In the context of the current full retrieval experiment, the chemistry transport model (CTM), namely Modèle de Chimie Atmosphérique à Grande Echelle (MOCAGE), serves as a key element of the pseudo-reality, providing the spatio-temporal distribution of key atmospheric aerosol chemical species. MOCAGE is a three-dimensional global and regional CTM initially developed by Météo-France (El Amraoui et al., 2020; Guth et al., 2016). It is used for data assimilation (El Amraoui et al., 2010), tropospheric-stratospheric composition (El Amraoui et al., 2020), air quality forecasting, and climate change research, among other areas. It can be run at several configurations with various geometric domains, resolutions, and for physical and chemical parameterization. Emissions, meteorological fields, and atmospheric composition are the primary inputs of the MOCAGE model.

MOCAGE's chemical composition is represented at 47 vertical levels, extending from the surface to 5 hPa, with vertical resolutions ranging from 40 to 400 m in the boundary layer and from 400 to 800 m in the free troposphere. The model's horizontal resolution depends on the scale: local (0.5°×0.5°), regional (0.1°×0.1°), and global (1 or 2°). Although MOCAGE cannot calculate meteorological variables, it can directly extract them from other models, such as the IFS model, ARPEGE, ERA-INTERIM, and ARPEGE-Climat. Modelled aerosol species for this study include black carbon (BC), organic carbon (OC), desert dust (Du), sea salt (SS), sulfate (Su), nitrate (Ni), and ammonium (Am). As described in El Amraoui et al. (2020), these species are represented using the schemes of Martet et al. (2009) and Sič et al. (2015), with the secondary inorganic aerosols (Su, Ni, Am) following the parameterizations of Guth et al. (2016). The atmospheric composition is further defined by the assimilation of MODIS Aerosol Optical Depth (AOD) data to constrain the aerosol fields within the 3D-FGAT variational assimilation framework (El Amraoui et al., 2022). Biomass-burning sources of BC and OC aerosols used in this study are the same as those used in Sič et al. (2016), and are based on a daily frequency from the Global Fire Assimilation System (GFAS) (Kaiser et al., 2012). More details regarding this specific model configuration and the treatment of aerosol sources can be found in El Amraoui et al. (2022).

By simulating the spatio-temporal distribution of the most abundant aerosol species and their interactions with atmospheric processes, MOCAGE serves as a realistic proxy for real-world observations. This synthetic environment enables testing and validation of retrieval techniques while providing a reliable standard (i.e., pseudo-reality) against which to compare them. In particular, MOCAGE outputs provide aerosol mass concentration profiles (in µgm-3) of six species: BC, OC, Am, Su, Du, and SS. MOCAGE data is extracted along the trajectories of the study's targeted geographical transects. As a first illustration, we use a transect observed by the CALIOP (Cloud-Aerosol LIDAR with Orthogonal Polarization) instrument on 15 May 2020 (see Fig. 3), measuring the aerosol vertical distribution over the Indian Ocean (Indian-O), Bangladesh, Tibetan Plateau (TP), Taklamakan Desert (TD), Kazakhstan, Russia, Arctic Ocean (Arctic-O), and then reaching Greenland. During this day, significant aerosol diversity is remarked along the trajectory, ensuring a robust case study for comparison and an adequate evaluation of the retrieval method's performance.

In this work, the pseudo-reality considers eight distinct aerosol chemical species: black carbon (BC), brown carbon (BrC), inorganic salt (IS), fine-mode water content (WC_F), quartz (Qz), iron oxide (IO), sea salt (SS), and coarse-mode water content (WC_C). These species correspond to those that can be optically distinguished with advanced satellite measurements (Dubovik et al., 2021) and they are derived from MOCAGE primary aerosols (BC, OC, SS, Du) and secondary inorganic aerosols (Ni, Su, and Am). The correspondence between MOCAGE outputs and the AEROCHEMPro species is summarized in Table 1 and described in the next paragraphs.

To obtain the aerosol properties considered in the pseudo-reality, the first step is to convert the dry mass concentrations (Cmass) of the species, calculated as outputs of the MOCAGE model, into dry volume mixing ratios (Cvol) using the known densities of each species (ρi). The general relationship used is: 1Cvol=Cmassρi where Cmass is the mass concentration in µgm-3 and ρi is the density of the species, expressed here in µgm-3. The species and their corresponding densities are listed in Table 1. Each species' mass concentration is divided by its corresponding density to yield unitless aerosol volume mixing ratios. This step is essential for comparing the contribution of different species to the total aerosol volume.

After calculating the volume mixing ratio of each species from the MOCAGE model at each altitude, some modifications of the aerosol chemical species are considered. They allow alignment with those to which polarimeter measurements are sensitive and that are intended to be retrieved using the AEROCHEMPro/GRASP algorithm. We consider concentrations of inorganic salts (IS), which are assumed to be the sum of the concentrations of those simulated by MOCAGE for ammonium and sulfate (as polarimeter measurements do not allow us to distinguish these two species). Black carbon (BC) and sea salt (SS) remain unchanged, with their respective volume mixing ratios carried over directly. A second carbonaceous species within the pseudo-reality is brown carbon (BrC), whose abundance is assumed to equal that of organic carbon for simplicity. For mineral dust, specific fractions are defined based on its composition: Iron oxide (IO) and quartz (Qz) are represented as 2 % and 98 % of the total dust concentration simulated by MOCAGE, respectively; these fractions follow the typical values adopted from Li et al. (2019). This redefinition enables consistent comparisons between aerosol species in the pseudo-reality and those retrieved from AEROCHEMPro/GRASP, thereby facilitating the analysis of aerosol properties and their radiative effects.

The choice of these aerosol chemical species is a refinement of previous works that exploits measurements of either the AERONET sun photometer or the POLDER multi-angular polarimeter (Dubovik et al., 2021; Li et al., 2019, 2020a, b). In these previous studies, the method is called GRASP/Component approach considering BC, BrC, fine- and coarse-mode non-absorbing soluble and insoluble particles, coarse-mode absorbing particles, and aerosol water content. In the current work, we distinguish between the water content for fine (BC, BrC, and IS) and coarse (SS) hydrophilic species. Non-absorbing soluble fine and coarse particles are identified as inorganic salts (IS) and sea salt (SS), while dust particles are composed of IO and Qz.

To estimate the mixing ratio of water content absorbed by aerosol particles, we first calculate the humidified volume of hygroscopic species based on ambient relative humidity (RH). Since aerosol particles take up water with increasing RH, we correct the dry volume mixing ratios accordingly, particularly for species such as ammonium sulfate and sea salt. The growth factor, GF, depends on the ambient RH and is applied to the fine and sea-salt modes. Note that this correction modifies the particle volume ratios to account for water uptake, but the particle radii themselves are not explicitly adjusted.

The hygroscopic growth factor GF(RH) is estimated using the Hänel (1976) parameterization, which quantifies the increase in particle size/volume with increasing relative humidity (RH). The growth factor GF_min, with respect to the particle size at RH_min, is defined as: 2GFmin(RH)=1-(RH/100)1-(RHmin/100)-γ Since direct comparisons require a standard reference RH, the growth factor is further scaled to a reference RH of RHref=40 % (without hygroscopic growth), following recommendations from previous studies: 3GFref(RH)=GFmin(RH)×1-(RHmin/100)1-(RHref/100)-γ where RH_min corresponds to the minimum RH in each pixel, and γ is the hygroscopic growth parameter value, as listed in Table 1. This method for calculating the growth factor follows Sicard et al. (2022), who applied it to a multi-wavelength lidar system to study aerosol hygroscopicity. It ensures consistency across different humidity conditions and captures water uptake of hygroscopic aerosol species, such as inorganic and sea salts.

For each hygroscopic species, the aerosol water content is determined as the difference between the ambient humidified volume and the dry volume: 4Cvol,water,i=Cvol,i(RH)-Cvol,i where Cvol (RH), the volume mixing ratio of species i at ambient humidity, is given by: 5Cvol(RH)=Cvol⋅GFref(RH) and Cvol,i is its original dry volume mixing ratio. The total water content in each mode is obtained by summing the contributions from all hygroscopic species within that mode.

The volume size distribution for each aerosol mode is derived by distributing the mode's total volume mixing ratios across multiple size bins. In this approach, the aerosol volume in each size bin is obtained by multiplying a predefined weighting factor (cm) – representing its relative contribution – by the modeled total volume for that mode. This ensures that the shape of the assumed log-normal distribution is preserved, while its magnitude adapts to the local aerosol load at each pixel. A total of seven size bins are considered: three for the fine mode, two for dust, and two for sea-salt aerosol modes. The corresponding mean radii and the geometric standard deviations are shown in Table 2. The weighting factors (cm) are chosen to mimic the average AERONET-retrieved size distributions reported by Dubovik et al. (2002). This configuration produces a total size distribution that resembles the average size distribution of the fine mode for urban and biomass-burning aerosols reported by Dubovik et al. (2002), peaking at a modal radius of approximately 0.17 µm and a geometric width of the log normal distribution of about 0.45. Likewise, the coarse mode closely resembles the characteristics of dust and oceanic aerosols described by Dubovik et al. (2002), peaking around 2.71 µm with a width of roughly 0.6.

In this study, aerosol sphericity is assumed for each species based on its physical properties, as reported in previous studies. We quantify it as a spherical fraction, where 1 represents pure spheres, and 0 represents pure spheroids (the assumed shape for non-spherical particles). Fine particles corresponding to inorganic salt, brown carbon, and black carbon are considered as optically spherical, thus their spherical fraction is taken as 0.9999, consistent with the approach of Dubovik et al. (2006). Similarly, the spherical fraction for sea salt is considered to be 0.9999, as they are known to be highly spherical due to their hygroscopic nature and formation processes (Dubovik et al., 2006). In contrast, dust species (quartz and iron oxide) account for the irregular shapes of mineral dust particles, assigning a near-zero spherical fraction of 0.01. These particles originate from natural mechanical processes, such as erosion and transport, that result in non-spherical shapes (Kandler et al., 2009).

To derive aerosol properties from passive measurements, it is necessary to account for surface reflectance properly. In our study and the GRASP environment, land surface reflectance is represented using the Bidirectional Reflectance Distribution Function (BRDF), which follows the kernel-driven Ross-Li model (Li and Strahler, 1992; Roujean et al., 1992; Wanner et al., 1995) in combination with a semi-empirical formulation of the BPDF (Maignan et al., 2009). Over ocean surfaces, the reflective properties are modeled similarly to earlier approaches used in the POLDER algorithm (Deuzé et al., 2001; Herman et al., 2005; Tanré et al., 2011). The Cox and Munk model (Cox and Munk, 1954) accounts for Fresnel reflection from the roughened sea surface.

While GRASP allows joint retrieval of surface and aerosol properties (Dubovik et al., 2021), we adopted a constrained single-pixel retrieval strategy for this baseline evaluation. Surface parameters were jointly adjusted in the state vector. On the other hand, they were initialized with spatially uniform values across the transects and subject to strong a priori constraints (Lagrange multipliers), thus significantly limiting their spatial variability during retrievals. Specifically, a high weight was assigned to the spatial variability constraint of the polarized surface component and the BRDF kernels to maintain stability (See Table 6 for details). This configuration effectively minimizes pixel-to-pixel surface variability, allowing the inversion to focus on the vertical aerosol speciation without significant interference from surface-aerosol signal mixing. We acknowledge that this approach represents an idealized scenario; while it establishes a theoretical performance baseline for AEROCHEMPro, future work will involve more flexible surface-atmosphere retrievals to explicitly capture the complexities of heterogeneous terrain.

LUCE lidar measurements are not expected to provide sufficient information to distinguish all the aerosol chemical species considered in AEROCHEMPro vertically. They are typically used to distinguish between aerosol types with clearly different lidar-derived optical properties (e.g., Qayyum et al., 2026). Therefore, we assume a simplified representation of the vertical distribution of the aerosol chemical species, categorizing them into three modes based on their size distribution and sphericity. A so-called “fine mode” is the mixture of black carbon, brown carbon, inorganic salt, and the water uptake associated with these species. A “dust mode” includes mineral non-spherical coarse compounds of desert dust, corresponding here to iron oxide and quartz. A third one, called “sea-salt” mode, is composed of spherical coarse sea salt and the water it absorbs. Each of these three modes is represented by an independent vertical profile of aerosol total volume mixing ratio, a size distribution, and specific mixing rules for determining the refractive index (see Sects. 2.2.2–2.2.4). This distinction between several aerosol modes' vertical profiles and their chemical composition is enabled by the multispectral synergism of radiance and depolarization measurements, which allow for a more detailed separation of aerosol species across fine, dust, and sea-salt modes.

We describe the abundance of each aerosol species as a volume fraction of each of the three aerosol modes. These fractions are computed by normalizing the vertically integrated volume mixing ratios of each species provided by the pseudo-reality by the total volume of the mode to which it belongs. A trapezoidal integration is performed over the vertical layers of the atmosphere, and the total column volume fraction (Frac_i) for a species is given by: 6Fraci=∫0ZmaxCvol,i(z)dz∫0ZmaxCvol,mode(z)dz where Cvol,i and Cvol,mode are the volume mixing ratios of the aerosol species i and the total volume mixing ratios of mode it integrates.

Moreover, calculations consider and normalize vertical profiles of each mode (Profmode(z)) independently. By normalizing the vertical distribution of a given mode (Cvol,mode(z)) with respect to its column-integrated total concentration (Cvol,mode,tot), the profile distribution is expressed as: 7Profmode(z)=Cvol,mode(z)∫0ZmaxCvol,mode(z)dz=Cvol,mode(z)Cvol,mode,tot where zmax is the height of the atmospheric top, here considered as ∼20 km, and a vertical step of 150 m.

For the size distribution, we use a log-normal representation pre-calculated in GRASP, where the size distribution is expressed by a set of size bins with associated volume fractions. Although earlier texts refer to “five lognormal modes”, our GRASP configuration actually implements three retrieved modes (fine, coarse spherical, and coarse non-spherical) whose size is represented by a total of seven size bins (independent parameters): three size bins representing the fine mode, two bins for dust (coarse non-spherical) mode, and two bins for the sea-salt (coarse spherical) mode. Thus, the theoretical number of independent parameters for the size distribution in this setup is 7 (3 fine + 2 sea-salt + 2 dust), which are jointly derived with the other aerosol properties in the inversion procedure. Each of the seven size bins corresponds to a log-normal size distribution with specific modal radii and widths presented in Table 2.

A volume-weighted (VW) mixing rule is used in this study as an empirical method for estimating the complex refractive index of an “internally mixed” aerosol mixture. This rule calculates an effective refractive index as a volume-weighted average of the real and imaginary parts of the index, based on the proportions of the individual compounds in the mixture. Within the GRASP/Component framework, the VW mixing rule has been integrated by Li et al. (2019). It is expressed as: 8nmode(λ)=∑i=1jFraci⋅ni(λ)∑i=1jFraci9kmode(λ)=∑i=1jFraci⋅ki(λ)∑i=1jFraci where nmode and kmode are the real and imaginary parts of the refractive index of the fine, dust, and sea-salt modes, and ni and ki those of the chemical species composing these modes.

The complex refractive index values for all aerosol species in the pseudo-reality, spanning wavelengths from 355 to 1650 nm, are listed in Table 3. These values are taken from Longtin et al. (1988), Bond and Bergstrom (2006), Querry (1987), Shettle and Fenn (1979), and Sun et al. (2007). In the current work, we consider these refractive indices for each species both in the pseudo-reality and the retrievals. Therefore, the performance analyses shown here are rather optimistic regarding the ability to distinguish different aerosol species. Other works considering aerosol typing from lidar alone (Qayyum et al., 2026) suggest a moderate effect on retrievals due to differences in intensive aerosol properties (size distribution and refractive indices) between the pseudo-reality and the retrievals. Future work will further analyze these aspects.

The active remote sensing sensor used in this study is the LUCE multiwavelength lidar (Di Girolamo et al., 2023, 2024), formerly called CALIGOLA (which stands for Cloud Aerosol Lidar for Global Scale Observations of the Ocean-Land-Atmosphere System). LUCE is a spaceborne multiwavelength lidar with potentially Raman capabilities developed by the Italian Space Agency (ASI), in collaboration with the National Aeronautics and Space Administration (NASA), designed to study interactions within the ocean–land–atmosphere system (Behrenfeld et al., 2023; Franco et al., 2023; Di Girolamo et al., 2023, 2024). The mission's scientific objectives include (1) investigating atmospheric and oceanic processes, (2) measuring key geophysical variables from space, and (3) satisfying stringent requirements on accuracy, precision, vertical and horizontal resolution, and sounding range. As no real measurements from LUCE are yet available, simulated lidar data are generated using the GRASP forward model configured to replicate LUCE's instrument characteristics. LUCE's laser emission operates at three wavelengths (354.8, 532.2, and 1064.4 nm) to measure the attenuated backscatter and depolarization ratios at these wavelengths (Di Girolamo et al., 2023, 2024), providing both daytime and nighttime measurements (see Table 4). An additional Raman channel in the UV is proposed for nighttime aerosol retrievals. However, we only consider the elastic channels of attenuated backscatter and the depolarization ratio of LUCE in the current work, as synergism with a polarimeter is only possible during daytime (when the Raman channel is not likely to provide useful information). LUCE combines high-energy multi-wavelength lasers – 157 mJ at 354.8 nm, 47 mJ at 532.2 nm, and 144 mJ at 1064.4 nm – with a large 1 m telescope, 51 Hz pulse rate, and 8–22 m surface footprint (Di Girolamo et al., 2023, 2024). This configuration delivers higher photon collection efficiency and significantly larger signal-to-noise ratio (SNR) compared to CALIOP (532 nm at 110 mJ, 1064 nm at 230 mJ, same aperture, 20.25 Hz PRF, ∼333 m shot spacing; Winker et al., 2009) and ATLID (355 nm at ∼35 mJ, effective ∼70 mJ, 0.62 m telescope, ≈0.302 m2 area, 51 Hz nominal, 25.5 Hz effective; Illingworth et al., 2015; Feofilov et al., 2023). LUCE's higher repetition rate than CALIOP improves along-track and vertical sampling, while its three wavelengths offer richer spectral discrimination than CALIOP's two and ATLID's single wavelength. The large aperture and high UV energy enhance aerosol detection in optically thin and high-altitude layers, especially under daytime background, whereas ATLID trades energy and aperture for high-spectral-resolution lidar (HSRL) capability, and CALIOP offers strong NIR returns but coarser sampling (Winker et al., 2007; Illingworth et al., 2015).

The polarimeter used in this study is an advanced UV-VIS-NIR-SWIR instrument designed to provide multi-angular polarized radiance measurements at multiple wavelengths. With a spatial resolution of 0.5 km and a 300 km swath, it enables large-scale, high-precision monitoring of atmospheric and surface features. The polarimeter captures radiance across eight spectral bands, ranging from 380 to 1570 nm, at 10 different viewing angles and two polarization states, significantly improving the retrieval of aerosol properties, including particle size, shape, and composition (adapted from a preliminary assessment by the AOS; AOS Study Team, 2021). Compared to earlier instruments like POLDER onboard PARASOL, which measures polarized radiance at 3 to 7 spectral bands between approximately 443 and 910 nm and at 14 viewing angles, the AOS polarimeter offers several essential improvements. Its spectral coverage extends from UV to shortwave infrared (SWIR), improving sensitivity to aerosol composition and particle size, especially for larger particles and complex mixtures. The increased number of spectral channels enhances spectral resolution. It improves the discrimination of aerosol and cloud properties. While the AOS polarimeter was designed to typically utilize 10 viewing angles – slightly fewer than POLDER – it features a specialized 670 nm channel with 60 viewing angles. This high-density angular sampling, combined with the extended spectral range into the SWIR, is expected to provide a particularly detailed characterization of the scattering phase function and aerosol microphysical retrievals. Additionally, enhanced SNR boost retrieval accuracy, particularly under challenging atmospheric conditions. Together, these advancements enable more detailed and accurate characterization of aerosol optical and microphysical properties, as well as enhanced cloud characterization compared to POLDER.

A realistic noise simulation is essential for evaluating the performance of retrieval algorithms under conditions that mimic actual observations. For the LUCE lidar system, noise is modeled by a theoretical estimation of the signal-to-noise ratio (SNR) in comparison with CALIOP, which is adjusted for environmental and instrumental factors. For the polarimeter, noise is introduced to the Stokes parameters I, Q, and U to simulate uncertainties in polarimetric measurements. This approach simulates noise variances encountered in advanced polarimeters, ensuring a satisfactory representation of instrument performance.

To simulate the noise characteristics of the LUCE lidar system, we implement a methodology based on theoretical relationships derived from prior studies and calibration datasets. This approach accounts for variations in SNR across different environmental conditions and wavelengths.

An approximate relationship between the SNR of LUCE and that of CALIOP (Cloud-Aerosol Lidar with Orthogonal Polarization) is established following (Trepte et al., 2024), based on theoretical estimations accounting for background sunlight and sensor detection noises. Due to larger laser energy, a larger collection area, and a lower orbit, the SNR of LUCE is estimated to be 7.5 to 16.9 times larger than that of CALIOP at 532 nm, specifically for the LUCE channels at 355 and 532 nm. Additionally, we use an empirical relationship between CALIOP SNR for different values of attenuated backscatter coefficients taken from Mao et al. (2022). Using these two relationships, we derive LUCE's SNRs as a linear function of backscatter coefficients. We assume the same relationship for 532 nm as for 1064 nm; the specific SNR values change with the backscatter coefficient at each wavelength.

In this work, we use LUCE lidar measurements with an average horizontal resolution of ∼3.4 km and a vertical resolution of ∼200 m. To reduce the impact of random noise and ensure reliable performance of AEROCHEMPro, the lidar signals are averaged over a larger horizontal window of 50 km and a vertical window of 200 m. The corresponding noise standard deviations are calculated from the original signal-to-noise characteristics and reduced by dividing by the square root of the number of averaged points. Finally, random noise with these reduced standard deviations and zero mean is added to the noise-free signals computed by the radiative transfer model.

The noise introduced in the polarimeter measurements is modeled to reflect the performance of an advanced multi-angle polarimeter design. For radiometric measurements, uncertainties are considered to be below 3 %. In contrast, uncertainties in the Degree of Linear Polarization (DoLP) are estimated to be less than 0.005, based on preliminary assessments by the AOS team (AOS Study Team, 2021). To evaluate the inversion performance against these mission requirements within our self-consistent simulation framework, the synthetic error model applies a 3 % relative noise floor to the total intensity (I) and an absolute noise floor of 0.005 directly to the linear polarized Stokes parameters (Q and U). This absolute configuration directly governs the target uncertainty of the mathematically derived DoLP. For bright viewing geometries where the normalized intensity baseline scales above unity (I>1), the absolute contribution of the standalone polarimetric components is suppressed, yet the combined total error on DoLP remains robust due to the simultaneous propagation of the intensity noise. Conversely, under low-signal conditions where I<1, the framework allows the simulated noise to dynamically exceed the 0.005 specification. This configuration provides a conservative signal-to-noise magnitude of the retrieval logic, guaranteeing that the framework evaluates and proves the stability of AEROCHEMPro under realistic instrument performance constraints.

The numerical inversion is implemented as a statistically optimized fit of observations using the multi-term least-squares method (LSM), which combines the advantages of various approaches (Dubovik et al., 2011, 2021). This strategy searches for a solution in a continuous space of solutions under statistically formulated criteria, optimizing the distribution of errors in the retrieved parameters, and it does not rely on pre-assumed potential solutions.

To achieve stable inversion, the numerical inversion module for single pixel fitting follows: 10fn∗=fn(a)+Δfn0i∗=Snai+Δ(Δai) where a is called the state vector, fn∗ is the LUCE and AOS polarimeter observation vector, which includes attenuated backscatter and depolarization values at 3 and 2 wavelengths, respectively, in addition to polarimeter radiance and polarization states at eight different wavelengths, fn(a) is representing the simulated observations by the forward model, and Δfn depicts the uncertainty of the observations.

In the case of AEROCHEMPro/GRASP single pixel retrieval, the retrieved state vector a is composed as follows: 11aT=(aV,afrac,aprof,asph,abrdf,abpdf,acox)T where aV, afrac, aprof, asph have the following structure, 11aa…=(a…F,a…D,a…SS) and abrdf, acox, 11babrdf=(aiso,avol,ageo),acox=(aiso,afr,as) where F, D, and SS denote the vectors corresponding to the fine, dust, and sea salt modes.

Where the elements of the unknown vector a correspond to:

aV(i) denotes the aerosol size distribution (expressed as the normalized abundance of 7 size bins associated with log normal functions dV(r)/dln⁡r with fixed modal radii and widths, see Sect. 2.2.2) in 3 modes (fine, dust, sea salt);

The second part of Eq. (10) describes the single-pixel a priori smoothness constraints applied to the inversion, where the 0i∗ denotes the zero vector, Δ(Δai) stands for the vector of the uncertainties characterizing the deviations finite differences from the zeros, and S is the matrix that includes the coefficients for calculating differences of m-th order (numerical equivalent of the derivatives of the same order). Smoothness matrix for a state vector described in Eq. (12) will have the following array structure (Dubovik et al., 2011, 2021): 12Sa=SV0000000000000000000000000000000Sbrdf0000000Sbpdf0000000ScoxaVafracaprofasphabrdfabpdfacox where 12aSv=SvF000SvD000SvSS12bSbrdf=Sbrdfiso0000000012cSbpdf=Sbpdfiso0000000012dScox=Scoxiso00000000 In Eq. (12), several blocks of the smoothness matrix Sa correspond to matrices that apply smoothness constraints to specific groups of variables.

afrac, aprof, asph rows remain zero, as vertical profiles, species volume fractions, and sphericity are unconstrained, resulting in zeros in the corresponding rows of Sa;

All details regarding the construction, parameterization, and smoothness orders of these blocks are summarized in Table 6 and described in detail in Dubovik et al. (2011, 2021). The errors Δ(Δa) are independent for each component of the vector (Δa)∗, and the smoothness matrix is expressed: 13γΔΩ=γΔΩ10000000000000000000000000000000γΔΩ60000000γΔΩ70000000γΔΩ8aVafracaprofasphabrdf,isoabpdf,isoacox,iso where Ωi=SiTWi-1Si uses the derivative matrices Si (i=1,…,4), aV, abrdf,iso, abpdf,iso, acox,iso.

By including the noise (explained in Sect. 2.2.6) for LUCE lidar and AOS polarimeter, the optimum solution considering the minimal error variance is obtained by minimizing the following cost function in Eq. (14): 14ψi(ai)=(fiobs-fisim(ai))TCi-1(fiobs-fisim(ai))+∑k=19γk(ah,i)TSh,kTWh,k-1Sh,kah,k=(Δfi)TWi-1(Δfi)+∑k=19γk(ah,k)TΩh,kah,k where fiobs contains the observed lidar and polarimeter signals of ith pixel. γk are Lagrange multipliers controlling smoothness, and Wk and Ωk is the weighting function and the smoothness matrices, respectively (Dubovik et al., 2011, 2021).

The minimization of Eq. (14) is performed using the Levenberg–Marquardt algorithm with a logarithmic convention to ensure the physical positivity of retrieved quantities. To ensure a robust global minimum is reached and to completely avoid dependence on the initial state, the state vector is initialized globally using a generic, spatially uniform “cold-start” configuration. This includes fixed initial chemical species volume fractions and a flat vertical profile distribution across all layers, as detailed in Table 6, ensuring the initialization is sufficiently generic and does not favor specific solutions. We apply a stringent convergence threshold of 1.0×10-9 for the relative change in the cost function, with a maximum of 15 iterations per pixel. Furthermore, the Jacobian matrix is computed with a finite difference scale of 5.5×10-5 to precisely capture the sensitivities between the aerosol species concentrations and the multifaceted observations.

The AEROCHEMPro/GRASP simultaneously retrieves multiple groups of variables – including aerosol size distribution, chemical composition, vertical profiles, particle sphericity, and surface reflectance – using physically consistent first guesses, bounds, and smoothness constraints. The main retrieval settings, including initialization values, bounds, and smoothness orders, are summarized in Table 6.

Figures 6 and 7 present a comparison between Level-1 synthetic measurements obtained from the pseudo-reality and the lidar profiles, along with polarimeter radiances fitted by AEROCHEMPro for the selected case study. They are both derived from the forward model (Dubovik et al., 2021); the first case corresponds to the aerosol concentration profiles from the pseudo-reality (Sect. 2.1), and the second to those retrieved by the AEROCHEMPro inversion scheme. The results demonstrate that the AEROCHEMPro method accurately reproduces both lidar and polarimeter measurements simultaneously by adjusting the concentration profiles of 3 aerosol modes.

Figure 6 compares measured and fitted profiles of attenuated backscatter at 355, 532, and 1064 nm, as well as volume depolarization at 355 and 532 nm, along a transect that covers several regions from the Indian Ocean to Greenland. Averaging over 50 km horizontally and 200 m vertically introduces random noise in the synthetic signals, reaching up to 8 % for attenuated backscatter and ranging between 10 % and 20 % for depolarization. The AEROCHEMPro method captures the spatial variability and vertical structures seen in the observed profiles. Key properties, such as elevated aerosol layers and high depolarization ratios, are accurately depicted, indicating the presence of non-spherical particles such as dust. The method can reproduce lidar signals under various atmospheric conditions, as demonstrated by a good match between the fitted and measured attenuated backscatter profiles at all wavelengths. Any minor differences are likely due to instrumental noise or assumptions in the retrieval process, but they are acceptable for practical use.

Figure 7 illustrates the agreement between measured and fitted polarimeter observations for all pixels and spectral channels across the selected transect, providing a direct verification of how the residuals relate to the injected noise levels. For the intensity component (I), the mean biases are negligible, with normalized absolute values ranging from 3.03×10-5 to 3.56×10-4, and standard deviations ranging between 1.28×10-3 and 2.97×10-3. These standard deviations align with the relative 3 % noise floor applied to the typical fractional normalized radiance levels of the scene. The normalized polarization residuals (Q/I) show minimal mean biases ranging from 3.62×10-5 to 1.22×10-3, with standard deviations remaining stable across wavelengths, hovering around 0.003 (from 2.71×10-3 to 3.30×10-3). This specification falls within the absolute noise floor of 0.005 assigned to the underlying Stokes parameters for the AOS polarimeter. As the simulations are performed in the principal plane, the U component is excluded from the analysis, as it does not carry independent information in this geometry. The consistency of the method is confirmed by the absence of systematic biases across the eight spectral bands.

This sub-section presents a point-by-point comparison between AEROCHEMPro retrievals of vertical aerosol volume mixing ratio distributions by mode and total column volume concentrations of each chemical species, as directly derived within the state vector adjusted through iterations, relative to the pseudo-reality as depicted in Fig. 8.

The profiles are organized into three aerosol modes: fine, dust, and sea salt. The fine-mode vertical profiles demonstrate the predominance of inorganic salt, followed by brown carbon, particularly over Bangladesh and eastern Kazakhstan. This distribution is attributed to anthropogenic emissions and transported pollution. The retrieval accurately represents the overall structure of the fine aerosol volume mixing ratio profile, including the vertical extent of the boundary layer. However, it slightly overestimates the volume mixing ratios at the surface, especially over Bangladesh and Greenland near the land surface, with a bias of roughly 2×10-5, likely resulting from vertical smoothing associated with lidar–polarimeter synergy. The total column volume concentrations of black carbon, brown carbon, and inorganic salt correspond closely with the pseudo-reality, indicating reliable retrieval of fine-mode species contributions. At the peak of the pseudo-reality concentration along the transect, inorganic salt and water content are retrieved with relative errors of 21.2 % and 25.6 %, respectively, above Bangladesh. The absolute overestimation of inorganic salt (+0.018 um3um-2) is largely balanced by the underestimation of water content (-0.018 um3um-2), indicating that while a minor partitioning offset sometimes exists between these two hygroscopic species, the total fine-mode volume is recovered with high accuracy. In the dust mode, the pseudo-reality shows significant dust activity across Bangladesh, the TP, and the TD, with enhanced iron oxide and quartz mixing ratios reaching up to around 12 km of altitude and extending into Kazakhstan and Russia at higher levels. The lofted dust layers span vertically from approximately 4 to 10 km, suggesting long-range transport. The retrieval captures the principal features of the dust plume, particularly its vertical extent and surface-level volume mixing ratio. The remarkable consistency between retrieved and pseudo-reality total column species volume concentrations for iron oxide and quartz validates the retrieval's ability to observe dust particles. Sea salt is primarily found at the surface, reflecting emissions from marine environments. Both pseudo-reality and retrieval show consistent patterns, with high sea salt volume mixing ratios limited to the boundary layer. A slight underestimation of sea salt mixing ratio off the coast of Bangladesh is observed, corresponding to background sea salt transported inland above this region, revealing very small biases under complex aerosol mixing conditions, such as with fine particles. Generally, the retrieval accurately resolves the horizontal distribution and total column concentrations of sea salt in most locations, indicating that it is appropriate for characterizing marine aerosol contributions.

Figure 9 compares the performance of AEROCHEMPro in retrieving essential aerosol properties, including aerosol optical depth (AOD), surface mixing ratio, total-column single scattering albedo (SSA), and total-column lidar ratio (LR) at 532 nm, differentiated by aerosol mode. For AOD and Csurf (Fig. 9a–d), the contributions of the fine, dust, and sea salt modes are presented as stacked area plots. In regions where a single mode dominates (such as the high AOD dust events), the areas for the remaining modes may appear negligible or invisible near the baseline, rather than being hidden by overlap. The figure shows a direct comparison between the retrieval results with the pseudo-reality dataset.

AEROCHEMPro-derived modal AODs exhibit a clear agreement in terms of patterns and absolute values with respect to pseudo-reality for all modes (Fig. 9a and b). The retrieval effectively captures fine-mode AOD peak amplitudes and occurrences, notably over Bangladesh, where anthropogenic and secondary aerosols are common. The method also reproduces high AOD values associated with dust plumes transported over Bangladesh. Sea salt retrievals correspond closely to the pseudo-reality over the Indian Ocean. AEROCHEMPro reproduces the expected spatial distribution of AOD at 532 nm across major aerosol species. Combustion-related species dominate over anthropogenic source regions, while mineral dust components are enhanced in arid regions such as Central Asia, and marine aerosols are prevalent over oceanic areas. Detailed species-resolved AOD distributions are provided in Fig. S1 in the Supplement.

Comparisons of surface mixing ratios (Csurf) in Fig. 9c and d further demonstrate the good retrieval's ability to quantitatively capture the spatial distribution of multiple species of aerosols. The method accurately retrieves enhanced surface volume mixing ratios for fine and dust modes near source regions. Still, it slightly overestimates them in Bangladesh, whereas the sea salt Csurf retrieval is clearly consistent with the pseudo-reality.

Figure 9e and f show the total-column single scattering albedo (SSA) retrievals that closely match the pseudo-reality dataset, reflecting variations in aerosol absorption properties over the transect. The total-column lidar ratio (LR) comparisons highlight the retrieval's ability to discern aerosol scattering and absorption features. Here, “total-column” SSA and LR refer to bulk optical properties derived from column-integrated scattering, extinction, and backscatter coefficients at 532 nm. Retrievals are consistent with pseudo-reality values in most locations. Still, a moderate underestimation of about 20 to 30 sr is detected stretching from the Tibetan Plateau to the Taklamakan Desert, which could imply some difficulties in distinguishing the scattering properties of coarse, elevated dust particles.

Vertical profiles of aerosol effective radius (reff) (Fig. 9g and h) display substantial spatial and vertical variability along the transect. Over the Indian Ocean, the effective radius increases near the sea surface, indicating that sea salt particles dominate at lower altitudes. The Tibetan Plateau shows a transition to larger coarse-mode particles, most likely transported dust. The profile of reff over Kazakhstan indicates a complex structure, with low-altitude layers denoting contributions from dust and mixed aerosol particles. This shows that the retrieval method generally aligns very well with the pseudo-reality dataset. However, there are minor overestimations of sea salt particle size over the Indian Ocean, at higher altitudes above the marine boundary layer, and a slight underestimation of reff at the surface above Kazakhstan.

Figure 10 shows a comparison of transects of vertical profiles of volume mixing ratio of individual aerosol chemical species using pseudo-reality data against those derived from AEROCHEMPro retrievals. This figure differs from previous figures (Figs. 8 and 9) by displaying direct comparisons of the state-vector variables. Figure 10 presents an AEROCHEMPro product derived from those variables. The aerosol profiles are independent for each aerosol species in the pseudo-reality, and they are estimated using simulations from MOCAGE-CTM. In contrast, the aerosol chemical species profiles, derived from AEROCHEMPro retrievals and assuming vertically constant mixing ratios per mode, are calculated by multiplying the fractions (Frac_i) of each chemical species by the corresponding modal concentration vertical profiles (Fig. 8b, f, and j). This derived product is indicated in the scheme of Fig. 4.

The results illustrate the unprecedented capacity of AEROCHEMPro to retrieve vertical profiles of the volume mixing ratio of multiple aerosol chemical species in good overall agreement with the pseudo-reality. Except in situations in which chemical species of the same aerosol mode display clearly different relative vertical distributions (not very often in this example), this derived product of AEROCHEMPro shows a good match for eight distinct transects describing the aerosol chemical composition.

Pseudo-reality BC and BrC profiles show high surface mixing ratios across Bangladesh and Kazakhstan, indicating emissions from biomass burning and industrial activities. The retrieval reproduces these near-surface plumes over most locations with high accuracy. Inorganic salt retrievals closely match pseudo-reality, particularly in Bangladesh and parts of Kazakhstan. However, overestimations occur between the surface and 2 km, along with a simultaneous underestimation of water content linked to fine particles, near coastal locations. Over specific and localized regions, the use of a single profile to represent the fine mode limits the ability to distinguish vertical variations in black carbon and brown carbon, as well as in inorganic salt and aerosol water content. Iron oxide and quartz, both dust mineral constituents, generate unique plumes in the pseudo-reality, especially over the Tibetan Plateau, Taklamakan Desert, and parts of Russia. The retrievals reproduce the shape and extent of these dust layers. Sea salt and its associated water content are very well represented in the retrieval, particularly near the surface over the Indian and Arctic Oceans. Generally, AEROCHEMPro retrieval of aerosol chemical species profiles accurately reproduces magnitudes and patterns, with minor limitations in differentiating species in complex mixtures, mainly resulting from vertically constant mixing ratios.

Figure 11 provides a comparative analysis of key column-integrated aerosol properties at 532 nm, specifically total-column single scattering albedo (SSA), aerosol optical depth (AOD), total-column lidar ratio (LR), and total-column effective radius (reffTC). The AEROCHEMPro retrievals show high agreement with the pseudo-reality for most optical properties, which indicates robust algorithm performance. SSA retrievals show a high correlation (R=0.96) and low root mean square error (RMSE=2.41×10-2), with a minor negative bias (mean bias error, MBE=-6.00×10-3). The retrieval spread widens at lower SSA values, notably below 0.9, when results become more dispersed in comparison to the pseudo-reality. This enhanced dispersion is often associated with low aerosol loading or the presence of absorbing aerosols like black or brown carbon. These conditions decrease the sensitivity of polarimetric measurements while increasing retrieval uncertainty.

The AOD comparison has excellent agreement, with a correlation coefficient around unity (R≈1.00), RMSE of 2.64×10-2, and negligible bias (MBE=1.01×10-2). This result demonstrates the algorithm's ability to accurately capture columnar aerosol loading. The lidar ratio, which varies with aerosol type and size distribution, shows a very high correlation (R=0.97) and an acceptable RMSE of 4.67. Most of the residual disparities occur in high LR cases, which are frequently linked to complex aerosol mixtures (fine dust mixed with smoke or marine aerosols). In these mixtures, different components may have similar column-integrated optical properties, and vertical overlap can reduce the retrieval's ability to distinguish their contributions, resulting in LR biases. The retrieval of reffTC performs well (R=0.98), though it has a negative bias (MBE=-0.05 µm). A slight underestimation is noticeable for particles with reffTC>0.4 µm, possibly due to the assumption of fixed, uniform size distributions within each aerosol mode (fine, dust, and sea salt) in the retrieval. See Table 2 for details. This simplification may not adequately account for the broad range of particle sizes in pseudo-reality, especially in mixed environments where the actual size distribution differs from the assumed modal.

Figure 12 presents global-scale AOD comparison results at 532 nm for fine aerosol species, including black carbon, brown carbon, inorganic salts, and their water content. Black carbon retrievals demonstrate the highest accuracy, as indicated by a high correlation coefficient (R=0.93), low root mean square error (RMSE=3.3×10-3), and a regression slope near unity (1.03). These results show the retrieval algorithm's effectiveness in detecting absorbing aerosols with distinct spectro-polarimetric signatures. Brown carbon retrievals display a moderately good performance (R=0.81), with an MBE of 1.81×10-3 and a slope of 0.99. An increased scatter at higher loading suggests a small bias in retrieving BrC contributions in regions with complex aerosol mixtures. This is most likely due to spectral overlaps with other weakly absorbing fine particles, which affects aerosol discrimination. Inorganic salts show a high R (∼0.89), with an MBE of 4.56×10-3 and a slope of 1.18, indicating a slight overestimation. This could be due to the retrieval algorithm compensating for the low LR and low absorption of sulfate-dominated particles. Water uptake associated with fine particles demonstrates good performance (R=0.85), a slope of 0.82, and a positive MBE of 3.6×10-3.

AOD retrievals at 532 nm for coarse particles, including iron oxide, quartz, and sea salt, along with the water content uptake by sea salt, are shown in Fig. 13. Iron oxide retrievals show a very high correlation coefficient of 0.99, a slope of 0.96 near unity (MBE=6.0×10-5), likely due to the elevated absorption signal of iron oxide in dusty conditions. Quartz retrievals are the most accurate among all species, achieving the best correlation (R nearly 1.00) and a slope near unity (0.97), with a low MBE of 2.9×10-3. This high accuracy shows the robust retrieval of the mineral dust signature, which dominates the coarse mode in arid regions. Sea salt retrievals show excellent performance (R=1.00), with a slope of 0.97 and MBE of -1.9×10-3. Water uptake by sea salt AOD shows a perfect correlation (R∼1.00), a slope of 0.97, and an MBE of -1.3×10-3.

In summary, AEROCHEMPro is highly efficient for distinguishing key chemical species in both fine- and coarse-mode aerosols. Absorbing and dust-like species, such as black carbon, iron oxide, and quartz, along with sea salt, showed the most accurate AOD retrievals.

Figure 14 depicts an examination of AEROCHEMPro' s retrieval capabilities for aerosol chemical species volume mixing ratios in the troposphere (covering altitudes from the surface to 12 km), emphasizing both its strengths and specific limitations. AEROCHEMPro performs well in retrieving most aerosol species, with R values greater than 0.67 for all species, showing a generally good agreement with pseudo-reality. For BC and BrC, the method indicates moderately good accuracy, with R values of 0.78 and 0.67, respectively, and slopes close to unity (1.06 and 1.04). MBE values of 8.92×10-9 and 3.86×10-8. Inorganic salt likewise performs well, with an R of 0.80, a slope of 1.19, and MBE of 8.42×10-8, indicating a slight overestimation. On the other hand, a slight underestimation of the volume mixing ratios of water content associated with fine particles, with R value of 0.85, slope of 0.80, and MBE of 5.10×10-8. Iron oxide and quartz are among the best-retrieved species, with R values of 0.98 each, slopes approaching unity (0.96 and 0.97, respectively), and MBE of (1.89×10-9 and 1.44×10-7), illustrating the robustness of AEROCHEMPro for non-spherical species. Similarly, sea salt and water content uptake associated with it have a very good agreement with pseudo-reality, with R values of 0.98 and 0.97, slope=0.96 and 0.94, and an MBE of 6.7×10-8 and 2.44×10-7, resulting in near-perfect scaling. Overall, while the approach performs very well in many situations, there are some areas, particularly brown carbon, where improvements might be made to improve accuracy. Future enhancements will focus on refining the a priori refractive index models used for organic species. This would better account for the inherent variability in these components and further reduce partitioning uncertainties between absorbing species.

Table 7 summarizes the global evaluation of AEROCHEMPro retrievals for aerosol chemical species volume mixing ratio at four atmospheric levels (0–2, 2–4, 4–8, and 8–12 km), based on layer-averaged values per pixel. The findings reveal altitude-dependent performance patterns, providing valuable insights for future refinements of the retrieval method.

The R shows strong agreement between derived values and pseudo-reality for most species. The highest correlations, always above 0.97, are seen for mineral species (IO, Qz) and sea salt (SS). In comparison, water content in fine species (WC_F) has lower correlations at higher altitudes, dropping to 0.55 to 0.63. MBE values are generally small, around 10-8 to 10-7, with no clear pattern of overestimation or underestimation across atmospheric layers. RMSE values get smaller with altitude for most species, which matches the lower concentrations found at higher altitudes, and are lowest for BC and IO. The slopes show a slight overestimation for BC, BrC, and IS in the lower layers, and a slight underestimation for WC_F, especially above 4 km. Overall, these results show consistent retrieval accuracy worldwide, with better agreement for coarse-mode species and somewhat lower performance for fine-mode water content aerosols at higher altitudes. The overall observed altitude-dependent performance is determined by signal strength, species abundance, and the information content of measurements at each atmospheric layer.