Accurate calibration is essential for the quantitative retrieval of the aerosol backscatter coefficient βa(z) from ceilometer measurements. The calibration strategy depends on instrument sensitivity, the availability of reference data, and the prevailing atmospheric conditions.

For the CHM15k instruments in E-Profile, calibration is performed using the Rayleigh method . This approach requires the identification of a high-altitude reference region, zref, typically located in the upper troposphere, where the aerosol backscatter contribution is assumed to be negligible (βa(zref)≈0). Under this assumption, the total atmospheric backscatter at this altitude is attributed exclusively to molecular scattering, such that β(zref)=βm(zref). The molecular backscatter coefficient βm(zref) is calculated using Rayleigh scattering theory together with standard atmospheric pressure and temperature profiles interpolated to the ceilometer altitude grid.

The lidar equation can be expressed in terms of the backscatter coefficient, β (Eq. ). In this formulation, P(z) denotes the received signal as a function of atmospheric optical properties and measurement range z, the exponential is the transmission term, while CL represents the system-dependent calibration constant accounting for instrumental characteristics. 1P(z)⋅z2CL=β(z)⋅exp⁡-2∫0zα(z′)dz′

The total backscatter coefficient β and the extinction coefficient α are expressed as the sum of aerosol and molecular contributions, i.e., β=βa+βm and α=αa+αm.

The CL is obtained by applying the backward Klett–Fernald solution using the reference altitude zref as boundary condition. Following , the aerosol backscatter coefficient is computed as 2βa(z)=Z(z)N(z)-βm(z),3Z(z)=z2P(z)exp⁡-2∫0z[Sa(z′)-Sm]βm(z′)dz′. In this formulation, the aerosol lidar ratio Sa(z) is fixed at 52 sr for calibration within E-Profile, while Sm≈8π/3sr and βm(z) denote the molecular lidar ratio and molecular backscatter coefficient, respectively.

The auxiliary function N(z) can be formulated in two equivalent ways depending on the adopted boundary condition. It can be expressed explicitly using the system constant CL, which is convenient for forward range integration starting at the lidar, or alternatively in terms of a prescribed reference aerosol backscatter value βa(zref), which leads to the standard backward integration from the reference altitude toward the instrument. In the E-Profile implementation, N(z) is expressed as 4N(z)=zref2P(zref)βm(zref)+βa(zref)+2∫zzrefSp(z′)Z(z′)dz′, where βa(zref)≈0 is assumed for Rayleigh calibration.

A major limitation of Rayleigh calibration arises from the low signal-to-noise ratio (SNR) in the upper troposphere, where ceilometer signals are significantly weaker . Therefore, rigorous filtering of clouds and residual aerosol layers is required to ensure the reliability of the selected calibration interval and to guarantee sufficiently high SNR conditions.

In operational practice, E-Profile applies a calibration procedure that combines long-term signal averaging with careful screening of the reference altitude to minimize the influence of residual aerosols and measurement noise. This procedure requires at least three hours of signal integration during nighttime under cloud-free conditions with negligible aerosol loading at the reference altitude. The resulting calibration constants are subsequently processed using a Kalman filter to derive a daily best estimate of the CL, thereby improving the long-term stability and consistency of operational aerosol profiling. A detailed description of the numerical implementation is provided by .

The forward iterative method solves the lidar equation by integrating upward from the surface, which eliminates the need for a high-altitude, aerosol-free reference and is therefore especially effective under cloudy or low SNR conditions where other methods are unreliable. However, the forward method requires a known system CL as input, requiring several hours of signal averaging to achieve sufficient SNR and is therefore being applicable to higher-power instruments such as the CHM15k (further details in Sect. ). This approach therefore assumes that the lidar system remains stable between the nighttime calibration and its application during the day.

The algorithm initializes the aerosol backscatter coefficient βa(z0) at the lowest reliable measurement height (z0=250m above ground level (a.g.l.)) and assumes that aerosol transmittance near the surface is approximately unity at infrared wavelengths. Consequently, the initial aerosol backscatter is taken directly from the measured attenuated backscatter (Eq. ). Although the CHM15k reaches full overlap only at approximately 800 m, the empirical temperature-dependent overlap correction described by substantially reduces artefacts above 250 m a.g.l. This correction is applied within the E-Profile processing chain and establishes 250 m a.g.l. the effective lower boundary for reliable aerosol retrievals in this study, consistent with the exclusion criteria applied throughout the analysis (Sects.  and ). 5βa(z0)=P(z0)z02CLTm2(z0)-βm(z0).

From this surface value, the algorithm integrates upward with steps δz. For every new layer zi=zi-1+δz, the extinction coefficient is first estimated with the constant lidar ratio Sa of 50 sr in Eq. (). 6αa(zi)=Saβa(zi-1),

Hence, the extinction contributed by that layer is computed in Eq. () 7τa(zi)=αa(zi)δz, and the two-way aerosol transmittance from the surface up to zi is calculated with Eq. () 8Ta(zi)=exp⁡(-2τa(zi)).

With this updated transmittance, the backscatter coefficient at the current level is recomputed from the total attenuated signal with Eq. (): 9βa(zi)=P(zi)zi2CLTm2(zi)Ta2(zi)-βm(zi).

This process is repeated until the relative change in αa(zi) falls below 0.01 % or 30 iterations are reached; the algorithm then steps to the next height and repeats the sequence up to the chosen top of the profile ztop. The convergence criterion of 0.01 % relative change in αa(zi) with a maximum of 30 iterations, follows and ensures numerical stability, throughout the retrieval.

This approach is particularly robust in aerosol-rich environments, where the backward method often fails due to the absence of a well-defined, aerosol-free reference region aloft. According to , a systematic uncertainty of 10 % in the CL leads to a systematic uncertainty 10 %–20 % error in retrieved βa within the planetary boundary layer (PBL), while a similar systematic uncertainty in the lidar ratio Sa contributes less than 5 % error. Since both sources of error propagate upward from z0, their effects are additive when biases have same sign and partially compensating when they are opposite. These findings highlight the method’s stability in complex aerosol scenes, provided that accurate calibration of CL is achieved, since errors propagate upward from the surface.

The analytical backward solution for retrieving the aerosol backscatter coefficient βa(z) from ceilometer measurements, as implemented in the E-Profile framework, is described by . In contrast to the forward retrieval approach, the backward method does not require prior knowledge of the system CL. Instead, CL is implicitly determined through the application of a boundary condition at a reference altitude where the aerosol contribution is assumed negligible. The explicit value of CL becomes necessary only when switching to a forward integration scheme.

The reference altitude (zref) was determined automatically within a fixed search window between 1500 and 5000 m a.g.l. using the A-Profiles package. Within this window, the algorithm identifies the vicinity of the minimum of a 3-bin running-average smoothed attenuated backscatter profile, selecting specifically the point where the raw signal most closely matches the smoothed signal. At this altitude, the aerosol backscatter is assumed negligible (βa(zref)=0) .

The molecular backscatter coefficient βm(z) is derived from atmospheric model profiles. The retrieval formulation follows the same mathematical framework used during calibration and is based on Eqs. () and ().

The auxiliary function Z(z) is constructed from the range-corrected signal together with a transmission correction that accounts for differential extinction between aerosol and molecular scattering (Eq. ). The function N(z) is defined according to Eq. ().

Under the Rayleigh calibration assumption βa(zref)=0, the first term of Eq. () simplifies to zref2P(zref)/βm(zref). This term provides the calibration reference by scaling the measured signal at zref to the known molecular backscatter. The backward solution is then obtained by integrating downward from the reference altitude zref toward the instrument. The aerosol backscatter coefficient profile is finally retrieved by combining Z(z) and N(z) and subtracting the molecular contribution, as expressed in Eq. ().

In operational applications, the backward retrieval is subject to several constraints. First, the relatively low optical power of ceilometers requires long signal integration periods to achieve an adequate signal-to-noise ratio (SNR) at the reference altitude zref, a condition most easily satisfied during nighttime due to reduced background noise. Second, the accuracy of the retrieved βa(z) strongly depends on the assumed aerosol lidar ratio Sa. Although advanced inversion schemes allow Sa to vary with altitude, a constant value of 50 sr is commonly adopted in operational ceilometer processing to convert backscatter coefficients into extinction profiles. Uncertainties in Sa or in the selection of zref can lead to nonlinear error propagation, particularly under multilayered or vertically inhomogeneous aerosol conditions . Despite these limitations, the backward approach provides a stable and self-calibrating retrieval of βa(z), as it does not require an externally determined calibration constant.

To evaluate the performance of the backward and forward methods in retrieving aerosol vertical structure, backscatter coefficient profiles are compared from both approaches against the synergistic GRASP_pac algorithm. The evaluation spans different aerosol regimes at the three European stations: Granada, Payerne, and Lindenberg (Fig. ).

Although both retrieval methods generally agree with the GRASP_pac reference in a general scenario, significant biases emerge under high aerosol load, particularly during coarse-mode events that are frequent at the Granada site (Table ). The forward method consistently shows smaller deviations and reduced variability than the backward approach, reducing the near-surface mean deviation by approximately a factor of three under coarse-mode-dominated conditions. This behaviour is consistent with the greater robustness of the forward retrieval in situations where the aerosol-free reference altitude required by the backward inversion is difficult to identify or may be affected by residual aerosol contamination.

At Granada, considering the full dataset, the backward retrieval shows a systematic positive bias of 0.027±0.194Mm-1sr-1 within the 0.25–2 km layer, whereas the forward retrieval shifts the mean to -0.028±0.079Mm-1sr-1, reducing the spread by more than half (Table ). Between 2–5 km, the backward results exhibit a slightly negative deviation with greater scatter (-0.005±0.109Mm-1sr-1), while the forward retrieval maintains a negative mean with relative dispersion below 35 %. The increasing negative deviation with height reflects the systematic error propagation characteristic of backward integration when the boundary value at the molecular reference altitude is poorly constrained . In contrast, the forward method removes this altitude-dependent trend and maintains uniform agreement throughout the troposphere (Fig. a).

Under high aerosol optical depth (AOD) conditions, the sign of the deviation reverses: the backward retrieval shows deviations of -0.454±0.304Mm-1sr-1 (0.25–2 km) and -0.420±0.310Mm-1sr-1 (2–5 km), while the forward retrieval reduces both to -0.168±0.167 and -0.161±0.169Mm-1sr-1 respectively. The substantially larger deviations observed for the backward retrieval are consistent with the increased difficulty of identifying an aerosol-free reference region under high aerosol loading, which enhances the propagation of errors throughout the profile. The forward approach is less affected by this limitation, maintaining similar deviations at both altitude ranges and closer agreement with the GRASP_pac reference under dust-rich and low-signal conditions (Fig. b).

For cases dominated by coarse-mode particles, the backward bias peaks at -0.551±0.262Mm-1sr-1 near the surface and -0.500±0.283Mm-1sr-1 at higher altitudes. The forward method again reduces the bias by approximately a factor of three (-0.200±0.158 and -0.191±0.165Mm-1sr-1) and maintains relative dispersion below 30 % at all heights (Table ). This results indicate that the forward method correctly retrieves the strong scattering from large particles in contrast to backward method.

Under fine-mode-dominated conditions, both algorithms underestimate the backscatter coefficient, though in different ways. For the 0.25–2 km layer, the backward method yields -0.123±0.271 and -0.032±0.125Mm-1sr-1 for the upper range, while the forward retrieval reduces both the mean deviation and dispersion by approximately ∼50% (-0.054±0.163 and -0.020±0.069Mm-1sr-1). The largest relative deviation occurs in the boundary layer, where fine aerosols are concentrated, and decreases with height. The forward method preserves this physically consistent gradient while tightening the spread, indicating a better representation of fine-particle vertical structure. It is also important to note that differences in the prescribed lidar ratio between the forward/backward retrievals and the dynamically adjusted lidar ratio in GRASP_pac may partly explain the discrepancies observed under high aerosol load for both fine- and coarse-mode-dominated cases.

At Payerne, the altitude-dependent behaviour is similar to that observed at Granada. The comparison of the full dataset shows a nearly constant negative backward bias with height (-0.129±0.167 and -0.077±0.116Mm-1sr-1 for the 0.5–2 and 2–5 km layers, respectively), while the forward retrieval centres the mean deviation profile near zero and reduces the dispersion 10 % with respect to backward for the 0.5–2 km layer (Fig. i). During high-aerosol-load episodes, the deviations remain negative, but the forward method keeps a deviation of -0.012Mm-1sr-1 and reduces the relative deviation by approximately 40 % while backward shows deviations greater than -0.170Mm-1sr-1 (Fig. j). Coarse-mode conditions amplify the vertical gradient with backward (-0.481±0.229, -0.383±0.229Mm-1sr-1; Table ), yet the forward retrieval reduces the mean deviation to -0.092 and -0.106Mm-1sr-1 (Fig. k). Fine-mode pollution introduces a stronger negative deviation in the 0.5–2 km layer (-0.207±0.211Mm-1sr-1) with backward that weakens with height; forward retrievals again exhibit narrower dispersion across all layers. Overall, Payerne reproduces the vertical patterns observed at Granada, confirming that the retrieval algorithm remains robust despite site-specific microphysical differences.

Lindenberg shows a behaviour broadly comparable to Granada, although with weaker systematic trends and a stronger influence of case-to-case variability. For the full dataset, both retrievals exhibit small negative biases, with the forward method yielding values closer to zero and a reduced dispersion compared to the backward approach (Fig. i). In contrast to Granada, no pronounced altitude-dependent increase of the backward bias is observed at this site. Under high aerosol load conditions, deviations remain moderate for both retrievals, while the forward method consistently reduces the spread of the differences, indicating improved stability (23 % vs. 44 %) in the 0.25–2 km layer. Coarse-mode-dominated cases exhibit larger negative biases, particularly for the backward retrieval, although these deviations are less pronounced than those observed at Granada. The forward method substantially mitigates these biases, reducing their magnitude by approximately 50 %–60 % (Table ); however, this result should be interpreted with caution due to the limited number of available cases (eight events). For fine-mode-dominated conditions, both retrievals tend to underestimate the GRASP_pac backscatter coefficient, but the forward approach maintains a noticeably narrower dispersion, particularly in the lower troposphere (relative variability of approximately 23 % compared to about 45 % for the backward method). Overall, the Lindenberg results confirm the improved robustness of the forward retrieval, while highlighting the role of variable air-mass history and more complex aerosol mixtures in shaping site-specific variability.

In summary, the forward method consistently outperforms the traditional backward approach across all aerosol regimes. Key improvements include eliminating the altitude-dependent bias characteristic of the backward kernel, reducing mean deviation significantly, and yielding smaller standard deviations in backscatter coefficient profiles. This enhanced performance is most evident under challenging conditions such as high aerosol load and coarse-mode particle dominance.

The profile-level results are confirmed by the scatter analysis. Across all three sites the forward retrieval consistently raises the correlation coefficient, reduces the RMSE, and brings the regression slope closer to unity relative to the backward approach, with the largest contrast emerging under coarse-mode-dominated conditions.

At Granada, the forward retrieval shows very close agreement with GRASP_pac over the full dataset (r=0.97, RMSE=0.08Mm-1sr-1, slope near unity), whereas the backward retrieval is markedly weaker (r=0.82, RMSE=0.19Mm-1sr-1). The backward slope below unity, combined with a small positive intercept, indicates that it underestimates backscatter at the higher values while slightly overestimating the lowest ones. This contrast widens sharply under coarse-mode-dominated conditions: the backward RMSE more than doubles (0.58 vs. 0.26 Mm-1sr-1 for the forward), and a large positive intercept reveals a systematic overestimation across the range that is most severe at low backscatter. This behaviour is consistent with the difficulty of identifying an aerosol-free reference altitude under high aerosol load, the key limitation of the backward approach. Under fine-mode conditions both methods improve and the gap narrows (backward RMSE 0.29 vs. 0.17 Mm-1sr-1), suggesting that the backward approach struggles most under coarse-mode-dominated condition.

Payerne shows the same overall behaviour. The forward retrieval shows strong agreement with GRASP_pac over the full dataset (r=0.90 vs. 0.72 for the backward), with a slope close to unity and only a small positive intercept, while the backward method again sits below unity, indicating a mild underestimation at the higher backscatter values. The coarse-mode contrast is reproduced, with the forward RMSE roughly half that of the backward approach (0.27 vs. 0.50 Mm-1sr-1). As at Granada, the backward method degrades most under high coarse-mode loading, where the difficulty of locating an aerosol-free reference altitude is greatest.

Lindenberg is consistent with both other sites, with the lower aerosol loading of this continental station reflected in generally smaller RMSE values across all regimes. The forward retrieval achieves close agreement with no appreciable bias (r=0.96, RMSE=0.07Mm-1sr-1, slope near unity), while the backward method (r=0.82, RMSE=0.16Mm-1sr-1) reproduces the Granada crossover pattern: overestimation at low backscatter and underestimation at higher values, with the transition occurring near 0.2 Mm-1sr-1. The coarse-mode subset again shows the strongest divergence between the two methods, though the small number of profiles available at this site means this result should be interpreted with caution.

Aerosol extinction profiles retrieved from ceilometer observations using the backward (Klett–Fernald) and forward (iterative) approaches are evaluated against the synergistic GRASP_pac products, which combine sun–sky photometer and ceilometer measurements and provide vertically resolved extinction without prescribing a fixed lidar ratio (Sect. ). The comparison is performed for three European sites and for different aerosol regimes (Fig.  and Table ).

Across all stations and conditions, the forward retrieval generally reduces the dispersion of extinction differences relative to GRASP_pac, while the backward method exhibits stronger altitude-dependent biases. For the complete dataset, relative variability is typically reduced by 30 %–60 % when using the forward approach, with the magnitude of the improvement depending on aerosol load and dominant particle size.

At Granada, the backward method shows pronounced biases throughout the tropospheric column, with mean deviations of 6.6±9.2Mm-1 below 2 km and 1.6±4.4Mm-1 between 2 and 5 km. The forward retrieval reduces these values to 3.2±4.0Mm-1 and 0.2±2.0Mm-1, respectively. Under high aerosol load and coarse-mode-dominated conditions, the backward approach strongly underestimates extinction, whereas the forward method reduces the magnitude of these deviations by up to 60 %–70 %, particularly above the boundary layer. For fine-mode dominance, both retrievals overestimate extinction near the surface, but the forward approach reduces the relative dispersion by about ∼50%.

At Payerne, deviations are generally smaller but exhibit a stronger dependence on aerosol regime. For the complete dataset, the backward retrieval underestimates extinction in both altitude ranges (-2.6±7.1Mm-1 below 2 km), while the forward method yields comparable mean positive deviations with reduced variability. Under high aerosol load, the backward method maintains a negative bias near the surface, whereas the forward retrieval shifts the mean deviation towards positive values and reduces the dispersion, particularly between 2 and 5 km. Coarse-mode-dominated cases again highlight the limitations of the backward approach, with negative deviations reduced by a factor of three to four when using the forward method.

At Lindenberg, extinction differences display weaker systematic trends but larger case-to-case variability. For the full dataset, the backward method exhibits a positive bias below 2 km (3.9±7.5Mm-1) that decreases with height, while the forward retrieval reduces both the mean bias and the dispersion. Under high aerosol load, the forward approach does not consistently improve the mean bias but reduces the relative variability by roughly a factor of two. Coarse-mode-dominated events produce the largest discrepancies, with strong underestimation by the backward method throughout the column (-23.9±3.9Mm-1 below 2 km), which is substantially mitigated by the forward retrieval. This result should, however, be interpreted with caution due to the limited number of coarse-mode cases at this site. Under fine-mode dominance, both retrievals overestimate extinction in the lower troposphere, but the forward approach maintains a markedly narrower dispersion at all altitudes.

Comparison against GRASP_pac reveals smaller relative dispersion for backscatter than extinction coefficients (Tables  and ). This difference can be attributed to the distinct sensitivity to the lidar ratio (Sa). Backscatter is retrieved directly from the measured signal with a weaker sensitivity, whereas extinction is computed as α(z)=Sa⋅β(z), inheriting both backscatter uncertainty and the full uncertainty in the assumed lidar ratio. Since GRASP_pac retrieves Sa dynamically, extinction deviations reflect the mismatch between assumed and actual aerosol-specific lidar ratios, explaining the larger dispersion. Although the GRASP_pac-retrieved lidar ratio could in principle be prescribed as input to the backward and forward inversions, we deliberately retain the fixed operational value of Sa=50sr for both methods. Adopting the GRASP_pac lidar ratio would improve the agreement by construction, but it would not represent the fully automatic, unsupervised E-Profile processing chain that this study sets out to evaluate; the impact of the fixed-ratio assumption is instead quantified through the sensitivity analysis in Sect. .

The scatter analysis for extinction reproduces a similar behaviour to that found for backscatter: at every station the forward retrieval improves the correlation, reduces the RMSE, and keeps the slope closer to unity than the backward approach, with the contrast sharpening under coarse-mode dominance.

At Granada, the forward retrieval agrees closely with GRASP_pac over the full dataset (r=0.96, RMSE=4.61Mm-1), showing substantially closer agreement with GRASP_pac than the backward method (r=0.81, RMSE=10.64Mm-1). The backward slope well below unity combined with a positive intercept indicates an overestimation of extinction at low values and an underestimation at higher values, with the crossover near 2 Mm-1, so the backward method fails to reproduce the extinction signal across the full dynamic range. The forward retrieval keeps a near-zero intercept, removing the low-end bias, although its slope of 0.80 implies a residual underestimation at the highest extinction values. The contrast sharpens under coarse-mode-dominated conditions, where the backward retrieval shows a large positive intercept (21.74 Mm-1) and a strong overestimation across the observed range (RMSE =24.64 Mm-1), whereas the forward retrieval reduces the RMSE by nearly 60 % (9.53 Mm-1) and recovers a slope close to unity, consistent with the profile-deviation analysis above.

Payerne is broadly consistent with Granada, with the forward retrieval again showing closer agreement with GRASP_pac than the backward method over the full dataset (r=0.90 vs. 0.72; RMSE=5.71 vs. 7.03 Mm-1), and both methods showing the same low-value overestimation/high-value underestimation. The coarse-mode regime, however, differs in character: the backward retrieval overestimates throughout the observed range (slope above unity, intercept 13.89 Mm-1) rather than crossing over, which likely reflects the interplay between the fixed lidar-ratio assumption and the distinct microphysics of the relatively infrequent coarse-mode episodes at this predominantly fine-mode site. The forward retrieval again roughly halves the RMSE (9.11 vs. 17.94 Mm-1).

Lindenberg reproduces the Granada pattern most closely. The forward retrieval agrees well with GRASP_pac over the full dataset (r=0.94, RMSE=4.86Mm-1) while the backward method (r=0.80, RMSE=8.23Mm-1) again overestimates at low extinction and underestimates at high, crossing over near 7 Mm-1. The coarse-mode subset shows the steepest backward overestimation of all three sites (slope 1.33, intercept 23.14 Mm-1), against which the forward retrieval reduces the RMSE by nearly 70 % (7.47 Mm-1); as noted previously, this result should be interpreted with caution given the limited number of coarse-mode profiles available at this site.

Overall, the comparison with GRASP_pac indicates that the forward iterative retrieval generally provides a more stable and vertically consistent representation of aerosol extinction than the traditional backward approach. The differences between the two methods are most pronounced under high aerosol load and coarse-mode-dominated conditions, where the forward retrieval exhibits smaller deviations and lower variability. These results support the use of the forward approach for operational aerosol profiling within ceilometer networks.

The column-level consistency of ceilometer aerosol retrievals is evaluated by vertically integrating extinction profiles derived with the Klett–Fernald backward and forward iterative methods to obtain aerosol optical depth (AOD), which is compared against AERONET Level 2.0 observations (Sect. ). Figure  summarizes the AOD deviations for the different aerosol regimes, while the associated statistical metrics are reported in Table .

For the complete dataset (first column in each panel of Fig. ), a good overall agreement is found between ceilometer-derived AOD and AERONET at all three stations. Forward retrievals exhibit mean biases close to zero at UGR, PAY, and LIN, indicating the absence of systematic over- or underestimation. The corresponding standard deviations are approximately ±0.01, comparable to the intrinsic uncertainty of AERONET AOD. In contrast, backward retrievals show larger mean biases and dispersion except for LIN, with standard deviations ranging from ±0.024 at Payerne to ±0.035 at Granada, i.e. roughly three times higher than those of the forward method. These results indicate the greater stability of the forward approach, which benefits from an independent calibration performed under cloud-free conditions with negligible aerosol loading at the reference altitude.

From an uncertainty perspective, the forward retrieval consistently yields lower absolute and relative standard deviations while maintaining a mean bias statistically indistinguishable from zero. According to the Guide to the Expression of Uncertainty in Measurement , this behaviour indicates negligible systematic uncertainty and confirms the adequacy of the adopted random uncertainty model. It should be noted that these evaluations were restricted to daytime hours, conditions under which the backward method faces additional limitations due to reduced SNR.

Under high aerosol load conditions (second column of Fig. ), deviations between ceilometer and AERONET AOD increase, particularly at UGR. There, the backward method exhibits the largest discrepancy, with a mean bias of -0.103 and a standard deviation of ±0.076, whereas the forward retrieval reduces both the bias and dispersion to -0.043 and ±0.050, respectively. Similar but smaller reductions in deviation and dispersion are observed at PAY (-0.033 vs. 0.005) and LIN (-0.018 vs. -0.008). In all cases, the relative standard deviation of the forward retrieval remains below 30 %, while that of the backward method exceeds this threshold, reflecting the greater sensitivity of the backward approach to contamination of the reference altitude under high aerosol load.

When filtering by particle mode, the largest AOD deviations are found for coarse-mode-dominated conditions. For the backward method, mean differences range from -0.116 to -0.184 across the three stations, corresponding to relative deviations exceeding -70%, although with more moderate dispersion (∼20%). The forward retrieval substantially reduces these mean deviations to values below -55% at all sites, while the standard deviation remains similar between both approaches (Table ). For fine-mode-dominated conditions, both methods exhibit smaller mean biases, with deviations generally below 0.020 at all stations, except for the backward retrieval at PAY. Nevertheless, the forward approach consistently shows lower dispersion (about 20 % compared to ∼60% for the backward method at UGR and LIN), highlighting its improved robustness.

The performance of the forward method is consistent with recent GRASP-MPLNET intercomparisons , which reported RMSE values of 0.022–0.038 at 532 nm. In the present study, forward retrievals yield RMSE values of 0.012 at Granada, 0.010 at Payerne, and 0.013 at Lindenberg, indicating agreement comparable to, or better than, that reported in previous studies using the same photometric reference.

In summary, the forward iterative method reproduces AOD more accurately and with significantly lower dispersion than the traditional Klett–Fernald backward approach across geographically distinct sites. The improvement is particularly evident under challenging tropospheric conditions, such as high aerosol load and coarse-mode particle dominance, where the backward method is more sensitive to reference-altitude contamination. The enhanced performance of the forward approach can be attributed to its refined calibration strategy, combined with the temporal stability of the ceilometer system, which allows calibration parameters to be reliably transferred over time. The systematic reduction of data dispersion, together with the absence of significant mean bias, supports the operational use of ceilometers coupled with forward retrieval algorithms for AOD monitoring in dust-influenced and polluted environments, and highlights the critical role of high-quality instrument calibration. The remaining discrepancies can be attributed to the use of a fixed lidar ratio of 50 sr, which is not optimal for all aerosol types.

The corresponding scatter plots are provided in the Supplement (Figs. S7 and S8). The scatter analysis against AERONET AOD reinforces these findings across all three sites, and the contrast between the two methods is the largest of the three quantities examined. For the full dataset the forward retrieval achieves high correlations, near-zero intercepts, and slopes close to unity, whereas the backward retrieval shows much lower correlations and slopes well below unity, indicating that it fails to reproduce AOD variability across the dynamic range. Because AOD is a column-integrated quantity, the reference-altitude error of the backward formulation accumulates vertically, which plausibly explains why the gap is wider here than for the backscatter and extinction profiles.

At Granada, the forward retrieval agrees strongly with AERONET over the full range of observed AOD (r=0.94, RMSE=0.01, slope 0.97, near-zero intercept), while the backward retrieval is only weakly correlated (r=0.54, RMSE=0.04, slope 0.47). The backward slope combined with a positive intercept indicates an overestimation of AOD at low values and an underestimation at higher values, with the crossover near AOD≈0.04. The same behaviour is reproduced at Payerne and Lindenberg, where the forward retrieval maintains r≥0.88 with slopes close to unity (0.89 and 0.94), near-zero intercepts, and RMSE=0.01, while the backward method stays below r=0.51 with slopes no higher than 0.41 and RMSE=0.03, showing the same low-value overestimation/high-value underestimation seen at Granada.

Under coarse-mode-dominated conditions the contrast is particularly pronounced. At Granada the backward regression is nearly flat and well above the 1:1 line (slope 0.21, intercept 0.19, r=0.17, RMSE=0.16), so the backward method is largely insensitive to AOD variability under dust conditions; the forward retrieval remains comparatively skilful (r=0.67, slope 0.87, RMSE=0.08). Lindenberg is consistent with this picture, the backward retrieval again losing skill (slope 0.44, r=0.26, RMSE=0.20); its forward slope of 1.28 suggests a moderate overestimation at higher AOD, in line with the profile-deviation analysis, though this regime rests on only N=63 matched cases and should be interpreted with caution.

Under fine-mode-dominated conditions the forward retrieval correlates well at all three sites (r=0.77-0.85, RMSE≤0.02), far above the backward method (r=0.11-0.43, slopes as low as 0.05), which reproduces fine-mode AOD variability very poorly. The forward slopes, however, remain below unity (0.56-0.83, lowest at Granada), indicating that even the forward retrieval underestimates fine-mode AOD at 1064 nm; this residual bias is most plausibly linked to the spectral conversion and Ångström exponent assumptions used to bring the retrieval to the AERONET reference wavelength. Overall, the lower dispersion and smaller deviations of the forward retrieval translate into stronger column-integrated agreement across aerosol regimes and sites, with its main remaining limitation associated with fine-mode scaling.

The column-integrated evaluation in Sect.  was carried out using a single fixed lidar ratio of 50 sr to convert backscatter into extinction. Although this value produced good overall agreement with AERONET, particularly for the forward retrieval, the residual deviations identified under high aerosol load, and most markedly under coarse-mode-dominated conditions, were attributed to the limitation of applying a single fixed lidar ratio across the contrasting aerosol regimes sampled at the three sites. To quantify how strongly this assumption propagates into the retrieved AOD, and to assess whether a composition-aware estimate can mitigate these regime-dependent biases, the forward and backward inversions were re-evaluated against AERONET using four lidar ratio configurations: fixed values of 30, 50, and 70 sr, together with a daily variable estimate derived from IFS–CAMS aerosol composition forecasts (Sa IFS; Sect. ).

The analysis follows the same aerosol regime classification described in Sect. . The results for Granada are shown in Figs.  (backward retrieval) and (forward retrieval); the equivalent figures for Payerne and Lindenberg are provided in the Supplement (Figs. S9–S12 and Tables S1–S6).

For the full dataset, Sa50sr and Sa IFS consistently yield the lowest RMSE across all three sites and both retrieval methods. For the forward method, RMSE values of 0.012 and 0.013 at Granada, 0.010 and 0.011 at Payerne, and 0.013 and 0.015 at Lindenberg are obtained for Sa50sr and Sa IFS respectively, with near-zero mean deviations at all sites. Sa30sr produces negative mean deviations at all sites, while Sa70sr produces positive mean deviations; at Granada, these reach -0.018 and +0.021 respectively for the forward method, corresponding to relative biases of -38.2% and +54.1%, substantially larger than those of Sa50sr (+0.001) and Sa IFS (+0.001). These results indicate that 30 and 70 sr are not appropriate as fixed operational values for mixed continental aerosol conditions.

Under high aerosol load conditions, the sensitivity to the assumed lidar ratio increases for both methods. For the forward method, Sa IFS yields the best or comparable performance to Sa50sr across all sites. The most notable difference is observed at Granada, where Sa IFS reduces the mean deviation from -0.043 to near-zero and the RMSE from 0.066–0.043 compared to Sa50sr. At Payerne and Lindenberg, Sa IFS and Sa50sr perform virtually identically, with near-zero mean deviations and RMSE values of 0.018 and 0.021 respectively. For the backward method, all lidar ratio configurations produce larger negative mean deviations under high load regardless of the assumed value, with Sa50sr yielding mean deviations of -0.103, -0.033, and -0.018 and RMSE values of 0.128, 0.047, and 0.045 at Granada, Payerne, and Lindenberg respectively, reflecting the growing influence of reference altitude contamination under elevated aerosol conditions.

Under high aerosol load and coarse-mode dominated conditions, the sensitivity to the lidar ratio is most pronounced. At Granada, Sa IFS achieves the best performance for the forward method, with a near-zero mean deviation and RMSE=0.053, improving over Sa70sr (RMSE=0.057) and substantially over Sa50sr (RMSE=0.085). Sa30sr performs worst, with a mean deviation of -0.126 and RMSE=0.134. The better performance of Sa70sr and Sa IFS relative to Sa50sr under these conditions is consistent with the fact that even during coarse-mode dominated episodes the atmospheric column at Granada is rarely pure dust, with residual fine-mode aerosol increasing the effective column lidar ratio above the pure dust value. For the backward method at Granada, all configurations yield large negative mean deviations ranging from -0.165 to -0.131, with RMSE values between 0.150 and 0.175, confirming that reference altitude contamination dominates over lidar ratio uncertainty for this method. At Payerne, the forward method results are consistent with Granada: Sa IFS reduces the mean deviation from -0.046 (Sa50sr) to -0.018 and the RMSE from 0.062–0.048, performing comparably to Sa 70 sr. For the backward method at Payerne, all configurations again yield large negative mean deviations, broadly consistent with Granada. At Lindenberg, no clear ranking among configurations is observed for the forward method; this result should be interpreted with caution given the limited number of coarse-mode cases at this site (N=63), reflecting the infrequent occurrence of such episodes at this predominantly fine-mode continental site. For the backward method, the same pattern of large negative deviations is observed regardless of the assumed lidar ratio.

Under high aerosol load and fine-mode dominated conditions, the results differ from the coarse-mode case. Sa 70 produces the largest positive mean deviations at all sites for the forward method, reaching +0.060, +0.045, and +0.032 at Granada, Payerne, and Lindenberg respectively, with the largest dispersion among all configurations. Sa30sr produces the largest negative mean deviations at all sites. At Granada, Sa IFS performs best for the forward method (mean deviation +0.012, RMSE=0.021), closely followed by Sa50sr (mean deviation +0.018, RMSE=0.024),with both showing comparable dispersion. For the backward method at Granada, Sa50sr and Sa IFS yield similar results (RMSE=0.041 and 0.040 respectively), with no configuration showing a clear advantage. At Payerne and Lindenberg, the forward method results are consistent with Granada: Sa50sr and Sa IFS perform virtually identically at Payerne (mean deviations +0.010 and +0.006, RMSE=0.017 for both), while at Lindenberg Sa50sr yields the lowest mean deviation (-0.002, RMSE=0.014) with Sa IFS performing comparably (mean deviation -0.008, RMSE=0.016). For the backward method at both sites, Sa50sr and Sa IFS again perform similarly with no configuration offering a clear advantage. The consistent performance of Sa50sr and Sa IFS across all three sites under fine-mode conditions suggests that 50 sr is a well-suited fixed value for these aerosol regimes, while the higher lidar ratios associated with fine-mode aerosol at 1064 nm are better captured by the IFS composition-based estimate.

In summary, the fixed lidar ratio of 50 sr provides robust results across all sites and aerosol regimes, with the lowest overall RMSE for the full dataset. Under coarse-mode dominated conditions, the fixed lidar ratio of 50 sr tends to underestimate AOD while under fine-mode dominated conditions it shows a small positive bias, reflecting the inherent limitation of a single fixed value across diverse aerosol regimes. The IFS-CAMS derived lidar ratio reduces these regime-dependent deviations, particularly under coarse-mode dominated conditions at Granada and Payerne, where it achieves near-zero mean deviations (-0.006±0.053 and 0.018±0.045 respectively) and RMSE reductions of up to 38 % compared to the fixed lidar ratio of 50 sr. At Lindenberg under coarse-mode conditions, no clear advantage of IFS–CAMS–derived lidar ratio over the fixed lidar ratio of 50 sr is observed, though this result should be interpreted with caution given the limited number of cases (N=63). For the backward method, the lidar ratio choice has a secondary impact compared to the reference altitude selection, which remains the dominant source of uncertainty under high aerosol load conditions regardless of the assumed lidar ratio value.