Considering an ensemble of poly-dispersed, randomly orientated aerosol particles of which each single particle follows the independent scattering process (Mishchenko et al., 2002), the extinction coefficient (α) and the ij entry of the phase matrix (Pij) can be respectively expressed as 1αλ,m=∫rminrmaxkαλ,m,rdVrdrdr,2Pijλ,m,Θ=1σscaλ,m∫rminrmaxkijλ,Θ,m,rdVrdrdr, where dVrdr signifies the VSD of the particle ensemble; kαλ,m,r and kijλ,Θ,m,r are the kernels corresponding to the extinction and phase matrix elements, respectively, which are functions of the wavelength (λ), the CRI consisting of the real part (mR) and the imaginary part (mI) and the particle radius (r). In addition, kij is also a function of the scattering angle (Θ). The σsca is the scattering coefficient derived in a similar way to α.

From Eqs. (1)–(2), the backscattering coefficient (β), which describes the scattering intensity in the backward direction, and the particle linear depolarization ratio (PLDR; δ), which describes the ratio of the perpendicular-to-parallel polarization components of the backscattered light, are given by 3βλ,m=σscaλ,mP11λ,m,π4π,4δλ,m=P11λ,m,π-P22λ,m,πP11λ,m,π+P22λ,m,π, respectively. It is well-demonstrated that P11≥P22 and the equality hold for spherical particles (Mishchenko et al., 2002). Therefore, the PLDR is an indicator of particle sphericity, with δ=0 for spheres and 0<δ<1 for non-spherical particles that are beyond the Rayleigh scattering regime.

The total volume concentration (Vt), effective radius (reff), and single scattering albedo (SSA; ϖ) are defined as 5Vt=∫rminrmaxdVrdrdr,6reff=∫rminrmaxdVrdrdr∫rminrmax1rdVrdrdr,7ϖ=σscaλ,mαλ,m. Together with VSD and CRI, they characterize the microphysical state of an aerosol particle ensemble and are referred to as the state parameters. Note that another definition of reff for non-spherical particles is (Foot, 1988) 8reff=3∫0∞vrdNrdrdr4∫0∞ardNrdrdr, where v and a are the volume and average projected area of a single particle. Saito and Yang (2022) pointed out that these two definitions are identical for spherical particles, while a bias of ∼ 10 %–20 % between Eqs. (6) and (8) will arise for non-spherical particles depending on the extent of non-sphericity if the volume-equivalent radius (i.e. the radius of the sphere with the same volume of the represented non-spherical particle) is taken as the size descriptor. Throughout this study, we adopt the Eq. (6) to calculate reff, consistent with all the studies that are selected for comparisons (Sects. 2.2.1, 6.3).

To effectively retrieve aerosol state parameters from lidar measurements, the BOREAL algorithm was developed. Here we recap on its inversion principle and retrieval process, while more detailed description can be found in Chang et al. (2022). The inversion is achieved by making maximum likelihood estimation (MLE) of the state parameters, which can be converted to a non-linear optimization procedure. Compared to other retrieval methods based on constrained linear inversion (e.g. Müller et al., 1999, 2019; Veselovskii et al., 2002, 2010), this strategy allows one to conveniently account for a priori constraints and simultaneously consider the uncertainties in measurements and in a priori constraints. Furthermore, VSD and CRI are retrieved simultaneously, which greatly simplifies the determination of solution space and thus improves retrieval efficiency. The relationship between the generalized measurement vector (y) and the state vector (x) that is composed of VSD representation and CRI can be expressed as 9yi=fix+εi, where i from 1 to 3 denotes lidar measurements (i=1), virtual measurements for the smoothing constraint on VSD (i=2), and virtual measurements for the a priori constraint on CRI. Each measurement vector is the sum of the measurement error (εi) and the theoretical truth that is calculated from the state vector using the forward model (fi). The forward model mapping the state vector to optical properties (i.e. f1) is based on the particle scattering theory and is referred to as the particle scattering model. The MLE results in the minimization of the following objective function: 10ϕx=∑i=13yi-fixTCi-1yi-fix, where Ci is the measurement covariance matrix corresponding to yi.

For the sake of clarity and completeness, we briefly introduce the retrieval process. The state vector has a size of 10, composed of the weights of 8 log-equidistance size bins representing the VSD, one for mR and one for mI. The retrieval is performed for a set of predefined inversion windows, rmin,rmax. Each inversion window does not necessarily have overlapped regions with others, and the union of all inversion windows covers a range of 0.05–15 µm where the measurements are considered to be sensitive to the state vector. For each inversion window, an individual solution to Eq. (10) is derived, and the corresponding Vt, reff, and SSA are successively calculated from Eqs. (5)–(7). After all the individual solutions are found, the qualified ones are selected by considering the fitting error and the shape of the VSD. The statistical quantities (mean and standard deviation) of the selected individual solutions are output as the final retrieved state parameters.

In this study, apart from the spherical model used in the preliminary version (Chang et al., 2022), two non-spherical models were integrated into BOREAL (Sect. 2.3). Furthermore, BOREAL is modified to be able to take into account the spectral variability of dust mI (Sect. 2.2.2).

This section reviews previous studies on dust microphysical properties including size, morphology, and mineralogy. It rationalizes the modification of BOREAL aiming at dust retrieval and provides bases for the choice of scattering models and the setup of sensitivity study.

Dust non-sphericity evoked studies that model dust particles with more complex shapes rather than spheres, such as ellipsoids (Meng et al., 2010), nonsymmetric hexahedra (Bi et al., 2010), super ellipsoids (Bi et al., 2018), deformations of spheroids and ellipsoids (Gasteiger et al., 2011), and other evolved irregular shapes (Kalashnikova and Sokolik, 2002). To derive the related scattering properties for different domains of the size parameter (x), these studies exploited combinations of computational methods such as the discrete dipole approximation (DDA; Yurkin et al., 2007), the finite-difference time domain (Yang et al., 2000), the advanced T-matrix method (Mishchenko et al., 2002), the invariant-imbedding T-matrix method (Bi et al., 2013; Johnson, 1988), the improved geometric optics method (IGOM; Yang and Liou, 1996), and physical geometric optical method (PGOM; Yang and Liou, 1997). To some extent, these studies improve the characterization of dust particles compared to simply assuming dust shape as sphere. On the other hand, however, they suffer from different limitations which impede further application to lidar observations and retrievals. For instance, in Meng et al. (2010), the model failed to consider the coherent backscattering enhancement (CBE) due to the use of IGOM, resulting in an underestimation of β by up to a factor of 2 (Huang et al., 2023; Zhou, 2018). Others are of high complexity and lack a complete published database, so it is difficult to apply them to remote sensing retrievals. In this study, we evaluate the performance of the following two non-spherical scattering models. The spherical model is also tested as a comparison, given the fact that most global aerosol models approximate dust as spherical particles (Gliß et al., 2021).

In this study, BOREAL combined with different scattering models is applied to invert real dust optical properties measured by LILAS (Lille Lidar Atmospheres) – a multi-wavelength Mie–Raman–polarization–fluorescence lidar system developed by the Laboratoire d'Optique Atmosphérique, Université de Lille. LILAS exploits the Nd:YAG crystal to emit laser pulses with a repetition rate of 20 Hz and pulse energy of 90, 100, and 100 mJ at 355, 532, and 1064 nm. Currently, the receiving channels are composed of the following: three pairs of parallel- and cross-polarized channels at 355, 532, and 1064 nm for the reception of elastic signals; two Raman channels at 387 and 530 nm; and a broad-band fluorescence channel centred at 466 nm. Such configuration allows simultaneous measurements of 3β+2α+3δ (PLDR at 355, 532, and 1064 nm) + 1βF (fluorescence backscattering coefficient centred at 466 nm). Detailed descriptions of data acquisition and error analysis can be found in Hu (2018), Hu et al. (2019), and Veselovskii et al. (2020). LILAS can be transferred as an individual lidar instrument to perform measurements in field campaigns (Hu et al., 2020; Veselovskii et al., 2016) as well as make stationary routine aerosol observations at the ATOLL (Atmospheric Observation in Lille) platform where various remote sensing and in situ instruments for atmospheric monitoring have been integrated into national and international observational networks such as the European Aerosol Research Lidar Network/Aerosols, Clouds, and Trace gases Research Infrastructure (EARLINET/ACTRIS) and PHOTONS/AERONET (Holben et al., 1998; Wandinger et al., 2016).

We focus on the comparison of extinction, backscattering, and absorption properties. Figure 2 shows P11 and P22 at 180° as well as SSA simulated with the spheroidal, IH, and spherical models for different reff, mR, and mI at 532 nm. The reff varies from 0.1 to 5 µm with a fixed Sg of 1.95 (mean value of Table 1), corresponding to a rV from 0.12 to 6.25 µm. The range of the CRI coincides with the envelop of the measurements of Di Biagio et al. (2019), with mR between 1.4–1.6, mI,532=0.004 in the first row (Fig. 2a–c) and mI,532 between 0.001–0.007, mR=1.5 in the second row (Fig. 2d–f). It can be seen that the P11(π) for non-spherical particles is smaller than that for spherical particles and presents less variability driven by the change in CRI. This contrast between spherical and non-spherical particles is caused by the surface wave (van de Hulst, 1958), of which the revision here indicates that the used non-spherical models are physically sound. On the other hand, IH and spheroidal particles in general produce similar P11(π), except for reff>1.5 µm, where the P11(π) for the IH particles is slightly larger. The magnitudes of P22 simulated with the two non-spherical models are similar for reff<0.5 µm; however, for reff>0.5 µm, the P22(π) of the IH particles is evidently smaller, which will consequently result in larger PLDR. The simulated SSA does not show much sensitivity to particle shape, as demonstrated by previous studies (Johnson and Osborne, 2011; Mishchenko et al., 2002; Müller et al., 2010); however, it decreases with the increase in reff or mI because that results in an increase in absorption (Bohren and Huffman, 1998; Tegen and Lacis, 1996).

Next, we compare the lidar-related properties. The size distribution range is selected the same as that in Fig. 2, while we utilize the values from D19 for the CRI (see Fig. 1). As a comprehensive database of the short-wave CRI of global dust samples, D19 has been used for sensitivity simulations by other studies (e.g. Ito et al., 2021). The simulated α and β at 532 nm, the extinction Ångström exponent (EAE), and backscattering Ångström exponent (BAE) over 355–532 nm are shown in Fig. 3. The solid lines and error bars represent the mean and 1 standard deviation of the spread results due to the variation of CRIs. It can be seen that the simulated extinction coefficient shows little sensitivity to CRI except at reff=0.1 µm. The magnitudes of α are always similar between IH and spheroidal particles, while those of spherical particles become lower with the increase of reff. This indicates the larger surface areas of the non-spherical particles than the spherical particles, which lead to higher mass extinction efficiency (Huang et al., 2023; Ito et al., 2021; Kok et al., 2017). The simulated backscattering coefficient presents dependence on both particle shape and CRI. In particular, spherical particles show considerably larger β than non-spherical particles, which is in line with the contrast of P11 shown in Fig. 2. A weaker backscattering is expected as particles become more aspheric and sharper (Kalashnikova and Sokolik, 2002; Mishchenko et al., 2002). Nevertheless, the β of the spheroidal particles is quite similar to that of the more aspheric and sharper IH particles. This is because the computational method used by the spheroidal model ignores the CBE effect (Sect. 2.3), resulting in a systematic underestimation of β. The EAEs simulated with all these models always show little sensitivity to CRI, and as reff increases, they first steadily decrease to zero and then show weak sensitivity to reff. The BAEs simulated with all these models also decrease with the increase in reff; nevertheless, they present increasing sensitivity to CRI as reff increases.

Figure 4 displays spectral LRs and PLDRs simulated with the spheroidal and IH models. The particle microphysical properties for the simulation are the same as those in Fig. 3. It can be seen that the LRs for the two particle types only show divergence at 355 and 532 nm for reff>1 µm, where the spheroidal particles have a larger LR. However, the LR variations against wavelength and particle size produced by the two non-spherical models are always consistent. For reff between 0.4–1 µm, the LRs slightly increase as the wavelength changes from 355 to 532 nm, while an obvious increase in the LRs is observed as the wavelength increases from 532 to 1064 nm. By contrast, for reff>1 µm, the LRs present a clear decreasing trend with the increase in the wavelength and, at the same time, are more affected by CRI. Furthermore, although it is not shown here, spherical particles have apparently lower LR than the non-spherical particles due to their stronger backscattering, as we have seen in Fig. 3a. The simulated PLDRs show strong size dependence varying with wavelength, indicating spectral PLDR contains useful information for particle size retrieval (Mishchenko et al., 2002). The influence of particle shape on PLDR seems to depend on the effective size parameter, xeff=2πreff/λ: the PLDR of IH particles is obviously larger than that of spheroidal particles when xeff is greater than 5. The higher PLDR produced by the IH particles could result from the more irregular shape with asymmetric surfaces that cause more complex inner reflections and thus more significant change in the polarization state between the incident and backscattered light (Liou, 2002). However, when xeff is less than 5, the sensitivity of PLDR to surface deformation disappears, and the PLDRs of both types of particles rapidly decrease with the decrease in xeff, consistent with the variations of P22(π) shown in Fig. 2b and d and the simulations of Gasteiger et al. (2011). The spectral variation of PLDR when reff>1 µm is another significant contrast between the two models: as the wavelength increases, the spheroidal-produced PLDR monotonically increases, whereas the IH-produced PLDR peaks at 532 nm or monotonically decreases. Furthermore, this simulation suggests that PLDR is more sensitive to particle size than CRI: the influence of CRI on PLDR emerges only if reff>1 µm at 355 nm for both types of particles.

We compared model-simulated optical properties with some real lidar measurements, which are compiled in Table 2. In particular, it shows the comparison between two triple-wavelength LR and PLDR measurements of Saharan dust (Gebauer et al., 2024; Haarig et al., 2022). The two measurements present consistent spectral variation of PLDR between 355 and 1064 nm and spectral variation of LR between 532 and 1064 nm, verifying the same Saharan source. However, the measurement by Gebauer et al. (2024) shows an overall smaller spectral PLDR and a larger LR₃₅₅ than those by Haarig et al. (2022), indicating the dust was polluted (Gebauer et al., 2024). Combining all these measurements, the measured EAEs vary in an overall range of ±0.3, showing no clear dependence on dust sources and transport processes. They can be reproduced by all the scattering models for reff>0.4 µm. The measured BAE varies in a larger range between -0.95 and 0.6 probably due to the variations of CRI. Except the polluted Saharan dust case, the measurements reveal a source dependence of dust BAE: Saharan dust has a systematically higher BAE than Asian dust, which might result from the difference in CRI induced by different mineralogic components in the source soil. In spite of the large range of the measured BAE, it can only be reproduced by the IH particles when reff is greater than 1 µm.

The measurements of LR generally decrease from 355 to 532 nm (except for the Saharan transported dust) and increase from 532 to 1064 nm. Such a spectral variation is well described by both spheroidal and IH models for the reff range 0.4–1 µm. When reff≥2 µm, the spheroidal model produces LR₃₅₅ too high, whereas the IH model produces more reasonable values compared with the measurements. On the other hand, both spheroidal and IH models can well reproduce the measured LR₃₅₅ when reff>0.4 µm. The measurements of PLDRs for Saharan and Middle East dust can be reproduced by spheroidal particles with reff between 0.4–1 µm, whereas it cannot be well reproduced by IH particles in this size range because the simulated values are either too high at 355 nm or too low at 532 nm. On the other hand, only the IH particles can reproduce the measurements of PLDRs for US and East Asia dust, of which the magnitudes and spectral variations are apparently distinct from those for Saharan and Middle East dust, probably because of the differences in dust purity and the fraction of larger particles (Gebauer et al., 2024; Hu et al., 2020).

The above simulations illustrate the distinctions and resemblances among the spherical, spheroidal, and IH models in generating particle scattering properties. The backscattering coefficient is the most prominent contrast between the spherical and non-spherical models, whereas the main difference between the two non-spherical models is observed in the PLDR. They indicate that the IH model might be the preferable one in terms of mimicking optical properties more consistent with real lidar measurements. A further discussion is presented in Sect. 6.

Recall that in Sect. 2.2.2, in order to account for the spectral dependence of dust imaginary refractive index, we use the derived regression relation to link mI,355 and mI,532 and fix mI,1064 to 0.001. Following from this consideration, two questions arise: (1) what will be the influence if we ignore the mI spectral dependency? (2) Since a visible dispersion of mI,1064 (2×10-4–2×10-3) is observed in Fig. 1b, is mI,1064=0.001 a reasonable assumption for the retrieval? To this end, we calculated the difference in optical properties due to different treatments of mI. The same microphysical properties as those in Figs. 3–4 are used for the calculation. As illustrated in Fig. 5, the solid lines represent the variations in α, β, and δ after setting mI,λ=mI,355 (λ=532 or 1064 nm), namely, dismissing the spectral dependence, while the dashed lines represent the variations after setting mI,1064=0.001. For comparison, we set measurement uncertainties as one-third of the estimated maximum measurement errors for LILAS (2019). It can be seen that at 532 nm, the variability of α is much smaller than the corresponding measurement uncertainty all the time, which means that ignoring the spectral dependence of mI has no influence on α532. Nevertheless, the variation in β rapidly increases as reff grows and exceeds the measurement uncertainty for reff>0.2 µm (at 532 nm) and reff>0.4 µm (at 1064 nm), respectively. The influence on δ is only significant for reff>3 µm at 532 nm and reff > 4 µm at 1064 for both spheroidal and IH models. On the other hand, the assumption of mI,1064=0.001 makes no change in δ1064; however, the variation of β1064 exceeds the measurement uncertainty for reff>3 µm for the spherical and spheroidal models and reff>4 µm for the IH model, respectively.

This simulation demonstrates that, on the one hand, it is necessary to consider the spectral dependency of mI in mineral dust retrievals, and on the other hand, the treatment presented in Sect. 2.2.2 is rational. Correspondingly, the retrieval of dust spectral imaginary part is converted to the retrieval of the monochromatic value at 355 nm. Hereinafter, unless explicitly stated, the imaginary part exclusively refers to mI,355, and we omit the subscript indicating the wavelength for simplicity.

One of the objectives of this study is to figure out how the inclusion of spectral depolarization measurements influences the retrieval accuracy. To this end, we study their sensitivities to reff and CRI (note that the PLDR is independent of Vt) using the error-normalized Jacobian (ENJ) matrix (Rodgers, 2000; Xu and Wang, 2015), which is defined as 13J̃=Sε-12JSa12, where J is the measurement Jacobian matrix for a specific state, Sε the measurement covariance, and Sa the a priori covariance of the state. The ENJ provides a direct comparison between measurement uncertainty and variability due to perturbations of the state vector. Accordingly, it is a measure of measurement sensitivity and tells whether the change in a state parameter is “detectable” by a measurement ({J̃}ij>1) or submerged by noise ({J̃}ij<1). In this study, the uncertainty in δ measurements is 5 % for all three lidar wavelengths (i.e. one-third of the estimated maximum error of δ for LILAS; cf. Hu et al., 2019). In practice, we set the a priori uncertainties of reff, mR, and mI to 20 %, 0.02, and 0.004, respectively.

Figure 6 displays the ENJ elements about spectral PLDR to reff, mR, and mI for the IH model (top row) and the spheroidal model (bottom row). The CRI used to generate ∂δ/∂reff is 1.52–0.0035 i (the mean value of D19), of which the real part is also used to generate ∂δ/∂mI, and the imaginary part is also used to generate ∂δ/∂mR. Furthermore, two effective radii, reff=0.5 µm (possible lower limit to allow both the IH and spheroidal models to reproduce most real lidar measurements) and reff=2 µm (near the median value in Table 1), are used to generate ∂δ/∂mR and ∂δ/∂mI to capture their behaviours under different sizes. It can be seen that both non-spherical models share some common features. The sensitivity of δ to reff decreases from positive to negative as reff increases, and δ1064 shows the highest sensitivity in both models when reff>0.2 µm. However, compared to the IH model, the PLDR sensitivity to reff decreases faster with particle size for spheroidal particles. The sensitivity of δ to mR decreases with the increase in reff, the increase in mR, and the decrease in λ. The sensitivity of δ to mI is overall negative, and its magnitude increases with the increase in reff and the decrease in λ, but it shows opposite variations with mI in different models. Also note that the mI here refers to mI,355, of which δ1064 is independent. The general findings from Fig. 6 are (1) δ1064 might be essential for size retrievals, while (2) δ355 could be helpful in imaginary refractive index retrievals. In addition, we point out that the accuracies of some state parameters, to which the incorporated δmeasurements do not have much sensitivity, can still be improved as a part of the integral retrieval. This will be further demonstrated in the next section.

In this section, we invert synthetic measurements without measurement errors included (error-free). We focus on the comparison of the results derived from (3β+2α+nδ) measurements, where n varies from 0 to 3 so as to evaluate the influence of different depolarization measurements on the retrieval. Figure 7 shows the retrieval errors of Vt, reff, mR, and mI for the reff values in Table 3 and the CRI of 1.4–0.001 i. The optical data are simulated and inverted by the IH model. Overall, the Vt and reff (for reff>0.3 µm) are underestimated, whereas the mR and mI overestimated, and this trend exacerbates with the increase in particle size. This could result from the retrieval process of BOREAL. As described by Chang et al. (2022), the initial guess for the VSD is a uniform distribution with a Vt usually smaller than the truth. During the iteration, the state vector (composed of VSD bin and CRI) should gradually converge, making Vt increase to the exact value. However, as particles become bigger, (3β+2α) measurements lose the sensitivity to the state parameters rapidly because of the fast decreases in the magnitudes (see Fig. 3), which, in turn, increases the underdetermination of the system. As a result, Vt often stops at an underestimated value, and the consequent effect is compensated for by biases of other state parameters (e.g. reff, mR and mI). As discussed in other studies, this is one of the main challenges of retrieving coarse aerosol particles from the conventional (3β+2α) dataset (cf. Burton et al., 2016; Müller et al., 2019; Veselovskii et al., 2002). On the other hand, it reveals BOREAL fails to provide effective constraints on large particles when measurement sensitivity gets weak, which is a crucial point to improve in future studies. The incorporation of δ largely improves the retrieval accuracy, especially for the reff in 0.5–2 µm where the δ shows higher sensitivity to reff as indicated in Fig. 6. The results also highlight the significance of δ1064 to the retrieval accuracy: the inversions including δ1064 are in the cluster with the best quality. Indeed, compared to other wavelengths, PLDR at 1064 nm shows the highest sensitivity to reff and mR, as indicated by Fig. 6. However, as reff>2 µm, the accuracies of δ-included retrievals also decrease due to the decrease in δ sensitivity, but they are still better than those of the non-δ-included retrievals.

Figure 8 is same as Fig. 7 except that the optical data are simulated and inverted by the spheroidal model. Some common features shared with Fig. 7 can be extracted. For instance, the envelops of retrieval errors show the same trend of variation with reff; the retrievals have the best accuracy when δ1064 is incorporated into the inversion set. Nevertheless, the most distinctive feature compared to the IH model is that for the spheroidal model, the incorporation of δ355 nearly makes no difference to the retrieval.

We zoom in to check the retrieved VSDs at two effective radii: reff=0.5 and 2 µm. The reason for selecting these reff values is mentioned in Sect. 3.3. The results for the IH model and the spheroidal model are shown in Figs. 9–10, respectively. For both scattering models, the monomodal shape of the VSD is successfully retrieved. It can be seen that the VSD retrieval deteriorates as reff increases from 0.5 to 2 µm, and a conspicuous improvement in accuracy is observed for the IH model when δ1064 is incorporated. The underestimates of Vt are mainly caused by the underestimates of Sg and the maximum VSD, while the underestimates of reff are mainly due to the underestimates of Sg and rV.

Figure 11 displays variations of retrieval errors with respect to CRI when reff is 2 µm for the IH model (the synthetic measurements are generated and inverted by the IH model). Here we focus on some representative features. Firstly, as mR or mI increases, the accuracy of non-δ (i.e. 3β+2α) inversion is gradually improved. This is because as the true CRI becomes larger (e.g. 1.6–0.009 i), it is less likely to retrieve a highly overestimated CRI because of the a priori constraint (mRa=1.5, mIa=0.005). This also leads to improvement in Vt and reff retrievals due to the compensation effect. When different δ measurements are considered, the difference between the retrievals with and without δ1064 included becomes smaller as mR or mI increases. However, it is clear that the inversions with δ1064 included receive the best accuracy when both mR and mI are low (e.g. mR=1.4, mI=0.001), and compared to the (3β+2α) inversion, a conspicuous retrieval improvement is found for any δ-included inversions as long as mR>1.5 and mI=0.009 do not simultaneously happen. Similarly, Fig. 12 shows the results for the spheroidal model. The results show a larger dispersion and variability as CRI changes. Compared to the IH model, the contribution of δ1064 to retrieval accuracy is less significant, especially in terms of the mI retrieval. However, it can still be identified when the true CRI is 1.4–0.001 i. Another noticeable feature for the spheroidal model is that as mR or mI increases, accuracies of the inversions only including δ355 are significantly improved and even are better than the inversions only including δ1064 when mR>1.5. This might explain the somewhat contradictory finding by Tesche et al. (2019) that δ355 brings more retrieval improvement than δ1064 because they utilized the spheroidal model, and all the retrievals in their study show mR greater than 1.5. A further discussion is presented in Sect. 6.2.

For both IH and spheroidal models, a retrieval is often of higher accuracy by only including δ1064 rather than δ355+δ532+δ1064, which is contrary to the intuition that the more incorporated measurements, the better the retrieval accuracy. The explanation here could be that due to the limited sensitivity of δ355 or δ532 to the particles with reff=2 µm, the incorporation of δ355 or δ532 in fact contributes more to the ill-posedness than the information contents (Twomey, 1977). Furthermore, we notice a sudden rise of retrieval error happens to the spheroidal model for two cases: (1) (3β+2α) inversion when the true CRI is 1.6–0.005 i and (2) (3β+2α+δ532) inversion when the true CRI is 1.6–0.009 i. We found that their low retrieval qualities are accompanied by fitting errors (Eq. 14) greater than 5 %, which are large compared to most of the cases where the fitting errors do not exceed 2 %. The reason is still unknown, but it implies potential instability of the spheroidal model in the inversion of lidar measurements.

As the reff gets lower (see Figs. S1–S2 in the Supplement for the case of true reff=0.5 µm), the variations of the retrievals against CRI generally keep the same trend for both IH and spheroidal models. However, it is noteworthy that due to the increase in sensitivity, there is a significant improvement in retrieval accuracy in the non-δ inversion. In addition, for the IH model, the accuracies of mI retrieved from δ-included inversions visibly decrease probably because of the reduction of δ sensitivity to mI as seen in Fig. 6c.

To have an overall point of view on these retrieval results, we grouped them by the reff range as Group 1 where 0.5 µm ≤ reff < 1 µm, allowing the scattering models to reproduce real lidar measurements for the submicron metre range (Figs. 3–4 and Table 2); Group 2 where 1 µm ≤ reff ≤ 2 µm, consistent with the reff ranges derived from most in situ results in Table 1 and lidar measurements are still sensitive to the state parameters to some extent; and Group 3 where 2 µm < reff,true≤5 µm, including the rest of the dataset. Tables 4–5 summarize the means ± standard deviations of the grouped retrieval errors when the (3β+2α+nδ) measurements are inverted with the IH (Table 4) and spheroidal (Table 5) model, respectively. We also list the results derived by inverting (3β+2α) with the spherical model in order to quantify the retrieval bias as ignoring particle non-sphericity. It can be clearly seen that the increase in particle size leads to an overall reduction of retrieval accuracy. Consistent with previous single-case comparisons, non-δ inversions result in underestimation of Vt and reff and overestimation of mR and mI due to the underdetermination and compensation effect. The incorporation of δ1064 largely improves the retrieval accuracy for all state parameters, especially for Group 2 and Group 3. Statistical results suggest that combining δ1064 with δ355 or δ532 could additionally improve the CRI retrieval for Group 1 and Group 2, while such an improvement cannot be achieved by singly using δ355 or δ532. Compared to the spheroidal model, the δ measurements represented by the IH model bring greater retrieval improvement, especially for CRI when reff belongs to Group 2. Finally, ignoring particle non-sphericity (i.e. inverting the synthetic measurements by the spherical model) severely underestimates mR and overestimates mI, which also leads to the largest retrieval bias for Vt and reff (both are underestimated). This is because the spherical assumption offsets β to higher values which have to be reduced by increasing mI or decreasing mR (see Figs. 2–3). Such a behaviour has been widely observed in previous studies (Müller et al., 2013; Veselovskii et al., 2010).

In this section, we investigate how well these scattering models work when inverting noise-contaminated measurements by adding simulated random noise to the synthetic measurements. We take the magnitudes of the maximum measurement errors of LILAS as references, which are 10 % for α355, α532, β355, and β532; 15 % for δ355, δ532, and δ1064; and 20 % for β1064 (Hu et al., 2019). We assume the independent Gaussian error for each measurement; thus, the corresponding standard deviations are 3.3 % for α355, α532, β355, and β532; 5 % for δ355, δ532, and δ1064; and 6.7 % for β1064. According to the results presented in Sect. 4.1, we dismiss the cases where reff>2 µm and the inversions using the spherical model as they do not have satisfying retrieval accuracy even for error-free measurements. Instead, we present the noise test results of two single but representative cases: (1) reff=0.5 µm, mR=1.52, and mI=0.0035 and (2) reff=2 µm, mR=1.52, and mI=0.0035. The span of reff covers the range allowing the scattering models to reproduce most real measurements in Table 2 (Figs. 3–4) and the range of most in situ measurements in Table 1. The selected mR and mI correspond to mean values of D19.

For each case and every possible δcombination as the input, Figs. 13 (for the IH model) and 14 (for the spheroidal model) show the standard deviations (SDs) of the retrieval errors derived from 1000 inversions of the noise-contaminated measurements. The SDs of εVt, εreff, and εmR for both models are comparable, but the spheroidal model shows a generally larger SD of εmI, particularly for reff=2 µm. Recall that we have demonstrated in Sect. 4.1 that compared to the IH model, the spheroidal model does not receive a significant retrieval improvement after the incorporation of δ measurements, especially for the mI retrieval. In this regard, we believe that the IH model works better in the retrieval process. In addition, for the IH model and reff=2 µm, measurement noise causes a larger retrieval dispersion if only δ1064 is included, while a combination of δ at other wavelengths can effectively reduce the retrieval dispersion. This could be related to the alteration of system ill-posedness due to changes in the measurement set and state parameters (Rodgers, 2000; Twomey, 1977). However, a clear suggestion is that it is important to combine measurements of both high and low sensitivities to suppress the influence of measurement noise. In this regard, we suggest using the IH model to invert at least (3β+2α+2δ) lidar measurements with one of the δ at 1064 nm in real applications.

The first case is from the Dust Aerosol Observation (DAO) campaign, where intensive field measurements were taken at Kashi (39.50° N, 75.93° E) located on the western edge of the Taklamakan Desert, one of the main sources of dust in Asia (Hu et al., 2020). On 15 April 2019, LILAS detected continuous aerosol layers extending from the boundary layer (BL) to around 3.5 km (Fig. 15), resulting in an obvious increase in daily aerosol optical depth (AOD) and decrease in EAE. Back-trajectory analysis and satellite observations indicate the aerosol layers are related to a dust activity in the Taklamakan Desert during 13–15 April (Hu et al., 2020). Figure 16 displays the time-averaged optical profiles between 18:00 and 20:00 UTC on 15 April. Below 2.3 km, all the optical properties are stable, and a well-mixed structure can be identified: the PLDR is larger than 0.3 at all wavelengths and shows a typical dust-type spectral variation (Haarig et al., 2022; Hu et al., 2020), indicating the presence of homogeneous pure dust. Between 2.3 and 2.8 km, the decrease in PLDR and increase in EAE suggest an increase in fine-mode particles, and the variation of LR implies an alteration of aerosol components, although the extinction and backscattering coefficients are still stable. Above 2.8 km, the extinction and backscattering decline rapidly due to the decrease in aerosol loading.

We averaged the optical properties every 200 m for the layer below 3 km and inverted the averaged optical properties with the IH, spheroidal, and spherical models, respectively. The (3β+2α) and (3β+2α+3δ) data are inverted, and the corresponding retrieved microphysical properties (Vt, reff, mR, mI) are shown in Fig. 17. The decrease followed by an increase in reff above 2.3 km, which is only reflected by (3β+2α+3δ) inversion, agrees with the variations of the δand EAE shown in Fig. 16. The stand-alone lidar observations cannot verify the exact involved aerosol species in that layer. Hu et al. (2020) pointed out the high possibility of the presence of anthropogenic aerosols that were lifted by convection and remained at higher altitudes as bigger dust particles settled down. Given that BOREAL is only developed for pure dust retrieval (Sect. 2.2.3) at the current stage, retrieval accuracy for this layer is therefore not guaranteed. For the retrievals below 2.2 km, when inverting (3β+2α+3δ) measurements, the two non-spherical models derive apparently distinct results so as to fit the input measurements: the spheroidal model derives smaller mR and mI, as well as larger Vt and reff. The influence of ruling out 3δ on the retrievals agrees with what we see in Sect. 4: on the one hand, the spheroidal-derived mR and mI become larger, accompanied by decreasing Vt and reff; on the other hand, the IH-derived result barely changes since the mR is retrieved close to 1.6 from the (3β+2α+3δ) inversion. Compared to the inversion of (3β+2α) using the non-spherical models, using the spherical model derives similar Vt and reff but evidently lower mR and higher mI. We also observed such a behaviour when mR is set to 1.6 in the simulation (not shown here). In a word, the variations of the inversions caused by different scattering models and input measurements are in line with our simulation results, indicating that the results from (3β+2α+3δ) inversion are least affected by retrieval uncertainty.

Figure 18 displays for different configurations the retrieved VSDs, SSAs, and the comparisons between the real and reproduced measurements in the layer 1.5–2 km. The selected layer is far from the polluted layer and thus is attributed to purer dust. All retrieval configurations derive monomodal VSD located in the coarse-mode range, except for the spherical model, which obtains a second mode at 0.4–0.5 µm. When inverting the (3β+2α+3δ) measurements, the IH-derived VSD has a similar volume median radius (rV ∼ 1.3–1.7 µm) to the spheroidal-derived VSD but a larger Sg, resulting in a smaller reff. Because of the implemented spectral relationship of dust mI, the retrieved SSAs present a typical spectral dependency of dust aerosols (e.g. Di Biagio et al., 2019), and the highest SSA corresponds to the lowest mI retrieved with the configuration (3β+2α+3δ, spheroidal). The error bars attached to the real measurements represent the corresponding maximum measurement errors. For all retrieval configurations, the difference between real and reproduced measurements is within the corresponding error bar as long as this measurement is incorporated into the inversion dataset. The inversion of (3β+2α+3δ) enables both IH and spheroidal models to reproduce the spectral variation of the measured PLDR. However, when it comes to the (3β+2α) inversion, the spheroidal model underestimates δ for all wavelengths, while the IH model performs much better: only a little overestimate of δ355 is observed.

An increase in aerosol activity above the BL was observed by LILAS operated at ATOLL/Lille since 14:00 UTC on 21 March 2022. The range-corrected signals and time-averaged optical profiles between 20:00 and 23:00 UTC are shown in Figs. 19 and 20. Different from Case 1, three stratified layers can be identified: Layer 1 below 2 km, Layer 2 centred at 4 km, and Layer 3 centred at 5.4 km. Dust could be abundant in the upper two layers (Layer 2 and Layer 3) due to their high PLDRs compared to the bottom layer (Layer 1). Figure 21 shows the HYSPLIT (Rolph et al., 2017) back trajectories of the three layers ending at 21:00 UTC on 21 March 2022 (2017). It can be seen that although the upper two layers both originated from the Saharan region, they went through different transporting pathways and times. The “dust RGB” products from the Spinning Enhanced Visible and Infrared Imager (SEVIRI) on board the Meteosat Second Generation (MSG) satellite (Lensky and Rosenfeld, 2008) (Fig. S3) reveal that the pathway of Layer 3 largely overlapped with that of dust lifted from northwestern Algeria between 14–15 March. The synoptic conditions (Kanamitsu et al., 2002) (Fig. S4) during these days show low-pressure cyclones over the region with daily averaged 10 m wind speed exceeding 7 m s⁻¹, which are favourable for dust emission. Both the trajectory and satellite images indicate that after the emission, the dust layer was elevated up to ∼ 6 km, crossed the Mediterranean, and dispersed over western Europe. Meanwhile, thick ice clouds continuously existed with the dust, implying the dust might undergo cloud processing. The dust enriched in Layer 2 was emitted probably during the daytime of 17 March near the border between Tunisia and Libya and elevated by the easterly trade winds (Khan et al., 2015). Although ice clouds kept forming above, the dust was not likely to go through cloud processing due to its lower altitudes. However, given the main lifting procedure happening over the sea and the longer time remaining over the Mediterranean compared to Layer 3, Layer 2 could contain more sea salt (Denjean et al., 2016). The different transport and ageing processes of these two dust layers can lead to contrasts of their microphysical properties and, consequently, the optical properties. As can be seen in Fig. 20, compared to the upper layer, the lower layer has a larger δ355, a larger LR₃₅₅, and a smaller BAE_355–532. However, here we only perform retrievals for Layer 3 because the measurement quality of Layer 2 is not guaranteed due to the relatively low aerosol loading.

Figure 22 displays the microphysical property profiles retrieved with different configurations for Layer 3. Moving from (3β+2α+3δ) to (3β+2α) inversion, changes in the retrievals are still in accordance with the simulation results. When the (3β+2α+3δ) data are inverted, the retrieved effective radii slowly decrease from 0.7 to 0.5 µm as altitude increases, in accordance with the gradual increase in EAE. Compared to Case 1, the IH and spheroidal models obtain less divergent results, and both retrieve smaller reff and larger mI in Case 2 when the (3β+2α+3δ) data are inverted. Such contrasts could result from the distinct spectral PLDRs of the two cases. Moreover, the IH-obtained mR (1.5–1.55) in Case 2 is smaller than that (1.58–1.6) in Case 1, consistent with the higher LR in Case 2 than in Case 1. The differences in optical and microphysical properties between the two cases might be explained by the longer ageing process of dust particles in Case 2 than Case 1, particularly the persistent dust–cloud interactions in Case 2, which could considerably modify dust size (Mahowald et al., 2014; Ryder et al., 2015; Weinzierl et al., 2017; Wurzler et al., 2000).

Figure 23 shows for different configurations the retrievals of VSD and SSA, as well as the comparisons between the real and reproduced measurements in the layer 5–5.5 km. Consistent with the change in mI, the retrieved SSAs are in general lower than those in Case 1. All the inverted measurements can be reproduced from the retrievals with biases smaller than their maximum errors. However, the IH model performs better in reproducing the intensive optical properties (LR and PLDR) than the spheroidal model. A more detailed discussion is given in Sect. 6.

Both simulations and real dust retrievals in this study demonstrate the significance of considering dust non-sphericity in the retrieval from lidar measurements. The spherical model severely underestimates mR or overestimates mI as it considerably overestimates β of non-spherical particles. Therefore, it should be replaced with a proper non-spherical model. Comparing the spheroidal and IH models, we may argue that the latter is preferable because spheroids are too “smooth” and “spherical” to be like real dust particles. However, note that it is unrealistic to take the exact shape of every single dust particle for optical and radiative transfer calculations. In this regard, all applicable non-spherical models are kinds of optically equivalent approximation. We may also argue that compared to the IH model, the spheroidal model ignores the CBE effect, which results in underestimates of β (Huang et al., 2023; Zhou, 2018). However, this in turn cancels the enhancement of β due to the more spherical shape of spheroids than IH particles (Gasteiger et al., 2011; Kalashnikova and Sokolik, 2002; Mishchenko et al., 2002). In remote sensing applications, we should focus on which model is able to mimic dust optical properties that are more consistent with real measurements. As mentioned in Sect. 2.3.1, the deficiency of the spheroidal model in mimicking particle backscattering properties has been found in many previous studies (Haarig et al., 2022; Müller et al., 2010; Noh et al., 2017; Shin et al., 2018; Tesche et al., 2019). In addition, Kemppinen et al. (2015) found that the assumption of ellipsoids (a general case of spheroids) cannot fit the measured scattering matrix of irregular dust-like particles unless the CRI was largely biased from the true value. Huang et al. (2023) found that neither the spheroidal nor the ellipsoidal dust optics could reproduce the scattering matrix of the Amsterdam-Granada Light Scattering Database (AGLSD; Muñoz et al., 2012) samples well. Although there have been increasing calls for using the IH model in dust scattering calculation (e.g. Castellanos et al., 2024), we only found a very recent study where a simple conclusion (without details) that the IH model is superior to the ellipsoid model in mimicking the scattering pattern of Martian dust analogues was given (Martikainen et al., 2025). In this regard, our study provides a direct comparison of lidar-related optical properties simulated by the IH and spheroidal models.

Our simulations show the differences between the two non-spherical models in producing lidar-measured optical properties largely depend on particle size. For reff<0.5 µm, only EAEs show a slight difference within the measurement error; thus, the two models are optically equivalent in this size range. As particle size increases, BAEs, LRs, and PLDRs simulated with the two models start to be divergent, among which the PLDRs differ most (IH particles produce significantly larger values than spheroids). In addition, it seems that the differences in LR and PLDR also depend on the size parameter, which also leads to distinct spectral variations produced by the two models as reff increases. More significantly, for reff>1 µm, a size range that is in line with most published in situ and sun photometer retrieval results (Table 1), our simulations found that the IH model can better reproduce the real lidar measurements of BAE, spectral LR, and spectral PLDR (Table 2). Therefore, in this regard, we confirm that the IH model performs better than the spheroidal model.

However, we have to point out that although we tried to establish representative dust microphysical properties for the simulations based on previous studies, the settings are still limited and cannot cover all the situations. For example, the VSDs are described using the monomodal lognormal distribution with fixed Sg=1.95, a value near the middle of the Sg range in Table 1; situations when Sg varies or a narrow submicron mode close to the coarse mode simultaneously presents, as many optical particle counter (OPC) instruments show (Ryder et al., 2013, 2018; Weinzierl et al., 2009, 2011), are not included. In addition, the simulations did not account for other aerosol types like sea salt and anthropogenic pollutants that can either externally or internally (by coating or chemical reactions) mix with pure dust particles, making the aerosol ensemble more spherical and less depolarized (Kandler et al., 2009; Matsuki et al., 2005; Ryder et al., 2015; Weinzierl et al., 2009, 2011; Zhang and Iwasaka, 2004). These omissions might explain the discrepancy between some model-reproduced and real measurements. For instance, although the PLDR spectral variation of Saharan dust can be reproduced by the IH model using the setup for Sg=1.95, reff>1 µm, the magnitudes from the model are evidently higher, especially at 355 nm. A future study will further compare these two non-spherical models by considering these possibilities.

Apart from the capability of mimicking dust backscattering properties, it is also essential to assess the retrieval performance of the IH and spheroidal models. That is, the accuracy and stability of the retrieval as different measurement combinations are inverted and aerosol microphysical states vary. From the results of both synthetic and real optical measurement inversions, we are quite certain that the lack of δ measurements causes a trend of underestimating Vt and reff and overestimating mR, mI. Retrievals from synthetic optical data (Sect. 4) reveal that spectral PLDR plays a significant role for improving the retrieval accuracy of large particles (reff>1 µm) under both of the non-spherical models. Among the three wavelengths, δ1064 is most essential since it shows the highest sensitivities to reff and mR. Although it is independent of mI, one can think of that δ1064 brings an effective constraint on particle size to the (3β+2α) inversion and thus improves the retrieval accuracies of all state parameters. This result is in contrast to that presented by Tesche et al. (2019). They concluded that δ355 is the most indispensable input measurement in lidar inversion, while δ1064 does not improve the retrieval performance. The main reason could be the fact that they only used the spheroidal model and retrieved real parts of CRI greater than 1.5 (a detailed demonstration has been given in Sect. 4.1). Moreover, the quite different experimental setup and retrieval outputs between the studies could also explain it. First, Tesche et al. (2019) used the spheroidal model and the retrieval methods similar to that of Müller et al. (2013) and Veselovskii et al. (2010). Second, they mainly focused on the retrieval of “spheroid fraction”, a parameter to characterize the mixing state of polluted dust. Finally, they did not consider the spectral variability of dust mI. We have discussed the potential limitations of these configurations in detail (Sect. 2), which render us confident in our result. In addition, our result is in line with the findings of Gasteiger and Freudenthaler (2014) and Shin et al. (2018), who attributed the greatest informational value to δ1064 measurements.

We cannot directly compare the synthetic retrievals across the models as the two non-spherical models are used to invert the synthetic measurements generated by their own. Nevertheless, the retrieval simulations signify that compared to the spheroidal model, the improvement attributed to δmeasurements is more manifest if the IH model is used. In particular, significant improvement in mI retrieval for large particles (reff=2 µm) was observed in the IH model. We also noticed that the combination of δ1064 with either δ355 or δ532 or both δ355 and δ532 allows (1) an improvement in CRI retrievals from a statistical point of view and (2) an effective suppression of retrieval instability caused by measurement noise. Accordingly, we conclude that the IH model is more suitable for lidar-based dust retrieval and suggest using the IH model to invert at least (3β+2α+2δ) measurements with one of the δ at 1064 nm to assure retrieval quality.

Inverting the real dust observations allows for direct comparisons of the results retrieved with the two non-spherical models. The two case studies in Sect. 5 verify the finding in the simulation that significant retrieval difference between the IH and spheroidal models is not expected if reff is retrieved to be ∼ 0.5 µm. In addition, when inverting the (3β+2α+3δ) data, the IH model tends to derive higher mR and mI than the spheroidal model because of their difference in producing spectral PLDR. It is also interesting to look at the LR at 1064 nm predicted by the (3β+2α+3δ) retrievals. In Case 2, both IH and spheroidal models derive the increasing LR from 532 to 1064 nm, while the magnitude of LR₁₀₆₄ retrieved with the IH model is more consistent with the measurements of Saharan dust (Gebauer et al., 2024; Haarig et al., 2022). In Case 1, the IH model found a quite distinct LR₁₀₆₄ compared to the spheroidal model, leading to different spectral variations. Unfortunately, we are currently in short of LR₁₀₆₄ measurements of eastern Asian dust for validation.

The lack of independent data (ideally collocated in situ measurements) to compare with the results retrieved from the real lidar measurements in Sect. 5 precludes a closure validation of this study. Alternatively, in this section we present the results of a comparison with corresponding AERONET retrievals and comparison with previous studies discussed in Sect. 2.2 for a preliminary verification.

Two AERONET retrievals closest to the places and times of the lidar measurements and passing the data quality check (Holben et al., 2006; Sinyuk et al., 2020) are selected for the comparison. However, we are aware that such comparison itself has many limitations. Firstly, in contrast to our lidar, which works at the night-time to ensure the quality of Raman signals, AERONET makes columnar measurements and works during the daytime. Given the high tempo-spatial variability of aerosol particles, they could target aerosols with different microphysical and optical properties. For example, the layer-averaged α532 from lidar measurements is 161 Mm⁻¹ in Case 1 and 64 Mm⁻¹ in Case 2, respectively, while the columnar-averaged α440 from corresponding AERONET measurements is 186 Mm⁻¹ in Case 1 (0–3.2 km) and 56 Mm⁻¹ in Case 2 (0–2 plus 3–6 km). Secondly, retrievals from AERONET are also based on optical inversion and obtained with the spheroidal model. This could render the lidar retrievals using the spheroidal model more consistent with the AERONET retrievals since they utilize the same scattering model. Accordingly, the comparison results must be interpreted with care. Table 6 lists state parameters retrieved with BOREAL under different configurations and those provided by AERONET retrievals for the two case studies. Figure 24 visualizes the VSDs and SSAs from the AERONET retrievals. In Case 1, AERONET derives a monomodal VSD with reff obviously larger than any BOREAL retrievals. As mentioned, it is no wonder that the (3β+2α+3δ, spheroidal) configuration obtains Vt, reff, mR, and mI that are closest to the AERONET retrieval since both retrievals exploit the spheroidal model. However, the SSA retrieved with the IH model shows a better consistency with the AERONET counterpart because the enhancement of SSA due to a smaller reff is compensated for by the weakening due to a larger mI. Compared to Case 1, BOREAL and AERONET retrievals are less comparable in Case 2 because the vertical aerosol distribution was more heterogeneous (we have demonstrated that the aerosols in the BL were of different types than the dust layer). The AERONET retrieval in Case 2 shows a bimodal VSD with the fine-mode volume fraction comparable to the coarse-mode volume fraction and an mR value much lower than the BOREAL counterpart. Due to the influence of the BL aerosols, it is hard to evaluate how well the AERONET-derived coarse mode and CRI represent the microphysical properties of the dust layer, let alone the time difference between the lidar and sun photometer observations, which further increases the uncertainty of the comparison.