How well can brightness temperature differences of spaceborne imagers help to detect cloud phase? A sensitivity analysis regarding cloud phase and related cloud properties

. This study investigates the sensitivity of two brightness temperature differences (BTDs) in the infrared (IR) window of the Spinning Enhanced Visible and Infrared Imager (SEVIRI) to various cloud parameters in order to understand their information content, with a focus on cloud thermodynamic phase. To this end, this study presents radiative transfer calculations, providing an overview of the relative importance of all radiatively relevant cloud parameters, including thermodynamic phase, cloud-top temperature (CTT), optical thickness ( τ ), effective radius ( R eff ), and ice crystal habit. By disentangling the roles of cloud absorption and scattering, we are able to explain the relationships of the BTDs to the cloud parameters through spectral differences in the cloud optical properties. In addition, an effect due to the nonlinear transformation from radiances to brightness temperatures contributes to the speciﬁc characteristics of the BTDs and their dependence on τ and CTT. We ﬁnd


Introduction
Passive spaceborne imagers, with their wide field of view and, in the case of geostationary satellites, high temporal resolution, allow global observations of clouds.These passive instruments typically use solar and/or infrared (IR) window channels to retrieve cloud properties.The advantage of pure IR-based retrievals is that they can be applied during both daytime and nighttime (Nasiri and Kahn, 2008;Cho et al., 2009).Such IR retrievals often use brightness temperature differences (BTDs) of IR window channels to detect clouds or retrieve cloud properties like optical thickness (τ ) or effective particle radius (R eff ) (e.g., Inoue, 1985;Krebs et al., 2007;Heidinger et al., 2010;Garnier et al., 2012;Kox et al., 2014;Vázquez-Navarro et al., 2015;Strandgren et al., 2017).
However, determining cloud parameters such as the thermodynamic phase from BTDs is a challenging task.Radiative transfer through clouds and the atmosphere is complex, with many parameters that can in principle influence satellite observations.Although radiative transfer models are capable of correctly accounting for all of these quantities, the relative importance of these parameters is often not fully understood.Ackerman et al. (1990) were the first to observe a correlation between BTDs in High-Resolution Interferometer Sounder (HIS) data and the different cloud phases as determined by concurrent lidar data.They proposed a trispectral technique to distinguish between ice, water, and clear sky using the BTDs between channels at about 8 and 11 µm ) and between channels at about 11 and 12 µm (BTD(11.0-12.0)).Strabala et al. (1994) expanded on their findings using MODIS airborne simulator data.They considered clouds of varying τ and found that distinguishing between ice and water clouds using these BTDs is difficult for optically thin clouds.Parol et al. (1991) and Dubuisson et al. (2008) studied the sensitivity of BTDs to effective radius R eff and particle shape for cirrus clouds.Parol et al. (1991) found that the BTD(11.0-12.0)for the Advanced Very High Resolution Radiometer (AVHRR) aboard the NOAA satellites is sensitive to whether cloud particles are spherical or non-spherical.Dubuisson et al. (2008) showed that the impact of different non-spherical ice crystal shapes on BTD(10.6-12.0)and ) from the Infrared Imaging Radiometer (IIR) aboard CALIPSO is small compared to their sensitivity to R eff .The effect of R eff on the BTDs was also considered by Baum et al. (2000), who further extended the trispectral method for MODIS phase retrievals by incorporating information about the horizontal variability of the BTDs.Similar to the study of Strabala et al. (1994), the radiative transfer simulations of Baum et al. (2000) primarily focused on low-level water clouds and high cirrus clouds and did not consider mid-level clouds.To bridge this gap, Nasiri and Kahn (2008) conducted a sensitivity study that also considered mid-level clouds for the MODIS .They showed that BTD(8.5-11.0) is sensitive to cloud-top height (CTH) and that this leads to limitations in the phase discrimination in the cloud temperature regime where both liquid and ice can exist.
These studies show that many different parameters influence the BTDs: cloud parameters considered in previous studies include thermodynamic phase, τ , R eff , ice crystal habit, and CTH.As outlined above, most of the studies so far, however, have each focused on only a small number of these cloud parameters; an overview of the relative impor-tance of all these cloud parameters is still missing.The influence of CTH or cloud-top temperature (CTT) on BTDs has especially not been studied in detail, with exception of Nasiri and Kahn (2008).Besides cloud parameters the amount of water vapor in the atmosphere (mainly above the clouds) also affects BTDs even in the (relatively) transparent spectral window region of 8-12 µm.This has been pointed out by several authors (Strabala et al., 1994;Nasiri and Kahn, 2008;Dubuisson et al., 2008), but the relative importance of atmospheric absorption compared to cloud parameters for BTDs has not been studied systematically.
In addition, the origin of the dependence of BTDs on cloud thermodynamic phase, as observed in satellite measurements and radiative transfer results, is not fully understood.Although phase retrievals are usually based on accurate radiative transfer calculations that take into account all radiative effects, it is argued that variations in the refractive indices of ice and water across the infrared window cause the BTDs to be sensitive to cloud phase (Finkensieper et al., 2016;Key and Intrieri, 2000;Baum et al., 2000Baum et al., , 2012)).However, besides these effects of the cloud phase, the phase also correlates with other cloud parameters like CTT and R eff , which in turn have large effects on the BTDs as mentioned above.It is not fully understood which cloud parameters dominate the response of the BTDs in given cloud scenarios.Additionally, traditional explanations of the phase dependence of BTDs have often neglected scattering effects, which as we will show can be substantial.Thus, it is not well understood which physical processes are responsible for the observed phase dependence of the BTDs.A full understanding of the satellite channel dependencies, however, is critical to designing optimal cloud (phase) retrievals and to understanding their limitations.
To compute BTDs, satellite radiances are first transformed into brightness temperatures (BTs).This transformation by means of Planck's radiation law is a nonlinear function.As nonlinear functions can lead to unexpected behavior, we expect that there are some effects of the nonlinear relationship between satellite radiances and BTs on BTDs.To our knowledge, the effect of this nonlinear relationship has not been analyzed before.
We use radiative transfer (RT) calculations to study two BTDs of the Spinning Enhanced Visible and Infrared Imager (SEVIRI) aboard Meteosat Second Generation (Schmetz et al., 2002): the BTDs between the IR window channels centered at 8.7 and 10.8 µm ) and between those centered at 10.8 and 12.0 µm (BTD(10.8-12.0)).These are the BTDs that are mainly used to identify cloud-top phase and determine (ice) cloud properties.Figure 1 shows an example scene from SEVIRI as an RGB composite and the two BTDs for the same scene.In this study we first investigate the effect of the nonlinear relationship between radiances and BTs on the BTDs.We then use the RT calculations to analyze dependencies and sensitivities of the BTDs with respect to all radiatively important cloud parameters, namely phase, CTT, R eff , ice crystal habit, and optical thickness (τ ) at 550 nm, disentangling effects of cloud particle scattering and absorption.We also consider the effect of water vapor in the atmosphere on BTDs by comparing the computed BTDs with scenarios without molecular absorption.The findings of these RT calculations are then used to analyze the information content of the BTDs with respect to cloud phase.Overall in this study we focus on the effect of cloud parameters; the effects of other parameters like viewing angle, surface emissivity, and atmospheric temperature profiles are not studied.
The aim of this study is twofold: first, it provides an analysis of the effects of all cloud parameters on the two BTDs, disentangling the interactions among the different parameters.Second, this study improves the physical understanding of the role of the different radiative processes leading to different BTD values.This helps to understand the information content of the BTDs with respect to the thermodynamic phase in order to better understand and improve the working principles of phase retrieval algorithms that use BTDs and to understand their uncertainties and limitations.We focus on the phase, but our results are also useful to better understand the dependencies of BTDs for other remote sensing applications where they are typically used, such as the retrieval of τ and R eff .Since BTDs also depend on atmospheric and surface parameters whose effects are not studied here, this study does not aim at explaining every phenomenon encountered with BTDs.However, understanding the effects of the cloud parameters helps to disentangle different physical cloud-related processes in all atmospheric or surface conditions.
Finally, we note that besides BTDs, there are other popular methods for retrieving cloud phase and other cloud properties, such as β ratios (Parol et al., 1991;Pavolonis, 2010;Heidinger et al., 2015).While this study is specifically aimed at BTDs, understanding the effects of different cloud properties on the radiative transfer through clouds is also useful to better understand the physics underlying β-ratio retrievals.

Physical background
To visualize relationships and dependencies between radiation at the top of the atmosphere (TOA) and cloud properties, the representation in the form of a causal diagram is very useful.Figure 2 shows cloud parameters that are related to the cloud phase, connected by arrows indicating causal relationships.Other factors influencing the radiation at TOA (in particular passive satellite observations), like satellite viewing angles, surface temperature, and atmospheric properties, are summarized under "other" in the diagram.
In this paper we use the terms "direct" and "indirect" influence of the cloud phase on the TOA radiation.Direct influence means the effect of changing the cloud phase while all other cloud parameters (R eff , CTT, τ , ...) remain the same (represented by the arrow from phase to TOA radiation in Fig. 2).The indirect influence of the cloud phase is represented by all other paths from phase to TOA radiation in Fig. 2. For example, the phase affects τ and R eff , which in turn affect TOA radiation.(Mayer et al., 2023).
In order to calculate the radiative transfer through a cloud with given cloud (microphysical) parameters, it is necessary to know how much radiation is absorbed, scattered, and emitted, i.e., the optical properties of the cloud.The translation from cloud (microphysical) parameters to optical parameters is given by the so-called single-scattering properties.The single-scattering properties are the volume extinction coefficient β ext , the single-scattering albedo ω 0 , and the scattering phase function p.As an alternative to β ext and ω 0 one can equivalently describe radiative transfer by the absorption coefficient β abs and scattering coefficient β sca , which can be easier to interpret.Definitions and physical interpretations of the single-scattering properties can be found in Appendix A. The interplay of the single-scattering properties, in combination with the cloud water path, determines how much radiation is transmitted through a cloud and, in combination with the cloud temperature, how much radiation is emitted from it.The single-scattering properties depend on the wavelength of the radiation and on the cloud parameters R eff , habit, and phase.They are shown in Fig. 3 for varying R eff and cloud phase.The variations of the single-scattering properties due to habit are mostly small in comparison and therefore not shown.Instead of p we show the asymmetry parameter g as a simpler measure to characterize the scattering process (see Appendix A).The spectral variations of β abs , β sca , and g translate into different BTD values for different cloud parameters.This will be investigated in detail in the next sections using radiative transfer calculations.

Radiative transfer calculations
Simulations for the three IR window channels of SEVIRI centered at 8.7, 10.8, and 12.0 µm were performed for a variety of water and ice clouds using the sophisticated radiative transfer package libRadtran (Mayer and Kylling, 2005;Emde et al., 2016;Gasteiger et al., 2014).The libRadtran package represents water and ice clouds in detail and realistically.It has been validated against observations and in several model intercomparison campaigns and has been extensively used to develop or validate remote sensing retrievals (e.g., Mayer et al., 1997;Meerkötter and Bugliaro, 2009;Bugliaro et al., 2011Bugliaro et al., , 2022;;Stap et al., 2016;Piontek et al., 2021b).The optical properties of water droplets are calculated using Mie theory.For ice crystals, we use the Baum et al. (2011) parameterization of optical properties for three different habits (general habit mixture, columns, rough aggregates).Simulations of TOA radiances for the SEVIRI IR window channels are made using the one-dimensional radiative transfer solver DISORT (Discrete Ordinate Radiative Transfer) 2.0 by Stamnes et al. (2000) and Buras et al. (2011) with parameterized SEVIRI channel response functions as described by Gasteiger et al. (2014).The complete permutation of τ , R eff , CTT and CTH, crystal habits, and phase was simulated and is listed in Table 1.The CTT is set to the atmospheric temperature at the altitude of the CTH and represents the temperature at cloud top.For simplicity we keep the cloud geometric thickness constant at 1 km; the impact of variable geometric thickness is discussed in Sect.6.3.We only consider singlephase (ice or water) and single-layered clouds.True mixedphase clouds and multilayered clouds are not considered.The simulation setup in terms of atmosphere, satellite and solar geometry, and surface type is summarized as well in Table 1.In this study, we focus on the influence of cloud parameters.Therefore, we choose a relatively simple setup for the atmospheric parameters, surface parameters, and satellite geometry, which is kept constant for all simulations.We use the US standard atmosphere (Anderson et al., 1986) and a surface temperature of 290 K.We place the simulations over the ocean where the surface emissivity is nearly constant for the three IR window channels and set it to 1.The satellite zenith angle (SATZ) is kept constant at 0°(nadir view).
To disentangle cloud effects from effects of the atmosphere, we also compute simulations with molecular absorption switched off.The libRadtran package further has the possibility of simulating the IR window channels for cloud layers for which scattering is switched off, meaning that the scattering coefficient in the simulation is set to zero while the absorption coefficient remains constant.This allows disentangling the effects of scattering and absorption in a cloud.

Effects of Planck's law: the BTD nonlinearity shift
Before analyzing the results of the RT calculations, we examine the effects of the nonlinear relationship between radiances and BTs on the BTDs.We call these effects the BTD nonlinearity shift.The BTD nonlinearity shift is purely due to the nonlinearity in the computation of BTDs and not due to wavelength-dependent optical properties of the cloud, which we will focus on in the next sections of this study.BTDs are calculated from measured radiances using Planck's radiation law, which describes the spectral radiance B λ of a black body emitting radiation at temperature T : where h is the Planck constant, c is the speed of light in a vacuum, and k B is the Boltzmann constant.The inverse Planck function accordingly maps spectral radiance R λ to the corresponding temperature, and is used to compute BTs from measured radiances in remote sensing.
The simplest version of the BTD nonlinearity shift can be explained using the Schwarzschild equation for radiative transfer.The Schwarzschild equation is a simple version of radiative transfer assuming no cloud scattering and no atmosphere.Its solution for one cloud layer is where R S TOA,λ is the radiance at TOA at a given wavelength λ with the superscript S for Schwarzschild, and τ λ is the optical thickness of the cloud for this wavelength.The first term in the equation is the transmitted radiance coming from the surface with the surface temperature T s ; the second term is the https://doi.org/10.5194/amt-17-5161-2024Atmos.Meas.Tech., 17, 5161-5185, 2024 radiation emitted by the cloud, assuming that the cloud layer has an approximately constant temperature T ≈ CTT.To demonstrate the BTD nonlinearity shift we set τ λ as equal for all wavelengths, τ λ = τ .Figure 4a shows the Planck function of the surface temperature, B λ (T s ), and the cloud temperature, B λ (CTT), in gray for exemplary values of T s = 290 K and CTT = 200 K.According to the Schwarzschild equation (Eq.3), R S TOA,λ lies between these two curves, approaching B λ (T s ) for τ → 0 and B λ (CTT) for τ → ∞. Figure 4a illustrates R S TOA,λ for τ = 0.5 (black line).From R S TOA,λ we can now compute the TOA BTs at the three IR wavelengths of interest as BT S λ = T λ (R S TOA,λ (τ )), where the superscript S again stands for Schwarzschild.The corresponding Planck curves, i.e., B λ (BT S λ ) for λ ∈ {8.7, 10.8, 12.0}, are shown in Fig. 4a as dashed colored lines.Recall that in this example calculation we have set a constant τ = 0.5, i.e., the same optical properties (transmittance and emissivity) for all wave-lengths (see Eq. 3).Naively, one might expect BTD = 0 (i.e., equal BTs) in this scenario.However, it is evident from the figure that the three BTs are different, with BT S 8.7 > BT S 10.8 > BT S 12.0 .Since these differences between the three BTs are not due to optical cloud properties, they must be caused by the nonlinear transformation from radiances to BTs.Hence, the BTD nonlinearity shift induces a BTD in situations where, naively, no BTD would be expected.
To get an overview of the BTD nonlinearity shift, we compute BTD S for both wavelength combinations (BTD(8.7-10.8)and BTD(10.8-12.0))from the results of the Schwarzschild equation (Eq. 3) for varying τ and CTT as Figure 4b shows the computed BTD S as a function of τ for different CTTs and a fixed T s = 290 K.These BTD S resem-ble an arc shape (similar to the well-known BTD arc from Inoue, 1985) and show higher values for lower CTTs, even though the amplitudes of their curves are smaller than for the full RT model, as we will see later.Thus, even if τ λ is the same for all three wavelengths, τ λ = τ , the nonlinearity of the inverse Planck function induces positive BTD S values and a dependence on the CTT.More generally, this dependence is mainly a sensitivity to the thermal contrast T = T s − CTT; however, for a fixed T s , as shown in the examples here, it reduces to a dependence on CTT.Notice that for these examples the BTD induced this way reaches up to 2.5 K and thus cannot be neglected.
In the next section we will discuss the effects of cloud properties on the BTDs due to the wavelength-dependent optical properties in the full RT model (described in Sect.3).The BTD nonlinearity shift adds to these effects and is therefore co-responsible for the (positive) BTD values and the CTT dependence of the BTDs, which we will discuss in more detail in Sect.5.6.In Appendix B we further analyze the BTD nonlinearity shift for the Schwarzschild model as well as the full RT model and disentangle this nonlinearity effect from the physical effects of wavelength-dependent optical properties on the BTDs in RT calculations.
This section can be summarized as follows.
-There is an effect (BTD nonlinearity shift) coming from the nonlinearity of the inverse Planck function that induces positive BTD(8.7-10.8)and BTD(10.8-12.0)values and a dependence on the CTT (or more generally the surface-cloud temperature contrast T ) in a simple RT model (Schwarzschild equation) even if cloud optical properties (transmittance and emissivity) are the same for all wavelengths.

Effects of cloud properties on BTDs
In this section we analyze the results of the RT calculations described in Sect.3. We start with the effects of scattering on the BTs of the three window channels separately.We then combine the BTs to BTDs and analyze them as functions of τ , phase, R eff , ice crystal habit, and CTT, focusing on the physical relationships between these cloud properties and the BTDs.In order to disentangle the effects of the different cloud parameters, we always vary only one or two parameters and keep the remaining cloud parameters at fixed "default" values, namely CTH = 6 km (corresponding to CTT = 249.2K) and R eff = 20 µm for both cloud phases and the general habit mix as the ice crystal habit.
The following conventions are used throughout this section: blue indicates the ice phase, and orange-red colors indicate the liquid phase.Solid lines represent a "normal" atmosphere with molecular absorption, and dashed lines mean that molecular absorption is switched off.

Effects of scattering on brightness temperatures
Scattering in the infrared window only needs to be considered for cloud particles; Rayleigh scattering by atmospheric molecules is negligible in the infrared window.The effects of cloud particle scattering on the BTs are shown in Fig. 5.It shows the difference between the BTs for a cloud with scattering and a cloud with scattering switched off for the three window channels, i.e., BT λ − BT no sca λ for each channel with center wavelength λ.This is shown as a function of τ (at 550 nm) for an ice and a water cloud with all other cloud parameters held constant.Switching off scattering in a cloud changes the optical thickness of that cloud, since only absorption now contributes to the extinction of radiation.However, to be able to compare scenarios with and without scattering for fixed cloud microphysics (same water content and R eff ), the τ parameter used for this figure is still the "original" optical thickness (with absorption and scattering).
All curves in Fig. 5 are negative everywhere, meaning that scattering is a radiation sink for all three wavelengths: part of the radiation coming from below the cloud is scattered back downwards.However, the amount of radiation lost to scattering is different for the different wavelengths.Scattering has a larger effect on the radiation at 8.7 µm than at 10.8 or 12.0 µm, as expected from β sca which is higher at 8.7 µm than at the other two wavelengths (see Fig. 3).For 8.7 and 12.0 µm, scattering by ice clouds is more significant than by water clouds; for 10.8 µm, scattering leads to a similar radiation loss for both water and ice clouds.Interestingly, scattering effects are visible even when the cloud is opaque (black, τ = 30).An explanation is that the observed radiance at TOA does not just come from the top of the cloud.Rather, it comes from the upper layers within the cloud (with decreasing intensity as one moves deeper into the cloud).Radiation emitted anywhere below the cloud top is still subject to scattering on its way to the cloud top.
Using different CTT or R eff values in the calculations (for both the liquid and the ice cloud) mainly changes the magnitude of the negative peaks but does not change the qualitative results shown in Fig. 5. Similarly, changing the ice crystal habit does not change the qualitative results and has only a small effect on the values shown.

Effects of optical thickness on BTDs
We begin the study of BTDs by analyzing the physical factors that drive the BTD behavior in relation to τ .Figure 6 shows BTD(8.7-10.8)and BTD(10.8-12.0)as functions of τ for both an ice and a liquid cloud and with molecular absorption switched on and off.
As τ approaches zero in all panels of Fig. 6, i.e., no cloud is simulated, the BTD curves with atmospheric absorption switched on (solid lines) do not go to zero.They remain above zero for BTD(10.8-12.0)and below zero for BTD (8.7-10.8).This is the effect of atmospheric absorption, https://doi.org/10.5194/amt-17-5161-2024Atmos.Meas.Tech., 17, 5161-5185, 2024  since radiation at 8.7 and 12.0 µm is more strongly absorbed by water vapor than at 10.8 µm: compare the curves with (solid lines) and without (dashed lines) molecular absorption for τ approaching zero.As τ increases, the curved shape of the BTD functions is (largely) due to the interplay of transmission and emission from the cloud.As discussed in Sect. 4 the BTD nonlinearity shift adds to these effects.Where trans-mission is dominant (small τ ), the spectral differences in extinction (see Fig. 3) lead to an increase in BTD values.Where emission is dominant (large τ ), BTD values are small, giving rise to the curved shape of the BTD functions (the wellknown BTD arc from Inoue, 1985).The BTD curves become constant at about τ 15.
To disentangle the effects of cloud absorption and scattering, Fig. 6a and b show the BTDs with cloud particle scattering switched off.As explained in the previous section, the τ parameter used for these figures is still the original optical thickness (with absorption and scattering).In Fig. 6c and d scattering is switched on.BTD(10.8-12.0) in Fig. 6a is positive, meaning that radiation at a wavelength of 12.0 µm is more strongly absorbed than at 10.8 µm and more radiation is transmitted through the cloud at 10.8 µm.This matches the absorption coefficient, which is higher at 12.0 than 10.8 µm (shown as an inset for the given R eff for convenience, as well as in Fig. 3).
Analogously, Fig. 6b shows that radiation at 10.8 µm is more strongly absorbed by the cloud than at 8.7 µm, especially for ice clouds.The stronger absorption at 10.8 compared to 8.7 µm can again be seen in the absorption coefficient (shown in inset and in Fig. 3).The spectral differences in the absorption coefficient are stronger between 8.7 and 10.8 than between 10.8 and 12.0 µm, leading to higher values of BTD(8.7-10.8)than BTD(10.8-12.0)(compare Fig. 6a  to b).For BTD(8.7-10.8),note that molecular absorption plays an important role even for optically thick clouds, decreasing BTD(8.7-10.8)everywhere by at least 0.5 K, since radiation at 8.7 µm is more strongly absorbed by atmospheric molecules (water vapor) than at 10.8 µm.
Switching on particle scattering (Fig. 6c), the BTD(10.8-12.0)values increase for ice clouds and stay about the same for liquid clouds.This will be further discussed in the next section (Sect.5.3).For opaque clouds (large τ ), the spectral differences in scattering effects lead to non-vanishing BTD(10.8-12.0)values for ice clouds (BTD(10.8-12.0)≈ 0.3 K).
For BTD(8.7-10.8),switching on scattering leads to a decrease, since scattering has a stronger effect at 8.7 compared to 10.8 µm (see Fig. 5).However, the increase in BTD(8.7-10.8)due to cloud absorption (Fig. 6a) outweighs this opposing scattering effect and the BTD(8.7-10.8)curve is still positive (when atmospheric absorption is not considered).Note the differences with BTD(10.8-12.0),where cloud absorption and scattering are concurrent effects, both leading to an increase in BTD(10.8-12.0).
The following list summarizes the most important results.
-Stronger absorption and scattering at 12.0 compared to 10.8 µm lead to positive values of BTD(10.8-12.0).
-These trends are consistent with expectations based on absorption and scattering coefficients.

Effects of cloud phase on BTDs
We now discuss the direct dependence of BTD(10.8-12.0)and BTD(8.7-10.8) on phase shown in Fig. 6.Direct depen-dence means that all other parameters such as R eff and CTT are held constant.BTD(10.8-12.0) in Fig. 6c has higher values for the ice phase than the liquid phase for all τ .Comparing the curves with and without scattering (Fig. 6a and c), we see that this difference between liquid and ice is mainly due to the different scattering properties of cloud particles at the two wavelengths: for liquid clouds the scattering has a similar effect at 10.8 and 12.0 µm, while for ice clouds radiation at 12.0 µm is scattered more than at 10.8 µm (see Fig. 5), leading to higher BTD(10.8-12.0)values for ice clouds.BTD(8.7-10.8)directly depends on phase only for small to moderate τ (τ 15), with higher values for ice than for liquid.This difference is due to absorption properties: the spectral difference in absorption between the two wavelengths is larger for ice clouds (see β abs in the inset of Figs.6b  and 3).Switching on cloud scattering reduces the differences between ice and liquid clouds in BTD(8.7-10.8)(compare Fig. 6b with d).The reason for this can be seen in Fig. 5: the effect of scattering at 8.7 µm is stronger for ice than for water, while it is similar for ice and for water at 10.8 µm.This leads to a stronger decrease in BTD(8.7-10.8)values for ice than for water clouds when scattering is switched on.However, overall the effect of absorption (leading to larger BTD(8.7-10.8)values for ice than for water) outweighs this contrasting scattering effect.
In summary, the most important findings are the following.
-There is a direct phase dependence of the BTDs due to the dependence of the single-scattering properties on cloud phase.
-For BTD(10.8-12.0),scattering is mainly responsible for the direct dependence on cloud phase.
-For BTD(8.7-10.8),absorption is responsible for the direct dependence on cloud phase, and scattering reduces the differences between the phases.

Effects of effective radius on BTDs
Figure 7 shows BTD(10.8-12.0)and BTD(8.7-10.8)as a function of τ and R eff for ice clouds (top row) and liquid clouds (bottom row) for the full RT model (i.e., scattering switched on).Note that the ranges of R eff values for ice and liquid clouds are different in order to simulate realistic cloud conditions.For low τ (τ 10), smaller R eff values lead to larger values for both BTDs.The effect becomes stronger in a nonlinear way as the R eff becomes smaller.This confirms previous results, for instance Dubuisson et al. (2008), who also found a strong and nonlinear dependence of BTDs on R eff .
The effect of R eff on BTD(10.8-12.0)physically results from the dependence of particle absorption on R eff : the spechttps://doi.org/10.5194/amt-17-5161-2024Atmos.Meas.Tech., 17, 5161-5185, 2024 tral differences of the absorption coefficient are larger for smaller R eff (see Fig. 3), resulting in lower transmission at 12.0 than at 10.8 µm and thus higher BTD(10.8-12.0)values for smaller R eff values.The effect of scattering on BTD(10.8-12.0) is similar for varying R eff and comparatively small (increases (decreases) the BTD by 0.5 K for ice (water) clouds).For the interested reader, Fig. C1 shows the sensitivity of both BTDs with R eff broken down into effects of absorption and scattering.
For BTD(8.7-10.8), the R eff dependence for small τ is, like the phase dependence, the result of two opposite effects: for smaller R eff , absorption increases for 10.8 compared to 8.7 µm, leading to an increase in BTD(8.7-10.8).On the other hand, scattering increases more for 8.7 than for 10.8 µm, leading to a decrease in BTD(8.7-10.8).However, the effect due to absorption is stronger and therefore the BTD(8.7-10.8)increases with decreasing R eff .Unlike BTD(10.8-12.0),BTD(8.7-10.8) is still dependent on R eff at large τ : here BTD(8.7-10.8)increases with increasing R eff , contrary to the R eff trend at small τ .The smaller the R eff , the more important this effect becomes.
The most important insights are summarized below.
-The BTDs depend strongly and nonlinearly on R eff .
-Physically this dependence is due to larger spectral differences in the absorption coefficient for smaller R eff .

Effects of ice crystal habit on BTDs
Figure 8 shows the sensitivity of the BTDs to ice crystal habits (in ice clouds).For both BTDs, rough aggregates lead to the smallest BTD values.For BTD(8.7-10.8),ice crystals with the general habit mix (ghm) lead to the largest BTD values, while for BTD(10.8-12.0),solid columns lead to slightly higher values.However, the sensitivity to ice crystal habits is relatively small ( 0.5 K) compared to other cloud properties.This confirms the results of Dubuisson et al. (2008), who showed that the habit has a small effect on BTDs compared to the effect of R eff , also for ice crystal shapes other than the ones considered here.The relative importance of different cloud parameters will be further discussed in Sect.6.

Effects of cloud-top temperature on BTDs
Figure 9 shows the sensitivity of both BTDs to CTT -and thus to CTH -for ice (top row) and liquid (bottom row) clouds.The results with molecular absorption switched off (dashed lines) show how much of this sensitivity is due to the atmosphere.Note that the CTT ranges for ice and liquid clouds are different in order to simulate realistic cloud conditions.
For BTD(8.7-10.8),molecular absorption is relevant for all τ values: clouds with high CTT, i.e., low CTH, have more absorbing atmosphere above cloud top, leading to more radiation absorbed at 8.7 compared to 10.8 µm.For BTD(10.8-12.0),this effect is less pronounced and molecular absorption is only relevant when there is a long path through the atmosphere (i.e., low CTH or small τ ).
At low τ ( 10), both BTD(8.7-10.8)and BTD(10.8-12.0)show a strong dependence on CTT that is not due to molecular absorption.Since the single-scattering properties are not CTT-dependent (see Sect. 2), this CTT effect on the BTDs is also not (directly) due to spectral differences in the single-scattering properties -in contrast to the effects of the other cloud parameters discussed above.Instead, there are more subtle reasons for this effect: in Sect. 4 we found that the BTD nonlinearity shift leads to a CTT dependence of the BTDs with higher BTD values for lower CTTs even when optical cloud properties are the same for all wavelengths.This explains part of the CTT dependence in Fig. 9.In Appendix B we further discuss the BTD nonlinearity shift, also allowing wavelength-dependent optical properties.It can be shown that for the Schwarzschild BTD S , spectral differences in the extinction coefficient are scaled by the difference between the surface and the cloud-top radiance, B λ (T s ) − B λ (CTT) (see Appendix B for a detailed discussion).Hence, the effects of spectral differences in optical properties on BTD S are amplified by larger T , i.e., differences between T s and the CTT.This is the main reason (be-sides the BTD nonlinearity shift) for the CTT dependence of the BTDs.Colder CTTs (or rather larger T ) thus increase both the BTD nonlinearity shift and the effects of spectral differences in optical properties.
The following list summarizes the CTT and CTH effects on the BTDs.
-For BTD(8.7-10.8),CTH has a large effect due to molecular absorption mainly above cloud top.
-Both BTDs show a strong dependence on CTT (or more generally on T ) with higher values for lower CTTs (larger T ).
-The BTD nonlinearity shift is co-responsible for the positive BTD values and the CTT (or T ) dependence of the BTDs, adding to the effects stemming from spectral differences in absorption and scattering properties.

Implications for phase retrievals
In the last section we analyzed the effects of cloud properties on the BTDs individually by varying only one cloud property at a time (besides τ ).In this section we combine the phase-related cloud parameters τ , R eff , ice habit, CTT, and thermodynamic phase for a sensitivity analysis of the BTDs.
From this analysis we determine typical BTD ranges for ice and liquid clouds and understand which cloud parameters are responsible for the phase information contained in the BTDs.
We analyze for which cloud scenarios we can distinguish between liquid clouds and ice clouds and when they overlap, allowing us to derive implications for phase retrievals.First, in Sect.6.For the R eff boundaries we select the upper and lower limits of all computed R eff scenarios (see Table 1).Additionally, as liquid clouds rarely have τ < 5, these values are omitted, since we focus for this sensitivity analysis on typical cloud scenarios.For ice clouds, different habits are shown as different markers.
Hence, the cloud parameters in Fig. 10 are chosen such that the majority of midlatitude cloud events for each phase lie between the very bottom and top blue curves for ice and the very bottom and top orange-red curves for liquid.
To verify that the computed ranges of BTD values are realistic, we compare the RT results with measured SEVIRI data using cloud-phase information from DARDAR.More details on this comparison and its results can be found in Appendix D. We find good agreement between the RT results and the measured SEVIRI data and conclude that the results of the RT calculations are realistic.
In Fig. 10 BTD(10.8-12.0)shows the highest sensitivity to τ , CTT, and R eff .shows the highest sensitivity to τ , CTT, and molecular absorption (closely linked to CTH).In comparison to τ , CTT, and CTH, the sensitivity to R eff is lower for BTD(8.7-10.8)and mainly relevant for small CTT.For both BTDs, the direct sensitivity to cloud phase, i.e., holding all other cloud parameters constant, plays mostly only a minor role: for BTD(10.8-12.0) the direct phase dependence is of the order of 0.5-1.5 K; for BTD(8.7-10.8) the direct influence of phase is only significant for small τ values ( 10) and then of the order of 1-2 K (see Sect. 5.3).
For a phase retrieval we need to know for which cloud properties liquid and ice clouds overlap and where they separate for both BTDs.The largest BTD(10.8-12.0)values in the typical cloud scenarios (about 2.5 to 5 K in Fig. 10) are only observed for optically thin and cold ice clouds with small R eff .Thus BTD(10.8-12.0) is useful to detect cirrus clouds, especially if they have small R eff (like contrails), and classify them as ice in a phase retrieval.However, our calculations show that certain liquid cloud scenarios with exceptionally low R eff and cold CTTs can also induce remarkably high BTD(10.8-12.0).This can lead to misclassification of these liquid clouds as ice.However, most liquid clouds have lower BTD(10.8-12.0),below about 2.5 K in Fig. 10.Since such low BTD(10.8-12.0)may also indicate ice clouds with "warm" CTTs and/or large R eff , or ice clouds with τ close to zero, a phase classification based on BTD(10.8-12.0)alone is challenging.The lowest BTD(10.8-12.0)values (about 0 to 1 K in Fig. 10) indicate optically thick clouds but otherwise do not contain much phase information.
As for BTD(10.8-12.0),large BTD(8.7-10.8)(around 1 to 5.5 K in Fig. 10) can indicate ice phase, since only ice clouds with low τ of about 1 < τ < 7 reach these values.Low BTD(8.7-10.8)(lower than about −0.5 in Fig. 10) can arise from very thin ice clouds (as BTD(8.7-10.8)decreases to about −2 K as τ goes to zero) or optically thick clouds.For optically thick clouds, BTD(8.7-10.8)decreases with higher CTT (due to lower CTHs and stronger molecu-lar absorption) and smaller R eff -both characteristics typical of liquid clouds.As a general guideline for optically thick clouds, lower BTD(8.7-10.8)indicates a higher probability of a liquid cloud.Overall, the phase information contained in BTD(8.7-10.8)originates mainly from its sensitivity to CTT for clouds with τ 10, while for optically thick clouds it stems mainly from its sensitivity to molecular absorption (closely linked to CTH) and (to a lesser extent) R eff .Only in cases of optically thin clouds (τ 10) is the phase information of BTD(8.7-10.8)additionally due to the direct phase influence on the (different) absorption properties of liquid and ice particles.
The main findings are summarized below.
-The sensitivities of the BTDs are complex.
-BTD(8.7-10.8)also provides CTH and R eff information for optically thick clouds, which can be useful for phase determination.
-For BTD(10.8-12.0),typical liquid and ice clouds overlap for most cloud scenarios, with the exception of cold, thin ice clouds.For BTD(8.7-10.8),liquid and ice clouds separate better, but the BTD values of the two phases are close when CTTs (CTHs) are similar.This phase separation is mainly due to the sensitivity of BTD(8.7-10.8) to CTT and CTH.

Sensitivity analysis for the combination of both BTDs
We perform a similar sensitivity analysis as in the last section for the combination of both BTDs.As in Fig. 10, Fig. 11 shows the BTDs for the same upper and lower boundaries of CTT and R eff but in the space spanned by BTD(8.7-10.8)and BTD(10.8-12.0).Along each line, τ increases from 0 to 30.To make the shape of the curves easier to understand, here liquid clouds with τ < 5 are also shown (in contrast to Fig. 10).
Figure 11 shows that the combined knowledge of both BTD(8.7-10.8)and BTD(10.8-12.0)leads to a better phase classification than considering BTD(8.7-10.8)and BTD(10.8-12.0)individually.For instance, liquid clouds at cold CTTs and small R eff (orange dotted line) separate from ice clouds in Fig. 11 as long as τ is not too large ( 10).In contrast, the same cloud scenario overlaps with ice cloud scenarios when only BTD(8.7-10.8)or BTD(10.8-12.0) is considered individually (Fig. 10).
In order to better showcase the range of BTD values for both phases and identify overlap regions, we use an additional type of plot: instead of showing only the boundary cases (as in Fig. 11), the left column of Fig. 12 shows (almost) all computed BTD values within the defined boundaries of CTT and R eff in the space spanned by the two BTDs.Only optically thick clouds (τ ≥ 10) with very low (< 233 K) or very high (> 273 K) CTTs are removed, i.e., the clouds that are easily categorized as liquid or ice by a CTT proxy such as BT 10.8 and for which a categorization by the BTDs is therefore not necessary.Liquid clouds are shown as round markers, while ice clouds are shown as crosses.The three panels in the left column of Fig. 12 vary only by their color code, which encodes τ , CTT, and R eff , respectively.They show that there is little overlap between the typical liquid and ice clouds (i.e., the clouds within the defined CTT and R eff boundaries).The only overlap is for very small τ (τ 1), since the BTDs approach the same values for all clouds, determined by atmospheric properties, as τ → 0 (best seen in Fig. 11).This means that a phase classification for typical liquid and ice cloud cases is possible in BTD(8.7-10.8)-BTD(10.8-12.0)space for τ 1 when atmospheric parameters are known.
However, Fig. 11 and the left column in Fig. 12 also show that liquid and ice BTD values are closest for clouds with similar CTTs.To further explore this issue and to test the limitations of a phase classification using the BTDs, the right column of Fig. 12 also shows BTD values for clouds outside the typical cloud boundaries.The three panels show the whole range of computed cloud scenarios (see Table 1), also including exceptionally cold liquid clouds and exceptionally warm ice clouds.Only the "easy" to distinguish cases (τ ≥ 10 and either CTT < 233 K or CTT > 273 K) are removed as before.The figures show that the overlap between liquid and ice clouds is significantly larger compared to the typical cloud cases (left column of Fig. 12).The clouds in the overlap region are mainly liquid and ice clouds which have similar CTTs in the mid-level temperature range, i.e., rather cold liquid clouds (CTT 260 K) and rather warm ice clouds (CTT 250 K).We discussed in the last section (Sect.6.1) that the CTT (and the closely linked CTH) is the most important contributor to the differences between liquid and ice clouds for both BTDs.It is therefore not surprising that phase discrimination for clouds with similar CTT is difficult even when knowledge of both BTDs is combined.Also note that additional information from BT 10.8 , which is often used as a proxy for CTT, does not help much in distinguishing between phases in these cases of mid-level CTTs.For the phase classification of these mid-level clouds, the R eff also plays a role: for R eff values that are rather large for the respective phase, the overlap occurs for all τ values; for R eff values that are rather small for the respective phase, the overlap occurs only for very small or very large τ values.
The most important results are summarized below.
-The combined use of ) and BTD(10.8-12.0) is better suited for phase discrimination than the two BTDs individually.
-The combined use of BTD(8.7-10.8)and BTD(10.8-12.0)can discriminate cloud phase for liquid and ice clouds in their typical CTT regimes as long as τ is not too small (τ 1) and when atmospheric parameters are known.
-Clouds in the mid-level CTT regime are challenging: if liquid clouds are particularly cold or ice clouds particularly warm, they often cannot be distinguished by the two BTDs.This is especially true for clouds with large R eff for the respective phase.

Sensitivity to additional cloud parameters: effects of geometric thickness and vertical R eff inhomogeneity
Cloud properties that have not been discussed so far are cloud geometric thickness and vertical inhomogeneities of microphysical parameters.Both can have an impact on BTDs (Piontek et al., 2021a;Zhang et al., 2010).To estimate how large these effects are, we performed a sensitivity analysis for varying cloud geometric thickness and for vertical inhomogeneities of R eff .Results of this analysis are shown in Figs.E1 and E2.We find that the sensitivity to both geometric thickness and vertical R eff inhomogeneity is small compared to other cloud parameters ( 0.5 K in most cases).This sensitivity does not significantly affect the regions in the space spanned by the two BTDs which are associated with the different phases and therefore has a comparatively small effect on a potential phase retrieval.

Conclusions
The aim of this study is to characterize and physically understand the relation of two IR window BTDs that are typically used for satellite retrievals of the thermodynamic cloud phase.As an example, we select BTD(8.7-10.8)and BTD(10.8-12.0)from SEVIRI, but the main findings can be generalized to other imagers with similar thermal channels.Although modern phase retrievals often rely not only on BTDs but also on other satellite measurements (Baum et al., 2012;Hünerbein et al., 2023;Benas et al., 2023;Mayer et al., 2024), it is important to understand the BTD characteristics and capabilities.This knowledge helps to design optimal cloud-phase retrievals and to understand their potential and limitations.
We present RT calculations that analyze the sensitivities of the two BTDs to cloud phase and all radiatively important cloud parameters related to phase, namely τ , R eff , ice crystal habit, CTT, and CTH.Previous studies of BTDs have tended to focus on only a small number of cloud parameters, and an overview of the relative importance of all cloud parameters and their interdependencies is still missing.We perform a sensitivity analysis of the BTDs, which to our knowledge has never been done for all cloud parameters combined.This provides an overview of the effects of all cloud pa-rameters and shows which parameters are responsible for the observed phase dependence of the BTDs, which is often used for phase retrievals (Ackerman et al., 1990;Strabala et al., 1994;Finkensieper et al., 2016;Key and Intrieri, 2000;Baum et al., 2000Baum et al., , 2012;;Hünerbein et al., 2023;Benas et al., 2023;Mayer et al., 2024).Even though the RT calculations were performed for a specific atmospheric and surface setup, the main insights of this study, including the physical understanding of the effects of cloud properties on BTDs and their relative importance, are valid for any atmospheric or surface condition.
To understand the behavior of the BTDs, we examine the effects of the nonlinear relationship between radiances and BTs through Planck's radiation law on the BTDs.This nonlinearity induces positive BTD values and a dependence on the CTT (or more generally the surface-cloud temperature contrast T ) in a simple RT model, even when cloud optical properties (transmittance and emissivity) are the same at all wavelengths.This effect is co-responsible for the arc shape of the BTDs as functions of τ and their CTT dependence, in addition to effects due to spectral differences in cloud optical properties.These spectral differences in cloud optical properties can explain the (remaining) dependence of the BTDs on the different cloud parameters.1), also including exceptionally cold liquid clouds and exceptionally warm ice clouds.The color codes of the three rows correspond to the color codes in the left column.Clouds which can be distinguished using a CTT proxy are again not shown.
We find that the dependence on phase is more complex than is sometimes assumed: although both BTDs are directly sensitive to phase (holding every other cloud parameter constant), this sensitivity is mostly small compared to other cloud parameters, such as τ , CTT, and R eff .Instead, apart from τ for which the sensitivity is well known, the BTDs show the strongest sensitivity to CTT (and the closely linked CTH).Since the CTT is associated with phase, this is the main factor leading to the observed phase dependence of the BTDs.Note that more generally, this CTT dependence of the BTDs is more accurately described as a dependence on the surface-cloud temperature contrast T , which reduces to a CTT dependence in our case with a fixed surface temperature.The direct phase dependence merely adds to the CTT effect, increasing differences between ice and liquid (for BTD(8.7-10.8)only for small τ 10).
The sensitivity analysis shows that it is straightforward to distinguish typical high ice clouds from low liquid clouds using the BTDs.However, it is challenging to distinguish a mid-level ice cloud from a mid-level liquid cloud -especially if the R eff is also similar.The combination of both BTDs increases phase information content and is therefore preferable in a retrieval.
This study was conducted for a simple fixed setup of the atmosphere, surface, and satellite viewing geometry in order to focus on the effects of cloud properties.If this setup is changed, we expect the cloud effects on the BTDs discussed in this paper to be superimposed by additional effects: for example, changes in water vapor content or satellite zenith angle shift BTD(8.7-10.8)due to its sensitivity to water vapor absorption.This shift is larger the more water vapor is above the cloud top and therefore depends on the CTH and the vertical atmospheric profile.A different type of surface with spectral differences in surface emissivity (for instance, a desert surface) shifts the values of both BTDs for optically thin clouds.For potential phase retrievals, these effects should ideally be taken into account.
This study focuses on liquid and ice clouds.We expect the BTD values of mixed-phase clouds to lie between ice and liquid values, as they represent a transition between the two.Depending mainly on the CTT and CTH, and to a lesser extent the R eff of mixed-phase clouds, their BTD values are expected to be closer to or further away from the liquid or ice BTD values.In that sense, we expect that BTDs can make a useful contribution to the retrieval of mixed-phase clouds and their composition.However, as the CTT, CTH, and R eff values overlap between liquid, mixed-phase, and/or ice clouds, we expect the regions of the different phases in the space spanned by BTD(8.7-10.8)and BTD(10.8-12.0) to also overlap, introducing ambiguity.The use of additional satellite channels containing, for instance, particle size or phase information is necessary to increase the phase information content for a retrieval.

Appendix A: Single-scattering properties
The single-scattering properties are the volume extinction coefficient β ext , the single-scattering albedo ω 0 , and the scattering phase function p.The volume extinction coefficient β ext describes how much radiation is removed through scattering and absorption (extinction) from a ray when passing through the cloud and can be expressed as where β sca and β abs are the scattering and absorption coefficient, with units of m −1 , measuring how much radiation is absorbed and scattered by cloud particles.Note that in this study τ is β ext at wavelength λ = 550 nm integrated over the path through the cloud; the optical thickness τ λ at other wavelengths λ is in general different from τ , depending on the other microphysical cloud parameters.The singlescattering albedo ω 0 is a measure of the relative importance of scattering and absorption, defined as Hence, as an alternative to β ext and ω 0 one can equivalently describe radiative transfer by β abs and β sca , which can be easier to interpret.The scattering phase function p( ) gives the probability of the scattering angle , i.e., the angle between the incident radiation and the scattered radiation.To understand radiative transfer through a cloud, the most important property of p is the angular anisotropy of the scattering process.This anisotropy is indicated to first order by the asymmetry parameter g, which is calculated from p as the mean cosine of the scattering angle .
If a particle scatters more in the forward direction ( = 0°), g is positive; g is negative if the scattering is more in the backward direction ( = 180°) (Bohren and Huffman, 2008).
Appendix B: Disentangling the BTD nonlinearity shift from effects of wavelength-dependent optical properties An instructive way to look at the BTD nonlinearity shift and to disentangle it from effects of wavelength-dependent optical properties is the following: to make the radiances at different wavelengths more comparable, we use the Planck radiance corresponding to the surface temperature T s as a reference.For typical atmospheric profiles (without temperature inversions), this Planck radiance B λ (T s ) is the maximal possible radiance in each wavelength, corresponding to τ → 0 (see Eq. 3).We express the TOA radiance as fractions f λ of https://doi.org/10.5194/amt-17-5161-2024Atmos.Meas.Tech., 17, 5161-5185, 2024 and therefore in f 10.8 .As a result, the Schwarzschild radiance fraction line in f 8.7 -f 10.8 space deviates from a linear to a concave line.This deviation is stronger for larger δτ (i.e., larger differences between τ 10.8 and τ 8.7 ), as well as for larger differences between the surface and the cloud-top radiance, B λ 1 (T s ) − B λ 1 (CTT).
As a last step of this analysis, we study the full RT model in f 8.7 -f 10.8 space.Recall that in the full RT model, in general, τ 8.7 = τ 10.8 = τ , where τ as usual refers to the optical thickness at 550 nm. Figure B1 shows the radiance fractions f 8.7 and f 10.8 computed with the full RT model for an ice cloud for varying τ and two different CTTs as solid blue and green lines.Molecular absorption is switched off for these examples.Note that this is an equivalent representation of BTD(8.7-10.8) to the corresponding CTT curves in Fig. 9.For increasing τ from 0 to 30, the radiance fractions of the full RT model form curves from f 8.7 = f 10.8 = 1 to the radiance fraction values corresponding to the blackbody radiance of their CTT.These curves are concave, as expected from our theoretical considerations above (see Eq. B7).This concave shape, as explained above, can be attributed to differences in the absorption coefficients of the two wavelengths, β abs,8.7 < β abs,10.8 .The concave shape results in higher BTD values compared to the Schwarzschild BTD S values, where τ 8.7 = τ 10.8 = τ (compare BTD(8.7-10.8)along the solid and dotted lines in Fig. B1).The figure also shows that the deviation from the linear Schwarzschild radiance fraction lines is larger for lower CTTs -in accordance with our theoretical considerations (see Eq. B7).
This leads to the following interpretation of Fig. B1: the Schwarzschild radiance fraction lines in Fig. B1 (dotted lines) represent the pure BTD nonlinearity shift, which induces positive BTD values even though τ is the same at all wavelengths.Adding spectral differences between the cloud optical properties "pushes" the radiance fraction lines into a concave shape and further increases BTD.Hence, the difference between the BTD(8.7-10.8)= 0 line and the Schwarzschild radiance fraction lines in Fig. B1 is due to the nonlinearity of the transformation from radiances to BTs; the difference between the Schwarzschild radiance fraction lines and the full RT model (solid lines) in Fig. B1 is due to the spectral differences in cloud optical properties.Lower CTTs increase both the BTD nonlinearity shift and the effects of spectral differences between the cloud optical properties.lead to increasing values for smaller R eff for both BTDs (top row of Fig. C1).
For BTD(10.8-12.0), the effect of scattering is similar for varying R eff and comparatively small (increases (decreases) BTD(10.8-12.0)by 0.5 K for ice (water) clouds; compare Fig. C1a and b with e and f).For BTD(8.7-10.8),scattering effects are stronger than for BTD(10.8-12.0)and depend on R eff : scattering leads to a stronger decrease in BTD(8.7-10.8)for smaller R eff (compare Fig. C1c and d with g and h).Since, however, the absorption effects are stronger, BTD(8.7-10.8)increases with decreasing R eff (Fig. C1g  and h). by the top cloud layer.The BTD signal is then dominated by the R eff of the top cloud layer (R eff,1 ).This makes a difference mainly for small R eff values (see min R eff curves in Fig. E2), as the BTDs depend nonlinearly on R eff (see Fig. 7). Figure E2 shows that these vertical R eff inhomogeneity effects on cloud emittance (dominant for large τ ) lead to larger overall effects on the BTDs compared to the effects on transmitted surface radiance (dominant for small τ ).

Figure 1 .
Figure 1.Example scene from SEVIRI on 11 June 2023 at 12:00 UTC.(a) RGB composite with yellow cloud colors indicating higher CTTs and white-blue indicating lower CTTs.(b, c) The two BTDs for the same example scene.

Figure 2 .
Figure 2. Causal diagram of cloud parameters that are connected to the cloud phase.Arrows indicate causal links.

Figure 3 .
Figure 3. Single-scattering properties: extinction coefficient β ext , single-scattering albedo ω 0 , and asymmetry parameter g, as well as absorption coefficient β abs and scattering coefficient β sca (computed from β ext and ω 0 ) as functions of wavelength for varying cloud phase and effective radius R eff .β ext , β abs , and β sca are scaled by the cloud water content, WC.Parameterizations for ice are according to Baum et al. (2011) and for liquid droplets according to Mie theory.For ice clouds, the "general habit mix" was used as the ice crystal habit.Vertical gray lines indicate the center wavelengths of the three IR window channels.

Figure 4 .
Figure 4. (a) Radiance at the top of the atmosphere (R S TOA,λ ) computed with the Schwarzschild equation (black line).Vertical gray lines indicate the center wavelengths of the three IR window channels with blue, red, and green dots at R S TOA,8.7 , R S TOA,10.8 , and R S TOA,12.0 , respectively.The dashed blue, red, and green lines correspond to the Planck curves of these three TOA radiances, i.e., B λ (T λ (R S TOA,λ )) for each wavelength, where B λ is the Planck function and T λ the inverse Planck function.The solid gray curves show the Planck curves of the surface temperature T s and the CTT as a reference.(b) Brightness temperature differences computed with the Schwarzschild equation, BTD S , as functions of τ for different CTTs and a fixed T s = 290 K.

Figure 5 .
Figure 5. Scattering effects on brightness temperatures (BT): difference between the BTs for a cloud with scattering and a cloud with scattering switched off for all three IR window channels, i.e., BT λ − BT no sca λ for each channel with center wavelength λ ∈ {8.7, 10.8, 12.0 µm}, for liquid and ice clouds as functions of optical thickness τ .

Figure 6 .
Figure 6.Brightness temperature differences BTD(10.8-12.0)and BTD(8.7-10.8)as functions of τ for cloud particle scattering (a, b) switched off and (c, d) switched on for liquid and ice clouds.Solid lines indicate a "normal" absorbing atmosphere, and dashed lines indicate that molecular absorption is switched off.

Figure 7 .
Figure 7. Effects of varying R eff on BTD(10.8-12.0)and BTD(8.7-10.8)as functions of τ for ice clouds (a, b) and liquid clouds (c, d).Solid lines indicate a "normal" absorbing atmosphere, and dashed lines indicate that molecular absorption is switched off.

Figure 8 .
Figure 8. Effects of varying ice crystal habit on (a) BTD(10.8-12.0)and (b) BTD(8.7-10.8)as functions of τ for ice clouds.Solid lines indicate a "normal" absorbing atmosphere, and dashed lines indicate that molecular absorption is switched off.

Figure 9 .
Figure 9. Effects of varying cloud-top temperature (CTT) on BTD(10.8-12.0)and BTD(8.7-10.8)as a function of τ for ice clouds (a, b) and liquid clouds (c, d).Solid lines indicate a "normal" absorbing atmosphere, and dashed lines indicate that molecular absorption is switched off.

Figure 10 .
Figure 10.Sensitivity analysis for each BTD varying the phase-related cloud parameters τ , R eff , habit, CTT, and thermodynamic phase: BTD(10.8-12.0)and BTD(8.7-10.8)for typical upper and lower boundaries of CTT and R eff for ice (blue colors) and liquid (orange-red colors) clouds.For ice clouds, different habits are shown as different markers.The figure shows typical BTD ranges for ice and liquid clouds.

Figure 11 .
Figure 11.Sensitivity analysis combining both BTDs and varying the cloud parameters τ , R eff , habit, CTT, and thermodynamic phase: blue lines show ice clouds, and orange-red lines show liquid clouds for typical upper and lower boundaries of CTT and R eff .Along each line, τ increases from 0 to 30.For ice clouds, different habits are shown as different markers.

Figure 12 .
Figure 12.Left column (a, c, e): BTD(10.8-12.0)and BTD(8.7-10.8)values within the defined "typical" boundaries of CTT.Round markers indicate liquid clouds; crosses indicate ice clouds.Clouds which can be distinguished using a CTT proxy like BT 10.8 , i.e., optically thick clouds (τ ≥ 10) with very low (< 233 K) or very high (> 273 K) CTTs, are not shown.The color code in the three rows encodes τ , CTT, and R eff , respectively.Right column (b, d, f): same as left column but for the whole range of computed cloud scenarios (see Table1), also including exceptionally cold liquid clouds and exceptionally warm ice clouds.The color codes of the three rows correspond to the color codes in the left column.Clouds which can be distinguished using a CTT proxy are again not shown.

Figure B1 .
Figure B1.BTD(8.7-10.8) in the space spanned by the radiance fraction f 8.7 and f 10.8 (defined as the radiance at TOA scaled by the Planck radiance of the surface with temperature T s = 290 K: f λ = R TOA,λ /B λ (T s )).The solid black line indicates BTD(8.7-10.8)= 0; the dashed black line indicates f 8.7 = f 10.8 .The blue and green lines show f 8.7 and f 10.8 values for varying τ at a given CTT: the dotted lines show f 8.7 and f 10.8 computed with the Schwarzschild equation (with τ 8.7 = τ 10.8 ), and the solid lines show f 8.7 and f 10.8 values computed with the full RT model.
FigureC1shows the sensitivity of both BTDs with R eff broken down into effects of absorption and scattering.The two rows show the same cloud scenarios, once with scattering switched off (top row) and once with scattering switched on (bottom row).The figure shows that the effects of absorption

Figure C1 .
Figure C1.Effects of varying R eff on BTD(10.8-12.0)and BTD(8.7-10.8)as functions of τ for ice clouds (blue) and liquid clouds (orangered) with scattering switched off (a-d) and switched on (e-h).Solid lines indicate a "normal" absorbing atmosphere, and dashed lines indicate that molecular absorption is switched off.
Information on these two parameters can give an indication about the cloud phase -e.g., clouds with small R eff are typically liquid clouds; clouds with very low τ are typically ice clouds.Ice crystal habit can influence the TOA radiation as well but is of course