This study is based on data from the novel ICOsahedral Non-hydrostatic model ICON model, e.g.. We use a simulation of a frontal case on 26 April 2013 over western Germany with rapidly increasing cloudiness developing to a completely overcast situation in the afternoon. Several light to medium rain showers occurred during that day, and ice clouds as well as snow in the upper atmospheric layers were observed. The case represents a spring day in the northern mid-latitudes. Choosing a tropical case or a much drier case, for example in the Arctic, will have an effect on both the Jacobians and the resulting information content. The Jacobians will peak at different heights, and the channels will observe the hydrometeor content depending on how far down they penetrate the atmosphere. Nevertheless, the principles of the observation and the interdependencies of the Jacobians can be made clear at the hand of this case.

The ICON simulation has a horizontal resolution of 650 m with 50 hybrid terrain-following vertical height levels to 22 km. It was performed in the framework of the BMBF project High Definition Clouds and Precipitation for advancing Climate Prediction (HD(CP)2) and was provided by the Max Planck Institute for Meteorology, Hamburg. The simulation complements the measurement campaign HOPE (HD(CP)2 Observation Prototype Experiment, Macke et al., 2017), which took place in April and May 2013 around Jülich in western Germany and focused on clouds and model evaluation e.g.

Details about the project and the campaign can be found on the project homepage http://hdcp2.eu (last access: July 2017) or on the data base SAMD (Standardised Atmospheric Measurement Data) homepage hosted at the Integrated Climate Data Center (ICDC) under http://icdc.cen.uni-hamburg.de/1/projekte/samd.html (last access: April 2018).

In order to perform an information content analysis a radiative transfer model is required to simulate the satellite measurements and the respective height-resolved Jacobians based on the atmospheric profiles simulated by ICON. We use the Atmospheric Radiative Transfer Simulator ARTS, version 2.3.296. ARTS is an open source detailed line-by-line radiative transfer model for microwave to thermal infrared radiation, which is capable of simulating polarized radiative transfer in all spatial geometries

See www.radiativetransfer.org (last access: 15 June 2018) for documentation and download.

ICON uses the two-moment microphysical scheme by , which offers more detailed information about the cloud microphysical properties than the commonly used one-moment bulk schemes. It simulates the mass mixing ratio (M) and number mixing ratio (N) of cloud liquid water, cloud ice, rain, snow, hail and graupel. As only very little graupel and hail was found in the simulation, we disregard them in the following. For the atmospheric radiative transfer simulator ARTS (Sect. ) the mass mixing ratios (unit kg kg-1) were converted to mass densities (kg m-3) by multiplying with the density of the atmosphere.

In the following, we refer to liquid cloud water mass density (liquid water content) as LWC, to cloud ice mass density as IWC, to rain mass density as RWC and to snow mass density as SWC. We refer to LWC and RWC as liquid hydrometeors and to IWC and SWC as frozen hydrometeors. The respective column integrated quantities, i.e. the paths are denoted as LWP (cloud liquid water path), IWP (cloud ice water path), RWP (rain water path) and SWP (snow water path). Note that even though the ICON model's microphysical parameterization requires a clear distinction between suspended and precipitating hydrometeors in each grid box, i.e. between LWC and RWC or IWC and SWC, this distinction can not be made in reality. Nevertheless we will discuss the cloud and precipitating hydrometeors separately in the remainder of the article, always keeping in mind that, in reality, there is a smooth transition between the cloud and precipitation hydrometeors.

For the simulation of the cloud radiative effect the size distribution and shape of the hydrometeors in terms of the mass–dimension relationship are of high importance. It is crucial to match the microphysical parameterizations of the radiative transfer model with those of the atmospheric model, especially the size distribution.

The size distribution in the two-moment scheme by is based on the hydrometeor mass. It employs a modified Γ-distribution with two free parameters as particle size distribution functions for each hydrometeor type. It is defined as follows: f(m)=Amνexp⁡-λmμ, where the independent size parameter is the particle mass m. The distribution parameters are A, ν, λ and μ and have to be provided by the scheme. In the actual version of 's scheme, ν and μ are fixed for each hydrometeor type (Table ) and A and λ are calculated prognostically (see for details of the calculation).

The size distributions from the two moment scheme for an idealized mean profile (purple) and a set of 90 individual ICON profiles (grey) are shown in Fig.  (for the definition of the idealized and the 90 profiles please refer to Sect. ). Note that the distributions are height dependent. They are shown at a height of 550 hPa, where both IWC and SWC exist in considerable amounts. The curves illustrate the sum of the distributions for IWC and for SWC, i.e. all frozen hydrometeors. For the mean profile, also the individual distributions for IWC and SWC are shown to illustrate to what extent they overlap. The two peaks, which are evident in the idealized and in some of the 90 profiles, result from the two different types of frozen hydrometeors. The one at smaller diameters belongs to IWC, the one at larger diameters belongs to SWC. As opposed to one-moment schemes, in which the distinction between IWC and SWC is done through a size threshold, in the two-moment scheme the distinction between IWC and SWC is done via the processes a particle has undergone. A particle is counted as snow if it has, for example, collided and joined with other hydrometeors (e.g. self-collection or collection of smaller hydrometeors). Single ice crystals are counted as cloud ice. Therefore, in the two moment-scheme cloud ice hydrometeors can be quite large in mass equivalent diameter and overlap with snow.

It is noteworthy that different schemes provide different size distributions. Compared to, for example, frozen hydrometeor size distributions for tropical cirrus clouds by (not shown), in the two-moment scheme the number densities for small ice particles are orders of magnitude smaller. Apart from the different meteorological situation, this is mainly due to the fact that processes creating small ice particles in the two-moment scheme are missing (Axel Seifert, personal communication, 2016). However, aircraft measurements have been criticized for having too many small particles due to shattering e.g. and the exact amount of smaller particles remains uncertain. For the millimetre and sub-millimetre range this is not critical because the sensitivity to particles smaller than 100 µm is small in this range .

It should be noted that aside from the differences in the size distribution also the mass–dimension relationship is a crucial ingredient for radiative transfer modelling e.g.. However, atmospheric models only implicitly assume such a relationship, for example to parameterize collision or fall speeds. Also, normally atmospheric models have no detailed information about the particle shape. Furthermore, different ice habits within one hydrometeor type (cloud ice, snow, hail or graupel) are not considered. This will introduce some errors in both the microphysical calculations of ICON and the radiative transfer simulations. A perfect matching of the atmospheric model and the radiative transfer model with regard to particle shape and habit would require more sophisticated assumptions in the atmospheric model.

The hydrometeor size distributions of the particles have been implemented in a discretized form into ARTS using the same distribution function as the two-moment scheme from . As a second variable representing the microphysical characteristics of the hydrometeors the particle mean mass m¯ was chosen. It was calculated by dividing the mass mixing ratio M by the number mixing ratio N, both taken from ICON. This has the advantage that the mass density Jacobians for a fixed particle mean mass (as opposed to a fixed particle number mixing ratio) correspond to the ones one would get from a one-moment bulk scheme. However, in the remainder of the article we will focus on the mass densities and only include the values for the mean particle masses m¯ in the final results for the information content to see if the channels higher than 183 GHz (see Sect.  for chosen set of channels) have a potential to measure cloud microphysical parameters such as hydrometeor size.

The scattering properties for the different hydrometeor types are defined as in . For LWC and RWC, spherical particles are assumed and for IWC soft spheres with a density of 900 kg m-3 are assumed. For spherical particles, the single scattering properties were calculated with Mie theory, using the program by . For SWC, the scattering properties were taken from the database of assuming sector-like snowflakes for channels up to and including 334 GHz and the data base of assuming aggregates for channels higher than that. Since the original database assumes a constant effective density for the aggregates and is also based on the earlier refractive index we use a corrected version of the database, in which the absorption is rescaled using the parameterization for the refractive index of ice. Rescaling is done by multiplication with imag(n)/imag(n0), where n and n0 are the refractive indices from and , respectively. The rescaling to obtain data for 183, 213, 243 and 266 K was applied. The scattering extinction and the phase matrix remain unchanged, which means that the rescaling only applies to the absorption (see for details).

We use the Discrete Ordinate ITerative (DOIT, ) method to calculate the scattering within ARTS. The Planck brightness temperatures were calculated for all side bands within the chosen set of channels (see Sect. ). We do not use an explicit sensor response function but perform monochromatic radiative transfer simulations for the centre frequencies of the side bands in each channel and use the mean of the two brightness temperatures. For clear sky, highly resolved (in terms of frequencies) tests showed that the error compared to this simplified treatment is less than 1 K . As the scattering properties of the hydrometeors change only marginally within the band width, a further increase of this uncertainty in the cloudy case is unlikely. A pencil beam with an incidence angle of 65∘ at the ground was used. For gas absorption we use the HITRAN (HIgh-resolution TRANsmission molecular absorption, ) database, the MT_CKD model version 2.52 for the continuum absorption of water vapour and the MT_CKD model version 1.00 for the continuum absorption of oxygen.

The calculation of the Jacobians by explicit perturbation (in contrast to the analytical calculation) J generally requires three steps: first, calculate the brightness temperature TBc for a specific channel c for the unperturbed atmosphere. Second, perturb one atmospheric quantity x, for example IWC, by a perturbation δ and again simulate the perturbed brightness temperature TBcδ for that channel. Third, divide the difference of the two brightness temperatures by the perturbation. If, as in our analysis, height resolved Jacobians are required, the perturbation has to be applied successively to each of the height levels k. Note that a perturbation at a distinct height level k strictly speaking means a perturbation of the respective quantity at the two adjacent height layers which the radiation passes through.

The Jacobian Jc,k at height k and for a channel c is thus given by the following equation: Jc,k=TBck,δ-TBcδ, with TBck,δ as the simulated brightness temperature if the quantity xk is perturbed, which denotes the value of x at the height level k.

For the analysis, we define δ as a relative perturbation of xk, as opposed to using an absolute value that is independent of the specific value of the xk. This is especially useful for the calculation of Jacobians for the hydrometeor profiles. First, the values of x over height span several orders of magnitude. The use of a relative perturbation ensures that the perturbation is always small compared to the original value, and linearity can be assumed. Second, the hydrometeor profiles are discontinuous and do not exist at all heights. Using the relative perturbation ensures that only that part of the profile where hydrometeors exist in the first place is perturbed. We use δ=1 % for each quantity (including water vapour, which in the following is referred to as H2O) and at all heights.

Relative Jacobians also correspond to the retrieval of a quantity in logarithmic space. Regarding 1+δ as the development of the natural logarithm for small δ it can be shown that δ=ln⁡(xk,δ)-ln⁡(xk)=Δln⁡xk, with xk as unperturbed value at height level k and xk,δ as perturbed value. The Jacobian then is as follows: Jc,k=TBck,δ-TBcΔln⁡(xk).

As stated above, this corresponds to a retrieval in natural logarithm space. In the remainder of the article, we will stay in the framework of a logarithmic retrieval entirely.

The Jacobians for each of the two sidebands (see Sect.  for the definition of channels and side bands) were calculated and the mean of the two Jacobians was used for the subsequent analysis. We use the same height grid in ARTS as in the ICON simulation. The Jacobians for the H2O volume mixing ratio (VMR), the hydrometeor mass densities M and the hydrometeor mean mass m¯ were calculated. For the analysis, Jacobians were used in units of K (100 %)-1 as calculated by ARTS. For the purpose of showing them in the following sections, they are normalized by the height layer thickness. Note that the height layers broaden with increasing height. This yields the unit K (% km)-1, which appears in the plots. Thus, the comparability of the Jacobians at different height levels is ensured. However, all calculations have been performed on the unnormalized values.

Note that Eqs. () and () only conceptually describe the Jacobian calculation. In practice, we do not make a fully independent TB calculation for each perturbation, since this is computationally very inefficient for the iterative scattering solver used . Instead, the scattering solver uses the result from the unperturbed scheme as a starting point. That result should be close to the result from the perturbed case already, because the profile perturbations are small. From that starting point, the perturbed Jacobians are calculated with far fewer iterations compared to a completely uneducated starting point, which makes the scheme far more computationally efficient.

The final component necessary to calculate the information content of a measurement is the a priori covariance error matrix Sa. In the ICON model framework, the matrix can be calculated directly from the model data as the covariance of the different quantities on different height levels. This means that we take the model mean state as a priori state, and the full variability of the model on the chosen domain (state domain) and simulation time as its uncertainty.

Certain assumptions have to be made in order to calculate the a priori covariance. First, only cloudy cases are considered. We assume that some kind of cloud detection has been done prior to the observation of the cloudy sky. To identify cloudy cases in the model, a threshold for the total condensed water path is applied, i.e. for the sum of the paths of all hydrometeors. We use the approximate detection threshold of 10-4 kg m-2. If the total water path exceeds that threshold, the corresponding profile is used in the calculation of the a priori error covariance.

Since relative Jacobians (see Sect.  ) were used, i.e. a retrieval in natural logarithm space, also the covariance needs to be calculated in ln space. To enable this calculation, zero values have to be removed from the hydrometeor profiles. For this purpose we set a threshold for each quantity for which the a priori error covariance is calculated. The choice of this threshold is a non-trivial task, and the threshold will affect the desired information content. We will address this in more detail in the respective result section. The smaller the threshold is, the larger the a priori variance is and the more information an observation will provide in comparison.

H2O is smooth and, above all, continuous. Therefore the numerical model threshold 10-20 kg m-3 is used for it. For the hydrometeor mass densities 10-7 kg m-3 is used. If we assume a detection limit for the water path of 1 g m-2 and a cloud thickness in the order of 1 km, then this value for the local mass density is a little bit smaller than the detection limit, depending on the real cloud thickness. Furthermore, that mass density threshold value is close to an internal threshold within the two-moment microphysics scheme (close, because the microphysics scheme employs mixing ratios instead of mass densities). For example, if the mass density of cloud ice is larger than that threshold, collisional growth can take place (Axel Seifert, personal communication, 2017). For the mean masses of the different hydrometeors, separate minimum values are used in the two-moment scheme. The thresholds employed are 4.2×10-15 kg for cloud liquid water mean mass, 10-12 kg for cloud ice mean mass, 2.6×10-10 kg for rain mean mass, and 10-10 kg for snow mean mass. The same thresholds for the mean masses were used in the calculation of the a priori error covariance. The chosen thresholds furthermore ensure that the relevant peaks, within the mass density distributions for LWC, RWC, IWC and SWC, which constitute the clouds or precipitation, are covered. The peaks are roughly located at -4.2 with a width of 0.7 in units of the decadal logarithm (LWC), -5.3 with a width of 0.8 (IWC), -5.0 with a width of 0.8 (RWC) and at -5.3 with a width of 0.9 (SWC). Therefore they are all well above 10-7, which was chosen as threshold. Because for the mean masses we used the model inherent thresholds for the distribution, this is naturally true for the mean masses of all hydrometeors as well (not shown). We are aware that those choices affect the results for the information content. It will be discussed further in Sect. .

Figure  shows the a priori covariance in ln space and the corresponding correlation matrix defined as Ci,j=Si,j/SiiSjj. The 25 block matrices give the covariance and correlation of pairs of model variables on their 49 height levels. Note that we have to skip the uppermost 50th height level from ICON because ARTS requires one level on top of the “cloud box”, which defines the cloudy area where scattering is calculated. Since the matrices are symmetric, only the lower triangle is shown for clarity.

The covariance naturally is largest at the height levels where hydrometeors reside and goes up to 12.2 in units of the natural logarithm on the diagonal of the LWC × LWC block matrix. For the other hydrometeors, the maximum reaches almost eight on the diagonal. The covariance for H2O is small compared to the one for hydrometeors (smaller than one). This reflects the much lower variability and smaller dynamical range of H2O compared to hydrometeors. Note that the scene investigated is a short period in spring time in the mid-latitudes. In other seasons, the a priori covariance will likely look different, especially for the hydrometeors. In winter, for example, snow can reach the ground and there would be a non-zero covariance down to the ground.

There is a rich structure of autocorrelations and correlations between different hydrometeors and over different height levels. The autocorrelation for the different hydrometeor types is mostly positive everywhere, which accounts for the thickness of the cloud layer and the falling precipitation. For SWC and RWC, for example, the correlation in the upper layers is negative. This may be interpreted in terms of the melting layer. Above the melting layer snow exists. It melts below the melting layer forming rain. Therefore precipitating snow in the upper layers will lead to rain in heights below the melting layer.

In the correlation plot also rich structures within the H2O related blocks become evident. H2O is positively correlated with hydrometeors within the clouds and precipitation regions, since the atmosphere is near saturation there. The negative correlation at lower regions may be due to evaporation in sub-saturated regions. Here, the hydrometeor mass decreases and the H2O mass increases. At higher regions above the clouds it may be spurious and stem from the fact that there are only numerical artefacts of very small amounts of hydrometeors in comparison to realistic amounts of H2O.

The covariance and correlations were calculated on the basis of 1.8 million near-realistic cloudy profiles from the ICON model simulation. With this covariance and correlation from ICON data, we automatically get the cross-correlations for the different hydrometeor types as well. However, one has to be aware that there are model inherent correlations due to the microphysical parameterization, which can cause some artefacts. Furthermore the choice of thresholds to remove the zero values in the profiles scales the covariance and therefore will affect the information content. The terrain following coordinates of ICON cause slightly larger covariances in the lower levels for H2O and rain, which are both present near the ground. Idealized covariance matrices might be constructed instead (e.g. ), but they have different downsides and also contain many assumptions, especially for the cross-correlation of hydrometeors. We therefore chose the model based a priori covariance matrix to perform the study, keeping in mind the downside that the model introduces some artificial correlations between the hydrometeors.

Radiometer channels as applied on the International Sub-millimetre Microwave Airborne Radiometer (ISMAR, ) and on the Microwave Airborne Radiometer Scanning System (MARSS, ) are used. They both were deployed on a recent flight campaign and cover channels from 89.0 up to 664.0 GHz. ISMAR will be extended with a channel at 874.4 GHz in near future. They include the AMSU-B channels and the ICI setup, with the exception that the ICI-1, ICI-2 and ICI-3 channels have a slightly different distance from the H2O absorption peak at 183.31 GHz (). We will later use the three 183.31 GHz channels from MARSS instead. The setup is further complemented by two channels at 23.8 and 50.1 GHz from the Dual-frequency Extension to In-flight Microwave Observing System (Deimos, ) to account for lower frequency precipitation channels, which are also part of MWI. The resulting set of channels, their side bands, and the respective instrument they belong to is given in Table .

With this setup it is possible to investigate the principle interdependencies of the information content on different hydrometeors from a set of channels, which is capable of observing liquid and frozen cloud as well as precipitation hydrometeors. However, it is also possible to put a special focus on the upcoming ICI instrument on MetOp-SG, which employs channels from 183 to 664.0 GHz to gain more detailed information about cloud ice, its microphysical properties and perhaps some more profile information than the instruments that are currently deployed in the different satellite missions.

To facilitate the analysis a mean profile (Fig. ) from 10 000 ICON profiles, which each are amongst the extremes for one specific hydrometeor or the humidity, was calculated. To choose the 10 000 profiles, the hydrometeor paths for each hydrometeor type and each atmospheric profile were calculated. To exclude unphysical outliers, which may be produced by the model, profiles with a hydrometeor path larger than the 99th percentile were disregarded. From the remaining profiles we chose 10 000 / 7 largest H2O paths, LWPs, IWPs, RWPs, SWPs and the paths for hail and graupel (the “divided by seven” stems from the seven quantities, over which the loop is done). This ensures that a considerable amount of each hydrometeor, except for hail and graupel, which only exist in very small amounts over the whole simulation, is contained in the profile. However, since the 10 000 atmospheres are not required to be extreme with regard to (or contain all) hydrometeor types at once, this gives us 10 000 cloudy profiles and on average a mean profile which is not extreme for any hydrometeor and which is comparably smooth. The mean profile that follows from these choices contains realistic amounts of hydrometeor masses. Cloud and precipitation are located in physically reasonable height ranges. However, it has to be kept in mind that this may lead to an unlikely combination of hydrometeors, such as LWC and SWC being present in the same atmospheric column. Therefore in Sect. , we will also show results from a set of 90 individual cloudy atmospheric columns drawn directly from the selected ICON simulation to consolidate the results from the idealized atmosphere.

This mean profile is used as a “base profile” containing all hydrometeors. From this base profile atmospheres with different combinations of cloud hydrometeors are constructed by taking out or putting in specific hydrometeors. A similar approach has been used by , who progressively put in cloud hydrometeors to quantify their respective influence on the brightness temperature in the 183 GHz channel of the humidity sounder SAPHIR on Megha-Tropiques. This study investigates if the information about one hydrometeor type depends on the presence of another. For example, we can have an atmosphere which contains only LWC, only IWC or one which contains the two hydrometeor types IWC and RWC. All in all 16 combinations including clear sky are possible. We are aware that not all combinations are physically possible and realistic, such as an atmosphere only containing SWC, but we want to understand how the measurement of one hydrometeor is in principle influenced by the others. Therefore all mathematically possible combinations are regarded.

The different atmospheres are denoted by an aX, where X contains the composition of the atmosphere. LWC is denoted by L, IWC by I, RWC by R and SWC by S. For example, the atmosphere containing IWC and LWC is called aIL. The clear sky case (“Vapour”) is called aV. Note that H2O is present in all of the atmospheres, even though it is not explicitly included in X in case hydrometeors are present. An overview over the atmospheric compositions is given in Table .

The brightness temperatures for the different atmospheric compositions are shown in Fig.  for an emissivity of ϵ=0.6. The brightness temperature spectra differ by up to 80 K, depending on the atmospheric composition and the measurement channel. In the window channels, the spread in the spectra is particularly large, while in the sounding channels near the absorption peaks the spread is smaller. For channels close to absorption line centres the spectra almost lie on top of each other because here the atmosphere is opaque due to H2O absorption.

Some of the different compositions such as aL and aLR or aIS and aIRS, have almost the same spectra except for differences in the window channels below 118.75 GHz. This implies that some hydrometeors are invisible to channels higher than that.

The brightness temperature is a result of the complex interaction of the radiation with the atmosphere. First, there is extinction, i.e. absorption, mainly caused by H2O and by the liquid hydrometeors and scattering, mainly caused by the frozen hydrometeors. Extinction determines how far down a certain channel can see. This also determines the sensitivity of a channel to an atmospheric component to some degree. If the channels do not reach down to levels where, for example, rain exists, naturally it is not sensitive to rain in that case. The whole hydrometeor path of a certain hydrometeor, which is in the pathway of the channel, contributes to the signal. It depends on the respective path of the hydrometeor and the sensitivity of the channel to that hydrometeor how big the contribution is. If we look at one particular hydrometeor, the combined signals of all other atmospheric components provide the radiative background for the signal of that particular hydrometeor. This may lead to a “shielding” of hydrometeors in the lower levels, because the water vapour path or hydrometeor path above those hydrometeors is so large that the atmosphere is entirely opaque for a channel. The signal from a certain hydrometeor can also be masked by other hydrometeors, that create a radiative background through absorption or scattering, which is similar to the radiative signal of the hydrometeor in question. In the following we investigate this further. We will have a closer look at the sensitivities of the brightness temperature to changes in the hydrometeor mass, namely the Jacobians J, in the absence and presence of other hydrometeors.

We first look at the Jacobians for H2O for the clear sky case aV and the all-hydrometeors case aILRS are analysed (Fig. ). H2O has the advantage that its profile is smooth and continuous, contrary to the hydrometeor Jacobians which per definition of the relative perturbation only exist where the cloud hydrometeors reside and which decrease to zero at the cloud edge with a steep gradient. With the chosen surface emissivity of ϵ=0.6, for aV, H2O gives a warming signal from the lower atmosphere (>500 hPa) at channels from 157.05 GHz downward and at 243.2 GHz. For channels from 183.31 GHz upward it gives a small cooling signal at higher levels (<650 hPa). Hereby “warming signal” (“cooling signal”) means that an increase of the amount of vapour or hydrometeor content leads to a warming (cooling) of the resulting brightness temperature at the top of the atmosphere. This is mainly due to the fact that the Jacobians for the sounding channels higher than 183 GHz peak higher up in the atmosphere than the Jacobians of the lower channels and that these higher regions are very cold compared to the ground.

This picture changes dramatically in the presence of all considered cloud hydrometeors (aILRS). Except for the most central frequencies of the sounding channels at 183.31 and 448.0 GHz the H2O signal in this case is entirely positive. The positive signal in between the channels at 157.05 and 23.8 GHz decreases to almost zero. The sensitivity of the measured brightness temperature to changes in H2O is highly dependent on the presence of clouds.

The H2O example illustrates the general principle of these interactions well. If the radiative background is cold, then the presence of a scattering or absorbing species tends to increase the brightness temperature. Conversely, if the background is warm, then the species tends to reduce the brightness temperature. For H2O at high frequencies, the presence of frozen hydrometeors in the upper troposphere, which have a cooling signal due to scattering, turns the scene from a warm background case to a cold background case.

Figure  illustrates Jacobians from atmospheres with one single liquid hydrometeor type each, i.e. aL and aR for both, LWC or RWC, along with the corresponding H2O Jacobians. As relative perturbation for the calculation of the Jacobians were used, (Eq. ), the cloud Jacobians only exist at those heights where LWC or RWC exist (cp. Fig. ). These heights are indicated in the figures for two different thresholds for the respective mass densities. Mainly the channels below 325.15 GHz (LWC) respectively 243.2 GHz (RWC) are sensitive to the liquid hydrometeors. The window channels at 23.8, 50.1 and 89.0 GHz give a warming signal at all heights, the outermost channel at 118.75 GHz (and 157.05 GHz for RWC) have a warming signal at lower levels and a cooling signal at upper levels. Note that a higher surface emissivity, i.e. a radiatively warmer surface, reduces the warming signal from LWC and RWC in the window channels, since the surface provides a warmer background in that case (not shown). The Jacobians for H2O change in the presence of LWC or RWC. Apart from 23.8 GHz the warming signal in the lower channels is considerably reduced compared to the warming signal from H2O alone.

The frozen hydrometeor types IWC and SWC generally give a cooling signal (Fig. ) as they mainly act as scatterers rather than absorbers in the selected channel range. Also, they exist at low ambient temperatures, and even their emission would cause a cooling signal. The upper channels above 157.05 GHz are sensitive to these hydrometeor types. For SWC a considerable signal also comes from the channels at 50.1, 89.0 and the outermost 118.75 GHz channel. The highest channels at 664.0 and 874.4 GHz are more sensitive to IWC than to SWC because the scattering efficiency in these two channels is larger for the smaller ice hydrometeors than for the larger snow hydrometeors.

The corresponding H2O Jacobians are considerably changed at channels above 157.05 GHz. The cooling signal from the clear sky H2O Jacobians (Fig. ) is turned into a warming signal except for the sounding channels closest to the absorption lines. This is in accordance with the findings of who found such a change of sign in the lowest-peaking SAPHIR channels near 183 GHz in the presence of high concentrations of snow above 500 hPa.

Next, we will go deeper into the interdependencies of the hydrometeor Jacobians. For this purpose, the view chosen in the previous figures is reduced, and only the Jacobians for the channels at 89.0 and 243.2 GHz are shown as line plots (Figs.  and ). In these channels we expect to get a signal from all cloud hydrometeors, while 89.0 GHz is more sensitive to the liquid hydrometeors and 243.2 GHz is more sensitive to the frozen hydrometeors. The Jacobians are shown for each cloud hydrometeor type for the cases where only that specific type is present, one other hydrometeor type is present, or all types are present in a combined plot. For the example of cloud ice these are IWC Jacobians for the cases aI, aIL, aIS, aIR, and aILRS.

For the LWC Jacobians (Fig. , top row), the lines group in two sets in both channels. In the 89.0 GHz channel, the signal is reduced in the presence of RWC in the lower levels. The presence of frozen hydrometeors does not alter the signal much. The pairs not including RWC, aIL and aLS, almost give the same Jacobian as aL, while both aLR and the all-hydrometeors case aILRS have a smaller peak and are very close up to about 700 hPa. Above that level, SWC has a greater influence. The Jacobian for aLS deviates from the one of aL and the all-hydrometeor case aILRS approaches the curve for the case aLS. The change of behaviour near 700 hPa is due to the height ranges where SWC or RWC occur, respectively. Near 700 hPa, the melting layer of the idealized atmosphere is located (see Fig. ).

The 243.2 GHz channel has its largest sensitivity for the detection of RWC higher up in the atmosphere than the channel at 89 GHz and therefore does not exhibit such a transition. The two cases aL and aLR which only contain liquid hydrometeors have negative LWC Jacobians, while the presence of any frozen hydrometeors results in positive LWC Jacobians. The largest signal comes from the all-hydrometeors case aILRS, the smallest positive one from the combination of LWC with IWC, i.e. aIL.

This again can be understood if we think of the other cloud hydrometeors as contributors to the mixture of signals from all heights and hydrometeors, which result in the respective brightness temperature. The paths of the other cloud hydrometeor types, the surface and the H2O create a radiative background for the hydrometeor type in question. At 89.0 GHz, the presence of RWC already increases the brightness temperature, therefore the emission from LWC only adds a smaller positive increment compared to an atmosphere where only LWC is present. At 243.2 GHz, the scattering by frozen particles decreases the measured brightness temperature such that the emission by LWC adds a positive increment instead of a negative one if no IWC or SWC is present. These effects are not linear and can not just be added up.

For RWC (Fig. , bottom row), in the 89.0 GHz channel we also find a grouping of the Jacobians similar to LWC. For RWC the sign of the signal depends on the height. The lower levels cause a warmer background, such that the higher levels' contribution is negative compared to it. In the 243.2 GHz channel, the signal from rain is negative with the exception of a small positive contribution near the ground. The addition of any of the other hydrometeor types decreases the amplitude of the Jacobian. Each hydrometeor type alone, LWC, IWC and SWC, gives a cooling signal and therefore causes a colder background in the mixture compared to the case where only RWC is present.

Figure  shows the corresponding figures for the frozen hydrometeors IWC and SWC for the two channels. Since the main interaction of the frozen particles with the radiation is scattering, the signal is robustly negative. In the 89.0 GHz channel, the IWC Jacobians for aI and aIS as well as the SWC Jacobians for aS and aIS are very similar. IWC and SWC only marginally influence each other in that channel. The addition of liquid hydrometeors below the frozen ones leads to a stronger signal for IWC and SWC, because the liquid hydrometeors provide a warmer background for the frozen hydrometeors. In the 243.2 GHz channel, the picture is almost the same. In this channel, however, the signals from the frozen hydrometeors are much stronger, and the combination of IWC and SWC results in a considerably stronger cooling signal for both cloud hydrometeor types. Therefore the Jacobians for the all-hydrometeors case aILRS lie between aIS and the other shown cases.

The amount of the information gained from the observation depends on the composition of the atmosphere. Figure  and Table  summarize ΔDOF for the 16 different atmospheric compositions, observed with the full set of channels and observed with the ICI channels. The analysis for particle mean mass (m¯) Jacobians was not shown in the previous sections, but we include the values for their information content in this section to show the potential of new sensors observing at frequencies of 183 GHz and higher to observe microphysical properties of the particles.

The main focus is on the detection of frozen hydrometeors, but in the following also the information content for liquid hydrometeors is included in the discussion. Liquid water retrievals at lower microwave channels are an established technique. For example a higher number of frequencies within the sounding regions between 50 and 57 GHz, which could be used in combination with the 118.75 GHz channels would be able to retrieve liquid cloud and precipitation, as was shown by . We lack channels in the region between 50 and 57 GHz, therefore in this analysis, we do not expect to have a great ability to detect liquid hydrometeors. We set our main focus on frozen hydrometeors. Nevertheless they will be discussed along with the information content on frozen hydrometeors and be treated as a side parameter for the detection of the frozen hydrometeors.

For the full set of channels, the total information content reaches up to values as high as 12.15 for aILRS and is lowest for the clear sky case aV with 3.44 (Table ).

Naturally, the more complex the atmosphere is, the higher the overall possible total information content is. The initial degrees of freedom for these cases are more numerous and a greater portion of the channels can be used to reduce them (Fig. ). The major part of the information comes from the frozen hydrometeors. IWC gains most information for both the mass density (3.10 on average) and the particle mean masses (3.28 on average). The information content for the mean mass of IWC is greater than the one for the mass density of IWC. It is necessary to keep in mind the specific choice of the a priori assumptions. The information content is in principle high for IWC. The proportion between the information content for the mass density and the mean mass of IWC may depend on the choice of the thresholds for the mass density and the mean mass used to calculate the error covariance in ln-space (see Sect.  and discussion at the end of this section).

SWC gains an information content of 2.57 on average. The mean mass of snow gains 1.73 on average. IWC and SWC compete for the information. If both are included, their information contents both decrease. The decrease is stronger for SWC in the presence of IWC than for IWC in the presence of SWC. This mirrors the behaviour of the Jacobians discussed above. If both frozen hydrometeor types are present, the absolute values of the Jacobians decrease. The spread of the information content for the different atmospheric compositions is slightly higher for SWC, but the minimum information content is high for both, 2.58 for IWC and 1.58 for SWC. The outer channels of the 118 GHz line, the window channels below, and the channel at 157 GHz add a considerable amount of information for LWC (1.69 on average) and some for RWC (1.08). For H2O under clear sky conditions, a maximum ΔDOF of 3.44 is gained, which decreases in the presence of clouds. Once hydrometeors are present in the atmospheric column, the information content for H2O is considerably reduced down to 1.21 for the case aILRS.

The overall picture is the same for both land and ocean, with only slight changes in the ranking of the total information content according to the total ΔDOF (not shown). For the higher land emissivities, the cases aILRS, aIRS are swapped. However, their information contents have very similar values in both cases and a small change in the information content easily leads to a slightly different ranking of the atmospheres.

Now the focus is set on ICI, which is designed to detect frozen hydrometeors. The set of channels is reduced to the eleven channels which correspond to this instrument (see Table ). Naturally, the resulting total mean ΔDOF of 6.19 is smaller than before because the number of channels is smaller. This reduction is mostly at the cost of information about the liquid hydrometeors, not the frozen hydrometeors, because the channels below 183 GHz are missing entirely in this case. For IWC the mean information content is only slightly reduced to 2.76, and for SWC to 2.27. For the particle mean masses, ΔDOF for IWC m¯ is reduced to 2.70 and the one for SWC m¯ is slightly reduced to 1.31 (Table ). The information content for liquid hydrometeors is considerably reduced to 0.42 for LWC and to 0.13 for RWC.

In the ranking according to the total ΔDOF (Fig. ) the atmospheres nicely separate into three groups. The atmospheres containing IWC build the group with highest total ΔDOF, those containing SWC but no IWC rank second. As before, the four atmospheres with the least information content are those without any frozen hydrometeors. Also information about the microphysical properties of the frozen hydrometeors is gained, although it is reduced compared to the full set of channels. For the purpose of ICI it is no disadvantage to leave out the lower channels. ICI's focus is on the detection of cloud ice and its ability to observe it on the global scale with a large spatial coverage seems to be unprecedentedly high.

We are aware that the results discussed in this section depend on the definition of the a priori covariance error. They especially depend on the choice of the lower threshold for the calculation of the covariance in ln space. If a very large threshold is assumed, the information content will be significantly diminished because there is only little variance left. For a very small numerical threshold, the variance will be large, and we will gain too much information. The dependence of the mean information content on the chosen threshold is shown in Fig.  for all hydrometeors for both mass density and mean mass. For cloud ice and snow, also the range between minimum and maximum ΔDOF is shown. The mean information content for IWC and SWC decreases from 4.7 down to about 1.5 for thresholds from 10-16 kg m-3 to 10-5 kg m-3. The mean information content for IWC m¯ and SWC m¯ increases by 1.6 or 0.6, respectively. The dependence of the mean information contents for the liquid hydrometeor mass density and mean masses is weaker. The spread of the minimum and maximum for SWC shows little dependence on the threshold for the mass densities. For IWC, the spread decreases for the mass density but increases for the mean masses. The threshold is only varied for the mass densities, but not for the mean masses. The dependence of the information content for the mean masses is due to the fact that a combined analysis of all variables is performed. Furthermore, the cross correlations between mass densities and mean mass will cause a change of the information content of the mean masses.

For this analysis, thresholds were chosen, which are as physically based as possible (see Sect. ). In particular, we use 10-7 kg kg-1 for the mass densities. Since this study is based on a spring time case from the mid-latitudes, the variance is likely smaller than one would expect if a whole year was taken into account.

So far only one single, smooth idealized cloudy profile was analysed. To consolidate the results from the previous section, we have randomly drawn 90 more realistic cloudy profiles directly from the 10 000 ICON profiles, which were used to create the mean profile (see Sect. ). The information content ΔDOF was calculated in the same way as before. Although an even greater dataset would be desirable, the calculation of the Jacobians with ARTS is numerically rather expensive and we had to trade extensive statistics against computing time.

Figure  gives an overview of the information contents for the different hydrometeor types depending on the respective hydrometeor paths in the atmospheric column. The results from the idealized atmosphere presented above are substantiated in this statistical approach. Naturally the system tends to higher information contents for higher mass contents of the respective hydrometeor. The values are in a similar range as they were found above, except for SWC. For SWC, the ΔDOF from the idealized atmosphere tends towards higher information contents than most of the realistic atmospheres, even though the path is well in the range of paths from those 90 atmospheres. This may be due to the fact that we tried to include all hydrometeor types in the idealized profiles. In most realistic profiles the combination of snow and liquid clouds is rare. Thus, in general we expect to gain slightly less information on SWC than was found for the idealized mean atmosphere.

For cloud ice and snow, high total ΔDOFs tend to occur for high integrated path values IWP and SWP. For the liquid hydrometeors, a relationship between high paths and high total information content is not found. On the contrary, for LWC the low total ΔDOFs tend to be at the upper end of LWP, where the cloud is mainly liquid and only little frozen water mass is present. If ice is present in the cloud, the liquid hydrometeors will be consumed quickly by the Bergeron–Findeisen process and riming, yielding lower LWPs but higher total ΔDOFs. The overall high information contents gained for frozen hydrometeors again points to the ability of sensors with such high microwave channels to observe ice and snow particles in clouds on a global scale robustly regardless of the atmospheric composition.

Some caution has to be paid with regard to the physical assumptions underlying the scattering and absorption properties of ice particles. For example, found that changes in the size distribution and scattering properties can shift the information content from IWC to solid precipitation. Also, contrary to this study, did not find their retrieval was sensitive to IWC using the same channels. They base their analysis on ICON simulations with a one-moment scheme, where the size distributions for IWC and SWC are more distinct and hardly overlap, and where the IWC distribution is shifted to smaller ice particles (Sect. ). Therefore the information content is distributed differently between IWC and SWC. In nature, this arbitrary distinction between IWC and SWC does not exist and we only gain information about the whole set of frozen hydrometeors at once, limited only by the size and amount of the particles, and depending on their shape.

In summary, the analysis of the model atmospheres with their different compositions shows satisfactory results. Despite the strong interdependencies of the Jacobians for cloudy conditions presented in Sect.  the information content about the frozen hydrometeors proved to be high, independent of the atmospheric composition. This is especially due to the channels at high frequencies, for which the Jacobians peak at different heights. Satellite missions such as ICI on MetOp SG, which employ a set of these high frequency channels therefore have a great potential to provide a robust retrieval of cloud ice and snow. For these frozen hydrometeors, even an estimation of a profile may be possible, because the channels give information about different heights in the atmosphere and we get ΔDOFs up to four for IWC, which corresponds to four different heights. Also, especially for cloud ice, consistently some insight into the microphysical properties is gained, i.e. about the mean particle mass.

To observe liquid hydrometeors, the lower channels from Deimos and Mars proved to be useful. These channels in the regions between 50 and 57 and around 118.75 GHz are employed on MWI on Metop-SG. MWI uses, amongst others, channels in these regions to retrieve precipitation over land and sea ().