Currently operational algorithms for satellite retrievals of cloud optical thickness (COT) and droplet size are based on bispectral measurements in absorbing and non-absorbing bands (Nakajima and King, 1990). Such algorithms are used, e.g., to generate data products based on the Moderate Resolution Imaging Spectroradiometer (MODIS) measurements (Platnick et al., 2003). Similar algorithm (along with its modified version relying on cloud droplet size derived from polarized reflectances) is also applied to measurements made by the airborne Research Scanning Polarimeter (RSP). These retrieval techniques have certain limitations owing to the influence of 3D structure of clouds (especially broken) on the reflected solar radiation, which is not accounted for in the look-up tables (LUTs) based on 1D radiative transfer models (Marshak et al., 2006; Zinner et al., 2010). The satellite data survey by Di Girolamo et al. (2010) found that the retrieved cloud reflectance is consistent with plane-parallel RT model only in 24% of cases, mostly limited to regions dominated by stratiform clouds at solar zenith angles less than 60° (for other regions or solar angles this frequency drops sharply to as low as a few percent). 3D radiative effects, such as escape of light through cloud sides and shadowing, can cause substantial underestimation of COT by 1D-RT-based retrieval techniques. Zhang and Platnick (2011) also reported significant overestimation of droplet size by the MODIS bispectral algorithm.

Correct COT is also important because of its role in calibration of the extinction coefficient fields computed using the tomographic technique designed by Alexandrov et al. (2021). The subjects of tomographic retrievals are isolated clouds such as Cu or CuCg (Tcu), which radiative properties are significantly impacted by 3D effects.

In this study (Part I of the series) we introduce the concept of linear correction of COTs of single-layer clouds retrieved using LUT based on 1D RT computations. This correction is simply multiplication of COT value by a renormalization factor depending on the cloud aspect ratio. We also present validation of this correction method based on 3D RT simulations. And finally, we provide theoretical justification of our technique based on a simple geometrical model, which allows for estimation of the effect of radiation escape through cloud sides on the cloud-top reflectance and subsequently on COT retrievals. We also extend this model designed for isolated cloud case to that for broken cloud field. In the upcoming Part II of the series we will present a comprehensive COT renormalization theory accounting for the relationship between reflectance and COT. We will also derive correction factors for essentially 3D (close to isotropic) scattering.

Cairns et al. (2000) suggested a renormalization of cloud optical properties allowing to use plane-parallel computations in the case of non-homogeneous cloud with horizontally varying droplet number concentration N c x = N c ̄ + ξ x . (1) Here N c ̄ is the mean concentration in the cloud and ξ(x) is the (zero-mean) fluctuation (x is the spatial coordinate). This renormalization means that accurate radiation fields for an inhomogeneous cloud still can be computed using 1D plane-parallel RT model (implying horizontal homogeneity of the cloud layer), but with adjusted mean cloud optical parameters (extinction cross-section, single scattering albedo, and asymmetry parameter). According to this methodology, the actual extinction cross-section σ _ext should be replaced by σ ̃ ext = σ ext 1 + V , (2) in plane-parallel computations. Here V = V a r ξ N c ̄ 2 (3) is the relative variance of the droplet number concentration, which is an effective measure of the inhomogeneity of the actual cloud field. Equation 2 shows that in order to mimic cloud inhomogeneity effects in plane parallel computations the value of σ _ext should be decreased by the factor 1 + V. This results in smaller reflectances compared to those computed with the actual σ _ext.

While derived for simplification of forward RT computations in general circulation models (GCM), this result can be adopted for use in cloud remote sensing. The retrieval of the extinction in an inhomogeneous cloud using a LUT based on plane-parallel computations yields the value of σ ̃ ext instead of the actual σ _ext. However, the latter can be derived from the former using Eq. 2 if the inhomogeneity parameter V can be estimated from the measurements.

In the context of airborne and satellite remote sensing the primary measured quantity is COT τ (rather than N _c , which is a more advanced data product (Grosvenor et al., 2018)). Thus, it is convenient to replace Eqs 2, 3 with τ ren = 1 + V τ meas . (4) and V = V a r τ meas ⟨ τ meas ⟩ 2 (5) respectively (angular brackets denote the ensemble averaging). Here we denote by τ _meas the COT retrieved from the measurements using a technique based on 1D RT (e.g., bispectral), while τ _ren is the “true” (renormalized) COT value. Note that since (1 + V) is a constant factor, the value of V can be computed using the measured COT τ _meas in Eq. 5 instead of the actual τ _ren. Note that V is closely related to the reflectance-based inhomogeneity parameter H σ 2 used by Zhang and Platnick (2011). Using that V a r τ = 〈 τ 2 〉 − 〈 τ 〉 2 , (6) the correction factor in Eq. 4 can be written as 1 + V = 〈 τ meas 2 〉 〈 τ meas 〉 2 . (7)

The COT retrievals for isolated clouds are affected by 3D effects that are caused by their geometry, such as light escape from cloud sides that decreases the reflectance at cloud top and leads to underestimation of the COT. Results for the path length statistics of random walks (Blanco and Fournier, 2006) and for the searchlight problem for a 1D cloud (Marshak et al., 1995) suggest that the aspect ratio of the scattering object plays a key role in controlling the path length statistics and the loss of radiation through cloud sides. In the case of clouds, scattering of light by large cloud droplets is strongly forward-directed with radiation staying close to the solar principal plane for clouds that are not too optically thick. In this case the radiative transfer can be assumed to be close to two-dimensional.

2D shape of a single-layer cloud is characterized by the cloud’s vertical and horizontal dimensions H = max h − min h a n d L = max x − min x (8) respectively. Here h and x are vertical and horizontal coordinates of the points at the cloud’s “boundary”. This definition is applicable to any cloud shape. Then the cloud’s aspect ratio is defined as A = H L . (9)

Motivated by the results presented in Section 2 we suggest to correct the isolated cloud’s COT using the factor similar to that in Eq. 4 but with the variability parameter V replaced by the cloud’s aspect ratio A: τ ren = 1 + A τ meas . (10) The larger is A, the more radiation passes through it’s sides causing larger negative bias in τ _meas. On the other hand, a plane-parallel cloud has L → ∞ and A = 0, so the correction factor equals to unity (i.e., τ _ren = τ _meas). Physical explanation of this model is provided in Section 5 below.

In order to test the above proposition, we performed COT retrievals from simulated reflectances using the operational retrieval algorithm designed for the RSP data analysis (Alexandrov et al., 2012). This algorithm uses nadir-view reflectances and is a modification of the legacy bispectral technique (Nakajima and King, 1990). In the modified version the droplet effective radius is retrieved from the polarized reflectance at 863-nm wavelength and no absorbing spectral channels are used. For the simulated datasets we imply that the r _eff is known and take its value (12 μm) from the cloud model to which the 3D RT computations were applied. Note that we use 863-nm LUT to retrieve COT values while the simulations described below are made at 555-nm wavelength. However, this should not have any noticeable effect on the retrievals since COT in this spectral range is almost wavelength-independent.

The simulations of the RSP measurements were produced by the 3D radiative transfer model MYSTIC (Monte Carlo code for the phYSically correct Tracing of photons In Cloudy atmospheres (Mayer, 2009; Emde et al., 2010)). Two different types of simulations were used: a smaller dataset based on clouds generated using Large Eddies Simulations (LES) and a more extensive dataset based on simplified box clouds with prescribed aspect ratios and COTs.

The appearance of the factor (1 + A) in Eq. 10 can be justified using a very crude geometrical model illustrated in Figure 4. We will call it “Block Model” or BM. This is not a radiative transfer model but rather a geometrical “supplement” to 1D RT model allowing for account of 2D/3D radiative effects. In the 2D variant of this model clouds are assumed to be rectangular L × H homogeneous columns, with plane-parallel case corresponding to L → ∞. In the 3D variant clouds are rectangular parallelepipeds (cuboids). We do not specify neither the microphysical optical properties of the droplets within the cloud, nor how the incident flux I _in gets into it. There is no absorption in the model, so the total outgoing flux I _out is equal to I _in and the total cloud albedo is Ω = I out I in = 1 . (12)

The main assumption of this model is that outgoing radiation is uniformly distributed along the cloud’s surface. Then the reflectance per unit boundary length/area is R = I out / S I in = Ω S = 1 S , (13) where S is the total perimeter/surface of the cloud. This is the quantity that is measured by a narrow-view nadir-looking sensor located above the cloud.

Due to radiation escape through the cloud’s sides, the measured reflectance R _meas = R is smaller than that from the infinite slab with the same COT. Thus, the COT τ _meas derived from R _meas using 1D-RT LUT will be also smaller than the actual one. This means that in order to facilitate accurate COT retrievals, R _meas should be appropriately increased, so that its renormalized value R ren = κ R meas (14) can be used in 1D-RT-based retrieval algorithm. Here κ ≥ 1 is the renormalization factor.

While no RT computations are performed within Block Model, it allows for estimation of the factor κ by comparison of radiation fluxes escaping the cloud in 1D (plane-parallel) and in 2D/3D cases: κ = R 1 D R (15) Here R is the cloud-top reflectance Eq. 13 “measured” above the actual 2D/3D cloud and R _1D is the one that would be measured if the cloud (with the same COT) was plane-parallel.

It is important to note that (unlike Eq. 10) and Eq. 14 relates reflectances, not COTs. It converts into τ ren = κ τ meas (16) at small values of τ and R, when they are roughly proportional to each other. We will show in Part II in the framework of general renormalization theory that Eq. 16 is indeed valid for all COT values and should be used instead of Eq. 14.

As we mentioned above, scattering of light by large cloud droplets is strongly forward directed with radiation staying close to the solar principal plane for clouds that are not too optically thick meaning that the transport of radiation is close to two-dimensional. In this section we will show that the correction factor (1 + A) in Eq. 10, which was validated using 3D RT simulations, can be derived from a simple heuristic 2D geometrical model. We then extend this model (initially designed for isolated cloud) to the case of broken cloud field where neighboring clouds radiatively interact with each other. The case of 3D scattering (close to isotropic) will be presented in Part II of this series. All clouds are assumed to be single-layer so there is no radiative interaction between cloud layers in vertical dimension.

Currently global COT datasets are derived from the single-angle measurements made by satellite-based imagers (such as MODIS, VIIRS, and OCI) using look-up tables based on 1D RT computations. Our 3D RT simulations presented in this paper showed that such inversions can underestimate COT by a factor of four (or possibly higher) due to presence 3D radiative effects (primarily radiation escape from the cloud’s sides) unaccounted for in 1D RT. This is the case for isolated clouds and for those in broken cloud fields. Such large bias in COT retrievals puts their value in question and may impair assessments of cloud effects on climate. Thus, finding a way for estimation of the biases and subsequent correction of the COT retrievals is a matter of the utmost importance.

This paper is Part I of the series targeting correction of airborne and satellite COT retrievals. In this study we introduced a correction (renormalization) procedure Eq. 10 for single-layer isolated cloud that is simply multiplication of the COT retrieved using LUT based on 1D RT by the factor (1 + A), where A is the cloud’s aspect ratio (the ratio between its vertical and horizontal dimensions). Then this assumption was successfully tested on two sets of 3D RT simulations: one based on three LES-generated marine Cu clouds (Figure 1) and the other using 12 simplified box clouds with prescribed COTs and aspect ratios (two of them presented in Figure 2). The COT retrievals in both cases were made from the measurements of nadir reflectances made by simulated Research Scanning Polarimeter (RSP) using the operational algorithm designed for the real instrument. The results of these computations presented in Table 1 and especially in Figure 3 demonstrate that the factor (1 + A) is an accurate estimate of the COT retrieval bias and can be successfully used for correction of the COT values derived based on 1D-RT LUTs.

Initially the factor (1 + A) was postulated by analogy with Eq. 4 derived to account for cloud inhomgeneity in COT retrievals. However, we then justified this choice using a heuristic geometrical model (Block Model) describing radiation escape from isolated cloud in comparison with that from infinite plane-parallel cloud (see Figure 4). This model was subsequently extended to the case of broken cloud field in the presence of radiative interaction between neighboring clouds (Figure 5). The resulting correction factor is defined by Eqs 21–23. This expression still needs to be validated using 3D RT simulations. In near future we plan to perform such simulations using a set of box-cloud fields with different characteristics. The results will be reported in our subsequent publications.

A more detailed theoretical basis of COT renormalization will be presented in upcoming Part II of the series resolving some issues left unaddressed in this study. In particular, we will introduce generalized renormalization theory and explain why the correction factors derived in the Block Model approach for reflectances should be applied to COT values instead. Also, while in this paper we assumed that the scattering of light is essentially 2D (restricted to the solar principal plane, which is an adequate assumption for large cloud droplets), in Part II we will extend Block Models for more isotropic 3D scattering in both isolated cloud and broken cloud field cases.

The NASA Plankton Aerosol Cloud ocean Ecosystem (PACE) mission that launched on 8 February 2024 carries a moderate resolution imaging spectrometer, the Ocean Color Imager (OCI), and two multi-angle polarimeters (SPEXone and HARP-2) one of which (SPEXone) is also a spectrometer. The main difficulty in applying our algorithm to observational data is providing appropriate constraints, or estimates, of the cloud aspect ratio. The passive (multi-angle) spectrometer observations of broken clouds in the oxygen A-band may provide sufficient information about path length statistics to constrain the aspect ratio of the clouds (cf. Hu et al., 2022). In future work we plan to test this approach against active radar and lidar observations from the ESA EarthCARE mission when its observations coincide with PACE.