structure factor models, which may then be tailored to particular applications.

## 6. Conclusion

This study introduces a novel backscattering model for soot aerosols, based on the Rayleigh-Debye-Gans for Fractal Aggregates (RDG-FA) approximation. This model provides simple analytical expressions for key lidar parameters specific to black carbon aerosols. The results show that this model can be used to calculate the differential backscattering cross-section, lidar ratio (LR), mass backscattering coefficient (MBC), along with the backscattering color ratio (CR) and Ångström exponent (BAE) at two wavelengths. Our findings demonstrate the model's capability to capture the evolution of these parameters throughout the life cycle of black carbon, from chain-like freshly emitted particles to more spherical aged particles.

While already verified on canonical DLCA aggregates [26], further validation of the RDG-FA backscattering model could be pursued using more accurate light-scattering methods, such as the DDA or MSTM, particularly for large aggregates where the validity conditions of the approximation may no longer hold. Testing the derived analytical expressions on a broader range of soot aggregates could prove beneficial to ensure the robustness of the model. Additionally, examining the internal electromagnetic fields through phasor analysis [48] could provide deeper insight into the limitations of the RDG-FA approximation. From this, corrections could be introduced to improve the RDG-FA backscattering model for a wider range of soot aerosols.

The findings of this work provide a foundation for future advancements in modeling the backscattering properties of soot particles. By extending our model to incorporate more accurate aging processes, thicker coatings, and restructuring, it could be integrated into lidar remote-sensing inversion schemes. Further refinement may incorporate advanced corrections, such as the Rayleigh-Debye-Gans theory for coated fractal aggregates (RDG-CFA) [101], or more realistic monomer morphologies, including necking and overlapping. These enhancements would enable a more accurate representation of light backscattering by aged black carbon aerosols, ultimately leading to improved retrieval and interpretation of lidar-derived aerosol properties.

### CRediT authorship contribution statement

**Romain Ceolato:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Jérôme Yon:** Writing – original draft, Methodology, Conceptualization. **Matthew J. Berg:** Writing – original draft, Methodology, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

We acknowledge funding by ONERA Project OPSIDIA (2022–2025). The authors are grateful to Chris Sorensen for his invaluable advice and insights on RDG-FA and q-space analysis, and to Gorden Videen for his guidance on light-scattering. We also thank Yoran Raynaud, Lucas Paulien, Andres Bedoya, Gérald Lemineur, and Manuela Hoyos for their support within the ONERA aerosol lidar group. Lastly, we acknowledge Michael I. Mishchenko, whose enduring legacy has been a source of inspiration for this work.

### Data availability

No data was used for the research described in the article.

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