Satellite retrievals approximate the microphysics of clouds by relying on an assumed droplet size distribution. The droplet size distribution is typically described by a modified gamma distribution defined by , and is given by: 1n(r)=Ar1-3veffveffexp⁡-rreveff, where n(r) is the number of droplets of radius r and A is a normalisation constant. This distribution is characterised by two parameters: effective radius (re) and the effective variance (veff). Physically, re is defined as the area-weighted mean radius of the droplet population: 2re=∫0∞r3n(r)dr∫0∞r2n(r)dr, whereas veff is a dimensionless parameter representing the variance of the droplet size distribution normalised by re2: 3veff=1Gre2∫0∞(r-re)2πr2n(r)dr, where G is the total geometric cross-sectional area per unit volume. While re is explicitly retrieved from short-wavelength infrared reflectance, veff must be assumed , with this study setting this at a constant value veff=0.10 to align with the operational assumption employed by the MODIS Collection 6 retrieval algorithm for liquid clouds. This fixed assumption introduces a systematic bias, as the veff of a cloud varies significantly with aerosol loading , entrainment , and precipitation onset . This is especially problematic when retrieving polluted clouds, as the relationship between the radiative properties and Nd is dependent on the dispersion parameter, k, which relates the volume mean radius (rv) to re : 4k=rvre3=(1-veff)(1-2veff) As veff reduces, k increases, ultimately approaching unity. From Eq. () it is clear that by fixing veff, k is also essentially fixed. This has implications for deriving Nd, typically calculated via the adiabatic assumption : 5Nd=3⋅LWP4πρwkre3H. Here H=500 m is the fixed geometric cloud thickness, used consistently in both cloud generation and Nd derivation to convert LWP to mean liquid water content (LWC = LWP/H). If the true k of polluted clouds exceeds the assumed k in Eq. (), the retrieval will systematically overestimate Nd. In contrast, LWP is relatively insensitive to the veff assumption. LWP is formulated as 6LWP=23ρwτre and is thus linearly dependent on re and optical depth (τ) only. Therefore, LWP is only affected by secondary biases through re, and not directly biased by the veff assumption. As such, it is expected that LWP will be retrieved more robustly. Under the adiabatic assumption, Eqs. () and () are mutually consistent: substituting Eq. () into Eq. () recovers Nd=τ/(2πkre2H). As a sensitivity comparison, the legacy pre-Collection 6 assumption of veff=0.13 is also evaluated throughout this study.

The retrieval of τ and re is commonly performed using the bi-spectral reflectance method set out by . This approach exploits the differing radiative properties of cloud droplets across the solar spectrum. A non-absorbing near-infrared band (0.86 µm) is used to obtain τ, as it is primarily sensitive to the cloud's total scattering cross section. For re, an absorbing shortwave infrared band (2.1 µm) is employed, due to the sensitivity of liquid water absorption to droplet radius.

The fixed veff assumption leads to a discrepancy between the scattering phase function assumed in the look-up-table generation and the true scattering behaviour of the observed cloud. The physical origin of this discrepancy lies in the calculation of the cloud's bulk scattering phase function, P‾(θ,λ), where θ is the scattering angle and λ is the wavelength. This function is derived by integrating the single-droplet Mie scattering phase function, P(θ,r,λ), over the entire cloud droplet size distribution, n(r), described in Eq. (). The bulk scattering phase function is calculated as : 7P‾(θ,λ)=∫0∞σscat(r,λ)⋅P(θ,r,λ)⋅n(r)dr∫0∞σscat(r,λ)⋅n(r)dr, where σscat is the scattering cross-section.

As shown in Eq. (), n(r) is a function of both re and veff, the resulting bulk scattering phase function, P‾(θ,λ), is also dependent on both parameters. This is illustrated in Fig. , which compares the bulk scattering phase function calculated across the plausible range of assumed retrieval veff (0.10, shown here, and 0.13) against the true polluted veff=0.05; the discrepancy scales with the assumed–true mismatch, growing larger as the assumed value moves further from the true polluted state. Across most scattering angles (20–140°) the percentage error between the two phase functions remains below 5 %, but it grows sharply toward the forward-scattering and backscattering limits, reaching approximately +31% near Θ≈5° and +18% near Θ≈175°, where diffraction and glory features are most sensitive to the underlying droplet size distribution. This indicates that an algorithm with a fixed veff encountering a cloud with a different veff will assume a biased bulk phase function, thus misinterpreting the backscattered light intensity. This may lead to a systematic retrieval error.

A population of 5000 clean clouds is generated with properties randomly sampled from ranges of typical non-drizzling marine stratocumulus clouds , with re in the range 10–18 µm and τ in the range 8–30. This sample size ensures dense coverage of the operational lookup table (LUT) space. The veff for these clouds is assigned one of veff=[0.1,0.13,0.15], a standard range for such clouds . For each clean case, a corresponding “polluted” case (representing a ship track) is generated by reducing re by a random factor between 15 %–35 % while adjusting τ in tandem to conserve LWP. This ensures that any change in radiative properties between the clean and polluted pair is due solely to the redistribution of water among a larger number of droplets. This is done to isolate the instantaneous Twomey effect from the adjustments. The polluted clouds are assigned a fixed veff=0.05 representing a strongly narrowed droplet size distribution as a lower bound to explore the maximum potential bias. While marine stratocumulus clouds generally exhibit broader distributions, the compilation of in-situ measurements by demonstrates that polluted air masses frequently produce highly narrowed distributions, including numerous recorded instances of the veff dropping below this bound. This choice is further supported by the empirical k-Nd relationship derived from aircraft observations by : at a typical ship-track droplet concentration of Nd≈100cm-3, their Combined Fit gives k≈0.84, which corresponds via Eq. () to veff≈0.05, with heavier pollution implying still narrower distributions.

Radiative transfer simulations are run using the DISORT algorithm in the libRadtran package driven by the pyLRT python wrapper for the 5000 clean/polluted cloud pairs. For each case, the top-of-atmosphere (TOA) reflectance function R is calculated following the definition of : 8R(τc;μ,μ0,ϕ)=πI(0,-μ,ϕ)μ0F0, where I(0,-μ,ϕ) is the reflected intensity at the TOA, μ is the cosine of the viewing zenith angle, ϕ is the relative azimuth angle, μ0 is the cosine of the solar zenith angle, and F0 is the incident solar flux density. The simulations are performed for two standard MODIS bands centred at 0.86 and 2.1 µm, across a range of solar zenith angles (20, 40, 60°). Ranges of viewing zenith angle (0, 20, 40, 60°) and the relative azimuth angle (0, 45, 90, 135, 180°) are also chosen, to represent the full combination of observation geometries. The scattering angle, Θ, is related to the solar zenith angle (θ0), the viewing zenith angle (θv), and the relative azimuth angle (ϕ) by the spherical law of cosines: 9cos⁡Θ=-cos⁡θ0cos⁡θv+sin⁡θ0sin⁡θvcos⁡ϕ. The clouds are modelled as single, vertically homogeneous 1D plane-parallel slabs with a geometric thickness of 500 m, a reasonable value for marine stratocumulus . The microphysical properties are determined using pre-calculated lookup tables generated using the libRadtran Mie tool . For these calculations, the cloud droplet size distribution is modelled using the modified gamma distribution defined in Eq. (). Within the libRadtran Mie tool, the shape of this distribution is dictated by the parameter α, which corresponds to the exponent of the r term in Eq. () and is related to veff by : 10α=1-3veffveff Each lookup table contains the bulk optical properties required to solve the radiative transfer equation: the extinction efficiency, single scattering albedo and the angular scattering phase function over a grid of re and wavelengths. Table  summarises the microphysical properties and viewing geometries used to generate the synthetic cloud population.

The simulated retrieval of cloud properties from the simulated reflectances mimics MODIS retrievals in two spectral bands . This is achieved by mapping TOA reflectances to re and τ onto a Nakajima-King grid . For the simulated retrieval, the grid in Fig.  is generated using the assumed veff of 0.10. The retrieval LUT spans effective radii from 2.0 to 20.0 µm in steps of 0.1 µm, and optical depths from 0.5 to 61, with a finer spacing of 0.25 between τ=0.5 and τ=10 to resolve the non-linear curvature of the Nakajima–King retrieval surface and coarser steps of 1.0 for τ>10 where reflectance saturates, following the variable-resolution LUT design used operationally for other bi-spectral retrievals . The LUT covers the same angle grid used for the forward simulations (SZA: 20, 40, 60°; VZA: 0–60°; RAZ: 0–180°). Cloud properties are retrieved by trilinear interpolation across viewing geometry, followed by linear interpolation within the Delaunay triangulation of the two-dimensional reflectance grid at each geometry node. The fixed veff assumption creates a systematic physical inconsistency when the veff of clouds varies, especially in cases such as ship tracks where veff can undergo a dramatic reduction . The percentage bias is calculated as the relative difference between the properties retrieved using the assumed veff=0.10 and the true properties used to generate each synthetic cloud. This study adopts veff=0.10 as its central retrieval baseline, in line with the current MODIS Collection 6 algorithm, which was revised from a previous value of 0.13 to better align with multi-angle observations . The legacy veff=0.13 assumption is retained throughout as a secondary sensitivity comparison, since it maximises the contrast between the assumed and true polluted state veff (0.05) and thus provides an upper-bound estimate for the retrieval bias relative to current operational algorithms. Importantly, this fixed-variance bias is bidirectional; while both the 0.10 and 0.13 baselines systematically bias the narrowed polluted clouds, tuning an algorithm to assume the narrowed 0.05 state would invert the problem, introducing a bias into the retrieval of the unpolluted background clouds.

The retrieval bias is first quantified for the directly retrieved variables, re and τ, using the central veff=0.10 retrieval baseline. Figure  displays the percentage bias retrieved re and τ as a function of the true re and τ, respectively. The two regimes, clean and polluted, are clearly separated by bias magnitude in re, but negligible bias is shown for τ in both regimes. For τ, the mean percentage bias in clean clouds is 0.07 % and in polluted clouds -0.15%. It is shown in Fig.  that optically thin clouds have a wider bias distribution in both regimes, with potentially compensating errors causing mean bias to be small. For re clean clouds, bias is negligible, with three distinct clusters.

Figure  demonstrates that the data points are clustered by veff, with the retrieval baseline 0.10 clustered around the 0 % bias line and 0.13 and 0.15 biases calculated as -1.05% and -1.56%, respectively. Thus, the mean bias in the retrieved re of clean clouds is calculated to be -0.87% (averaged over the synthetic population as defined in Sect. 2.3). Polluted clouds, however, show a more substantial bias percentage with a mean bias of 2.30 %. This occurs because a polluted cloud transition is assumed to involve a simultaneous increase in τ to maintain a constant LWP as re decreases. The discrepancy in the true veff state explains why the bias profiles for the clean and polluted regimes remain distinct, even when compared to the same re value. As a sensitivity comparison, adopting the legacy veff=0.13 baseline instead nearly doubles the polluted re bias to 3.43 %, since the contrast with the true polluted state (veff=0.05) is larger. It is shown in Fig. b that the specific veff value has little effect on τ bias, which is to be expected given τ is known to be insensitive to the shape of the droplet size distribution.

The sensitivity of the bias to SZA in the polluted re regime is further investigated, as shown in Fig. . In Fig. a the clean clouds are shown coloured by three chosen values of SZA, 20, 40, and 60°. The data shows that for clean clouds, the mean bias changes by 0.09 % as the sun moves from 20 to 60°. Thus, it is concluded that there is no significant contribution to the bias from SZA. Figure b, showing the polluted regime, however, displays more separation due to SZA. Analysing the contribution to the mean re of each SZA value shows this is a secondary contribution to the overall bias, with a mean bias of 2.47 %, 2.41 %, and 2.00 % for SZA values 20, 40, and 60° respectively. This small SZA dependence arises from the change in scattering angle observed by the satellite, changing the discrepancy between the assumed and true phase functions. This demonstrates that lower SZA have an increased bias over higher SZA, but the mean effect is quite small and can be ignored in the context of the veff bias.

The sensitivity of re to Viewing Zenith Angle (VZA) and Relative Azimuth Angle (RAA) in the polluted regime is demonstrated in Fig. . It is found that the overestimation of re is consistent at approximately 2.4% at all VZA, aside from VZA =60° at 1.84 %. It is important to note that while the mean bias shows a small decrease at VZA =60° the precision degrades, with the standard deviation increasing to 4.97 % from 0.8 %–2.1 % for the other geometries. The bias shows a small dependence on RAA, ranging from 2.01 %–2.64 %. The precision of the retrievals is stable across most azimuths, although the standard deviation increases to 5.97 % at RAA =180°, coinciding with the backscatter geometries (Θ→180°) discussed further in Sect. 3.2; RAA remains a secondary driver of bias compared to the veff induced retrieval bias.

As discussed in Sect. 2.1, LWP and Nd products are derived from the directly retrieved products. The errors in re and τ propagate to the derived variables LWP and Nd and are amplified or dampened depending on their specific functional form. LWP is calculated using the linear relationship shown in Eq. (). Due to the linear relationship between re and τ in LWP, the bias is of a similar magnitude to re but is slightly damped due to the small negative error in τ. Figure a illustrates the distribution of bias against the true LWP amount for clean and polluted clouds. The non-linearity in re seen in Fig. a is lost through the propagation, with the distribution appearing unstructured. The mean bias in the polluted regime is calculated to be 2.15 %, slightly less than the re bias due to the small compensating bias in τ. This indicates that LWP is relatively robustly retrieved with respect to uncertainties in veff, with the wrongly assumed droplet size distribution having a small effect on bulk water content. The response of Nd, however, is much more severe due to its direct dependence on k defined in Eq. (), which is, in turn, a function of veff. Nd is proportional to re-3, meaning that the overestimation of re exerts a negative pressure on the derived Nd, causing an underestimation of the concentration. Nd is inversely proportional to the k parameter. The baseline retrieval assumes a distribution of 0.10 corresponding to a lower value of k (kassumed≈0.72). However, the true polluted cloud has a narrower distribution, corresponding to a k value closer to unity (ktrue≈0.86). The artificially small kassumed causes the algorithm to overestimate the concentration of droplets due to the inverse dependence on k. This overestimation is illustrated in Fig. b. A clear separation is seen between the clean and the polluted regime, with a substantial overestimation being seen in the polluted cases. This overestimation is calculated to be a mean bias of 13.91 %, a large error. It is noted that the standard deviation of this bias is quite large at ±22.14, showing that the precision is relatively low, with the possibility of excessively high bias in some Nd retrievals. As a sensitivity comparison, adopting the legacy veff=0.13 baseline instead produces a substantially larger polluted Nd bias of 24.56 % (±23.75%), since the contrast with the true polluted state (veff=0.05) is greater; the central veff=0.10 baseline therefore roughly halves, but does not eliminate, the Nd overestimation relative to the legacy assumption.

Separation can also be seen in the clean regime of Fig. b. Figure c displays Nd bias coloured by veff, and illustrates that there is not only substantive bias in the polluted regime, but also clean regime veff values that do not align with the algorithm assumption. Clouds with a veff of 0.13 have their droplet number concentration underestimated by an average of 8.26 %, whereas clouds with a veff of 0.15 are underestimated by 14.51 %. This not only shows that fresh ship tracks with a very narrow distribution have their Nd significantly overestimated, but that clean clouds broader than the assumed baseline have their Nd correspondingly underestimated. Evaluating this range of veff values allows for a bounding of the bias associated with the fixed veff assumption. Although this bias range is substantial, this indicates that Nd is highly sensitive to discrepancies between the assumed and true veff values.

Further analysis of the Nd retrieval bias reveals the scattering angle (Θ) has a strong influence on the bias variance. For the vast majority of the simulated polluted clouds, the Nd overestimation varies between a minimum of +11.8% (Θ≈152°) and a maximum of +36.6% at the backscatter peak (Θ=180°). At this geometry, the re bias drops to -0.30%, removing the compensating bias effect of the re-3 term and maximising the error driven by k. Outlier analysis indicates that at extreme swath-edge geometries in the forward-scattering regime (e.g., SZA =60°, VZA =60°, Θ=60°), the retrieval algorithm becomes highly unstable. At these specific angles the algorithm occasionally underestimates re by over 70%, which due to the inverse-cube dependence, yields overestimations of Nd exceeding 1200%. These extreme outliers are rare: across the full synthetic population of ∼ 10 000 cases, approximately 0.01 % (one case) exceeds the 1200% threshold, and excluding it shifts the mean polluted Nd bias by less than 0.3 percentage points (from 13.91 % to 13.63 %), confirming the mean estimate is robust to outlier removal. Nonetheless, the single outlier and the broader heavy tail drive the standard deviation to ±22.14%. Standard quality-control filters on retrieved re and τ do not remove this case, as it falls within physically plausible retrieval bounds. Dedicated geometric filtering of extreme forward-scattering angles is required. These geometries are not confined to the synthetic experiment. MODIS views up to viewing zenith angles of approximately 60–67° at swath edge. Combined with solar zenith angles around 60°, common outside near-noon overpasses, this range includes the forward-scattering geometry identified here. A practical filter would flag retrievals with both high viewing zenith angle and low scattering angle, matching the specific combination shown above to produce the largest bias.

Table  summarises the bias statistics for each variable with their standard deviations, for both the central veff=0.10 baseline and the legacy veff=0.13 sensitivity comparison.

The results above compare only two discrete points on the assumed-veff axis: the current MODIS Collection 6 baseline (0.10) and the legacy pre-Collection 6 assumption (0.13). To characterise how the bias scales continuously as the assumed veff approaches the true polluted state (veff=0.05), the retrieval is repeated for an intermediate assumption of veff=0.07, using the same synthetic cloud population and viewing geometry as the 0.10 and 0.13 cases. Figure a shows that all four bias metrics decrease monotonically as the assumed veff is reduced toward the true polluted value: the polluted Nd bias falls from 24.56 % (veff=0.13) to 13.91 % (veff=0.10) to 7.31 % (veff=0.07), while re and LWP bias fall from approximately 3.4 % and 3.2 % to 0.4 % and 0.4 %, respectively, over the same range. This confirms that the magnitude of the fixed-veff bias scales with the discrepancy between the assumed and true polluted veff, rather than being an artefact specific to either baseline examined above. Notably, Fig. b shows that the raw standard deviation of the Nd bias does not fall monotonically alongside the mean: it increases from ±22.14% (veff=0.10) to ±35.90% (veff=0.07) despite the smaller mean bias. This increase is driven almost entirely by the heavy tail of extreme-geometry cases discussed in Sect. 3.2, rather than by a broad loss of precision: when the 1st–99th percentile core of the distribution is considered, the spread in fact narrows monotonically, with standard deviations of 1.7 %, 2.5 %, and 3.4 % for assumed veff of 0.07, 0.10, and 0.13, respectively. The amplification of the tail is consistent with k(veff) becoming increasingly steep at low veff (Eq. ), such that the small number of severely biased re retrievals at extreme forward-scattering geometries translate into proportionally larger fluctuations in k, and hence Nd, as the assumed distribution narrows. Table  summarises the sweep.

The analyses above, in common with the polluted-cloud population as a whole, assign every polluted case a fixed true veff=0.05, regardless of its true Nd. This is a deliberate idealisation that isolates the retrieval bias from any assumption about how strongly droplet dispersion and concentration actually covary in nature, but it also means the reported bias magnitudes implicitly assume a perfect, noise-free correspondence between a cloud's true Nd and its true veff. Empirical k-Nd relationships derived from aircraft data, including the fit underlying , are subject to considerable scatter about their central curve, and the strength of this coupling may be weaker than the fitted relationship alone would suggest. It is therefore necessary to investigate how robust the reported bias magnitudes are if the true veff-Nd coupling is much noisier than the deterministic case assumed above. The strength of this coupling may also depend on scale. This study's paired ship-track construction implicitly assumes that factors other than the aerosol perturbation are held fixed between the clean and polluted cases. If this assumption does not hold in practice, for example if meteorology or background aerosol still varies somewhat within a pair, the weak aggregate k-Nd correlation seen in aircraft data may also be relevant at the individual ship-track scale, not only as a diluted, large-scale artifact. This motivates testing the weaker coupling case as a plausible scenario rather than only as a conservative bound.

To test this, the fixed true veff=0.05 assigned to each polluted case is replaced with a true k drawn from the Combined Fit curve evaluated at that case's true Nd, plus Gaussian scatter calibrated so that the resulting correlation between true k and true Nd falls to r≈0.10. This value is chosen as a bounding case rather than derived from a specific dataset; the qualitative conclusion below is not sensitive to the precise value chosen, only to the fact that the coupling is much weaker than deterministic. The scatter is clipped to a physically plausible range for cloud droplet spectra (veff≲0.3) before recomputing the true Nd from the true re and τ. This procedure holds the retrieved quantities (re, τ, and hence Nd retrieved) fixed at their values from the original forward-simulated retrieval, since these depend on the true top-of-atmosphere reflectances, which were generated at veff=0.05 and are not recomputed here. The result below should therefore be read as a bound on how sensitive the reported bias magnitudes are to the assumed strength of the veff-Nd coupling, rather than as a fully independent re-simulation.

Table  compares the deterministic-truth bias statistics already reported in Table  against this weak-coupling case (r≈0.10, mean noisy true veff≈0.08, standard deviation ≈0.08). The mean polluted Nd bias falls substantially under the weaker coupling relative to the deterministic case at every retrieval baseline (13.91 % to 3.80 % at veff=0.10; 24.56 % to 13.62 % at veff=0.13), and the standard deviation increases by roughly 50 %–100 % at every baseline. The correction, which reduced the deterministic-truth bias to near zero (Sect. 4), instead systematically overcorrects by 5–11 percentage points under the weaker coupling. The apparent success in the deterministic case follows, in part, from testing the correction against a truth constructed from the same curve it corrects toward. This indicates that the polluted Nd bias magnitudes reported throughout this study, and the apparent effectiveness of the correction, are best interpreted as an upper bound that assumes a tight coupling between droplet dispersion and concentration. If the true coupling is closer to the weaker correlation tested here, both the bias and the benefit of correcting for it are smaller and less certain than the deterministic case alone would suggest.

The results presented here are derived from a controlled, idealised sensitivity experiment isolating the effect of the fixed veff assumption. Several simplifications constrain the extent to which the specific bias magnitudes can be generalised to operational retrievals. First, the clouds are modelled as single-layer, vertically homogeneous, plane-parallel slabs with a fixed geometric thickness of H=500 m. Real marine stratocumulus and ship-track clouds exhibit vertical structure in liquid water content and droplet size, as well as variable geometric thickness. Both are known to introduce additional retrieval biases beyond those considered here . The sensitivity of the reported bias magnitudes to H has not been tested.

Second, this study does not reproduce the full operational MODIS retrieval chain, including cloud masking, multilayer and partly-cloudy pixel flags, and other quality-control logic. The experiment is deliberately restricted to the bias introduced by the veff assumption alone, isolated from these other sources of operational uncertainty.

Consequently, the reported bias magnitudes should be interpreted as a bound on the sensitivity to this one assumption under idealised conditions, rather than as a prediction of the net bias in any specific operational Nd product. In an operational setting this bias would act alongside other established uncertainty sources in bi-spectral retrievals, including three-dimensional radiative effects, sub-pixel heterogeneity, and instrument calibration . A quantitative intercomparison of these contributions is beyond the scope of a single-assumption sensitivity test. The magnitude of the Nd biases reported here suggests that the fixed-veff assumption warrants consideration alongside these established sources, particularly in ship-track studies where the clean-to-polluted contrast in veff is largest. While no published study isolates the fixed-veff contribution to retrieval bias specifically for ship tracks, two recent aircraft-based evaluations provide broader context. find no significant bias in MODIS-retrieved re against in-situ measurements across nondrizzling-to-drizzling marine stratocumulus, consistent with the near-zero clean-regime re bias found here. report Nd discrepancies of ±50% or more between MODIS and aircraft measurements in marine stratocumulus, indicating that the veff-driven Nd bias identified here (14 %–25 %) is plausibly a substantial, but not dominant, contributor to the total observed retrieval uncertainty.