Abstract. Droplet concentration (N d ) retrievals from passive satellite retrievals of cloud optical depth (τ ) and effective radius (r e ) usually assume the model of an idealised cloud in which the liquid water content (LWC) increases linearly between cloud base and cloud top (i.e., at a fixed fraction of the adiabatic LWC) with a constant N d profile. Generally it is assumed that the retrieved r e value is that at the top of the cloud. In reality, barring r e retrieval biases due to cloud heterogeneity, etc., the retrieved r e is representative of that lower down in the cloud due to the vertical penetration of photons at the shortwave infra-red 5 wavelengths used to retrieve r e . This inconsistency will cause an overestimate of N d (referred to here as the "penetration depth bias"), which this paper quantifies. Here we estimate penetration depths in terms of optical depth below cloud top (dτ ) for a range of idealised modelled adiabatic clouds using bispectral retrievals and plane-parallel radiative transfer. We find a tight relationship between dτ and τ and that a 1-D relationship approximates the modelled data well. Using this relationship we find that dτ values and hence N d biases are higher for the 2.1 µm channel r e retrieval (r e2.1 ) compared to the 3.7 µm one (r e3.7 ).
10The theoretical bias in the retrieved N d is likely to be very large for optically thin clouds, nominally approaching infinity for clouds whose τ is close to the penetration depth. The relative N d bias rapidly reduces as cloud thickness increases, although still remains above 20 % for τ <19.8 and τ <7.7 for r e2.1 and r e3.7 , respectively. Californian stratocumulus regions produce fairly large overestimates due to the penetration depth bias with mean biases of 35-38 % for r e2.1 and 17-20 % for r e3.7 . For the other stratocumulus regions examined the errors are smaller (25-30 % for r e2.1 and 11-14 % for r e3.7 ). Significant time variability in the percentage errors is also found with regional mean standard deviations of 20-40 % of the regional mean percentage error for r e2.1 and 40-60 % for r e3.7 . This shows that it is important 20 to apply a daily correction to N d for the penetration depth error rather than a time-mean correction when examining daily data. We also examine the seasonal variation of the bias and find that the biases in the SE Atlantic, SE Pacific and Californian