Mathematical misdefinition of adjusted cloud-fraction change limits isolation of aerosol effects

It has been proposed that aerosol effects on stratocumulus cloud fraction (CF) vary over the diurnal cycle, with nighttime processes playing a dominant role (1, 2). This is supported by a recent study of Pugsley et al. (3), in which a Lagrangian framework is used to track cloud evolution along trajectories over the Southeast Pacific, and an “adjusted” cloud-fraction change metric (ΔCF_adj) is introduced to isolate the aerosol effect. A monotonic increase of ΔCF_adj with droplet number concentration (N_d) is found with a stronger gradient at night compared with the daytime, and the ΔCF_adj gradient is attributed to aerosol-driven precipitation suppression acting most effectively at night (3). The authors further imply that the timing of marine cloud brightening deployments will be crucial for their efficacy.

However, Pugsley et al.’s (3) conclusion fails to consider the influence of meteorological confounding, a critically important factor for isolation of aerosol effects (4–6). The equation (ΔCF_adj = ΔCF_{(for a given Nd–CF bin)} – mean ΔCF_{(for the corresponding initial CF)}) in Pugsley et al. (3) is not well-defined in general because ΔCF_{(for a given Nd–CF bin)} and mean ΔCF_{(for the corresponding initial CF)} are conditioned on different sample spaces. By subtracting the mean ΔCF conditioned on the corresponding initial CF (row-wise averaging) from the ΔCF in each (N_d, CF) bin, this method (3) neglects the strong correlation between N_d and CF, such that conditioning on CF alone is insufficient to eliminate confounding when predictors are statistically dependent. Formally, if ΔCF = f(N_d, CF, Z) + ε, where Z represents meteorological covariates that simultaneously drive N_d and ΔCF, such as lower-tropospheric stability, SST gradients, and boundary layer depth (7–9), then ΔCF_adj = f − E[f | CF] + ε. Unless f is additively separable in N_d and (CF, Z), this transformation does not recover ∂f/∂N_d but instead induces apparent N_d gradients even when no causal aerosol effect exists. Within any CF bin, high-N_d and low-N_d parcels systematically sample different meteorological environments, so ΔCF_adj retains a residual gradient of meteorological origin. Here, we extend the SI framework of Pugsley et al. (3) by introducing a latent variable Z that jointly drives N_d and ΔCF, with γ = 0. We show that applying identical binning and row-wise subtraction to this zero-aerosol-effect system reproduces ΔCF_adj patterns qualitatively indistinguishable from Pugsley et al. (3): monotonically increasing sparklines, blue-to-red heatmap gradients, and a stronger nighttime signal (Figs. 1 and 2), arising because meteorological control is stronger at night (9, 10), not because of aerosol-driven precipitation suppression.

Fig. 1.

Comparison of ΔCF_adj vs. N_d between published sparkline data from Fig. 2 of Pugsley et al. (3) and synthetic data with γ = 0, for daytime (A) and nighttime (B). Synthetic sparklines are rescaled to match observed amplitude and normalized to zero at the lowest N_d bin. The synthetic model introduces a latent meteorological driver Z ~ N(0,1) that jointly controls N_d [ln(N_d) = 5 + 0.5Z + ε₁] and ΔCF [ΔCF = c₀ + βZ + δ(CFi − 0.5) + ε₃; β = 0.06 (day), 0.10 (night)], with no direct N_d effect (γ = 0). CFi depends on both Z and N_d. Noise follows Pugsley et al. (3) (σ_CF = 0.03, σ_Nd = 50 cm⁻³). ΔCF_adj retains a residual meteorological gradient. Supporting experiments [SI Eqs. S1–S3 and Fig. S3 in Pugsley et al. (3)] omit this pathway by drawing noise independently of N_d. The qualitative agreement demonstrates that the observed gradients are consistent with meteorological confounding alone.

Fig. 2.

ΔCF_adj heatmaps from the synthetic γ = 0 model (see Fig. 1 caption), computed using the identical binning and row-wise subtraction of Pugsley et al. (3). Despite the complete absence of aerosol coupling, the heatmaps show the characteristic blue-to-red gradient with increasing N_d for both daytime (A) and nighttime (B), with a stronger nighttime signal mirroring figure 2 of Pugsley et al. (3). The model was run with N = 500,000 samples; bin edges: 23 logarithmic N_d bins (10 to 600 cm⁻³) × 20 linear CF bins (0 to 1); bins with <30 samples excluded.

We strongly agree with Pugsley et al.’s (3) conclusion that stratocumulus clouds exhibit important diurnal variability. However, we believe that ΔCF_adj cannot distinguish aerosol-driven signals from meteorological confounding when shared drivers of N_d and ΔCF are present, causing the day–night asymmetry in Pugsley et al. (3) to be equally consistent with diurnal variations in meteorological sensitivity rather than aerosol-driven precipitation suppression. Robust causal attribution requires methods that explicitly condition on meteorological state, such as multivariate regression with meteorological covariates.