Synergy of Active- and Passive Remote Sensing: An Approach to Reconstruct Three-Dimensional Cloud Macro- and Microphysics

This paper presents a method to retrieve three-dimensional cumulus cloud macroand microphysics measured by remote sensing instruments on the German research aircraft HALO. This is achieved by combining our hyper-spectral pushbroom spectrometer specMACS with active and passive remote sensing instruments, such as a lidar, a microwave radiometer, a radar and dropsondes. Two-dimensional cloud information such as cloud size, optical thickness, effective radius and thermodynamic phase are retrieved by specMACS with established remote sensing methods. Information of the other active and passive remote 5 sensing instruments with a smaller field-of-view are mapped to the wider specMACS swath following Barker et al. (2011). The combination of specMACS with passive and active remote sensing quantities, for example, the Cloud Top Height from lidar measurements, allows new possibilities: three-dimensional cloud macrophysics can be reconstructed. Applying a sub-adiabatic microphysical model constrained with measurements allows to extend the measured quantities to a three-dimensional representation of microphysics. A consistency check by means of a three-dimensional radiative transfer simulation of the specMACS 10 observations of these derived three-dimensional cloud fields shows good agreement.

scene and depended on the method. Wolf et al. (2019) also mentioned that these values are highly affected by shadows on cloud edges which could not have been taken into account (see also e.g., Marshak et al. (2006)). Shadow effects can be eliminated in our case by combining the lidar with our pushbroom spectrometer specMACS (spectrometer of the Munich Aerosol Cloud Scanner, Ewald et al. (2016)) which was done following the approach of Barker et al. (2011). Barker et al. (2011) described for the first time a method to combine one-dimensional, satellite-based active remote sensing data with the pushbroom spec- 5 troradiometer MODIS aboard the Aqua and Terra satellites. This algorithm is the cornerstone for the combination of passive and active instruments in our research.
The hyper-spectral pushbroom spectrometer specMACS allows the determination of horizontal two-dimensional information such as cloud fraction and cloud size distribution (see Appendix B), stereoscopic cloud top height , 10 optical thickness, effective radius and the thermodynamic phase. This retrieved specMACS data was then combined with other passive and active remote sensing instruments in form of a microwave radiometer, a lidar and a radar, following the approach of Barker et al. (2011). The cloud top height from the WALES lidar (Wirth et al., 2009) and the cloud bottom height calculated from dropsonde data create the possibility to reconstruct single-layer cumulus clouds in all three dimensions. Using the liquid water path from the HAMP radiometer (Mech et al., 2014) derived by Jacob et al. (2019a) and the cloud droplet effective radii 15 retrieved from specMACS as constraints for a simple microphysical model allows us to resolve the three-dimensional cloud microphysics. In addition, the radar is used to determine multiple cloud layers.
The purpose of this study is to reconstruct three-dimensional macro-and microphysical cloud properties consistent with all available remote sensing observations for the observed trade wind cumulus and stratocumulus fields. These datasets will form 20 the basis for a systematic analysis of radiative diabatic effects of these cloud fields.
Section 2 will give an overview of the NARVAL-II campaign, the instrumentation and the radiative transfer software libRadtran. Section 3 describes the theoretical cloud macro-and microphysical properties and the simple sub-adiabatic microphysical model used to reconstruct the three-dimensional cloud microphysics. Section 4 describes how passive and active remote sens- 25 ing instruments are combined, which data needs to be excluded and presents our three-dimensional cloud reconstructions.
Finally, Section 5 discusses and concludes our research.

Instruments, Alignment and Radiative Transfer Software
The NARVAL-II campaign took place close to Barbados for four weeks in August 2016 . Ten research flights were conducted with the German research aircraft HALO. The NARVAL-II campaign is closely related to the multi- 30 aircraft campaign called NAWDEX (North Atlantic Waveguide and Downstream Impact Experiment) which took place in Iceland during autumn of the same year (Schäfler et al., 2018). HALO had in both campaigns the same remote sensing instruments aboard with their sensor eyes looking downwards. The instruments are: the pushbroom spectrometer specMACS (Ewald et al., 2016), the one-dimensional spectrometer SMART (Wendisch et al., 2001), the lidar system WALES (Wirth et al., 2009), the radiometer and cloud radar HAMP (Mech et al., 2014), the HALO internal sensor system BAHAMAS and dropsondes.
They will be described in the following.

specMACS
The spectrometer of the Munich Aerosol Cloud Scanner (specMACS) was developed at the Ludwig-Maximilians-Universität 5 of Munich (LMU) and used to derive optical and macro-and microphysical cloud properties (Ewald et al., 2016). It is the main instrument allowing to map nadir remote sensing information with a limited field-of-view to the wider image swath for a synergistic retrieval. It is a hyper-spectral pushbroom spectrometer based on two line cameras measuring radiation from 400 to 2500 nm with a spectral bandwidth ranging between 2.5 and 12 nm. The spatial dimension allows to resolve the twodimensional cloud structure. The across-track field-of-view is 32.7°for the visible near-infrared (VNIR) camera and 35.5°for 10 the shortwave infrared (SWIR) camera produced by SPECIM (Specim, Spectral Imaging Ltd.). VNIR has 1312 and SWIR 320 spatial pixel on its swath. Additionally, specMACS has one two-dimensional RGB camera with a larger field-of-view. The temporal resolution during the campaigns was 30 Hz and the instantaneous field-of-view is across-track about 1.4 mrad for the VNIR and about 3.8 mrad for the SWIR. The along-track instantaneous field-of-view is about 2 mrad for both cameras. This allows, for example, to resolve 30 m surface features along the across-track line of 8.7 km from 15 km altitude. Recently, 15 specMACS got extended by a wide-field-of-view polarization imager.

HAMP
The HALO Microwave Package (HAMP) was used to retrieve the LWP (Liquid Water Path) among other quantities. It consists of passive microwave radiometers with 26 channels with frequencies ranging between 20 and 183 GHz and an active 35.5 GHz cloud radar. Seven K-band channels ranging from about 22 to 31 GHz and the 90 GHz channel are used to derive the LWP. This is possible since the brightness temperature increases at about 22.2 GHz with increasing liquid water. The Passive and Active Microwave Transfer code PAMTRA (Maahn and Löhnert, 2017) and dropsonde profiles are used to create simulated HAMP measurements (Jacob et al., 2019a). Comparing simulations with measurements using a linear regression model (Mech et al., 2007) allows the LWP determination. The LWP values have an absolute accuracy of about 20 gm −2 for LWP values below 5 100 gm −2 and an accuracy of 20 % for values above (Jacob et al., 2019a). The field-of-view of the microwave radiometer K-Band has a diameter of 5.0°and the temporal resolution is 1 s.

WALES
The airborne multi-wavelength WALES lidar instrument (WAter vapor Lidar Experiment in Space, Wirth et al. (2009)) is a combined water vapor Differential Absorption (DIAL) and High Spectral Resolution Lidar system (HSRL, Esselborn et al.

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(2008)). It allows direct and simultaneous measurements of water vapor mixing ratios and aerosol optical properties from aircraft to ground level. Particle extinction at 532 nm as well as particle linear depolarization and particle backscatter at 532 and 1064 nm can be retrieved. The high vertical resolution of 15 m allows detecting cloud top heights with high vertical accuracy (Gutleben et al., 2019). WALES data with 5 Hz temporal resolution corresponding to a horizontal resolution of approximately 170 m at typical aircraft speed are used in this study.

Dropsondes, BAHAMAS and External Quantities
Dropsondes are radiosondes that are dropped from an aircraft measuring temperature, pressure, relative humidity and horizontal wind velocity and direction. The dropsonde system (HALO-DS, Hock and Franklin (1999)) of the German Aerospace Center and the corresponding dropsondes are used to get the atmospheric temperature and pressure profiles as well as the surface wind velocity for our radiative transfer simulations. Moreover, the surface temperature and the dew point are used 20 to calculate the Lifted Condensation Level (LCL) and the corresponding Cloud Bottom Height z cbh . The dropsondes were released at irregular time steps, ranging from several minutes to hours, depending on the synoptic situation and flight security.
In each flight between 10 and 50 dropsondes were released resulting in an estimated spatial resolution ranging roughly from 50 to 400 km. 25 The Basic Halo Measurement and Sensor System (BAHAMAS) provided us highly accurate aircraft position and orientation data with a temporal resolution of 100 Hz such as velocity, altitude, location, as well as the principal axes roll, pitch and yaw. These data are necessary, for example, for the determination of the CTH (Cloud Top Height), transformations of camera coordinates into Cartesian coordinates and radiative transfer simulations. 30 5 https://doi.org/10.5194/amt-2020-49 Preprint. Discussion started: 11 March 2020 c Author(s) 2020. CC BY 4.0 License.
External quantities such as the Solar Zenith Angle (SZA) and Solar Azimuth Angle (AZI) are important for the retrieval of the optical thickness and for developing a shadow mask. Both angles are calculated using the PyEphem package which is known for high-precision astronomy calculations (Rhodes, 2011).

Coordinate Systems
The WGS-84 (World Geodetic System 1984) is used as the earth coordinate system which includes the flattening of the earth 5 (see e.g. technical report, DMA (1991)). It is used as reference system for the GPS (Global Positioning System) and is, for example, the standard global reference system of the U. S. Department of Defense (Groten et al., 1988). The reference coordinate system is a Cartesian coordinate system with NED (North-East-Down) coordinates using the WGS-84 surface as as fundamental plane. It is located at height 0 m on the WGS-84 ellipsoid at arbitrary coordinates (e.g. at 19°north and 49°west for NARVAL-II). The north axis is parallel to the longitudes and the east axis is parallel to the latitudes.

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The local reference coordinate system is also a reference coordinate system in Cartesian coordinates. The only difference is that this one is located relatively close to the observer which is in our case the HALO position. It is used for reconstruction of the three-dimensional cloud macro-and microphysics. Data of WALES, specMACS and HAMP were transformed into our reference or local reference coordinate system as described above. The aircraft HALO has its own coordinate system. Inside 15 the HALO, the coordinate systems of BAHAMAS, WALES as well as the VNIR and SWIR cameras are located.

Sensor Positions
The specMACS could not be aligned perfectly inside the HALO aircraft because of time constrains. The viewing direction of the SWIR camera was aligned about 2.6°towards the HALO aircraft nose and the center pixel of the SWIR camera was not oriented perfectly vertical downwards along the yaw axis of the aircraft. The WALES lidar instead looks as good as technically 20 possible vertically downwards along the yaw axis of the aircraft. That means that the SWIR camera will see a cloud earlier or, in rare cases, later than the WALES lidar by a time offset of ∆t cth depending on the CTH. Figure 2 shows schematically the alignment between WALES and the SWIR camera inside specMACS. In the following, the angular offset between the instruments will be corrected. A not-corrected angle offset in viewing direction might influence the results particularly. 25 Firstly, the time offset needs to be found: We calculate vector x which is the projection of vector p defined as withê WALES as the WALES viewing vector onto the SWIR viewing directionê SWIR As can be seen in Equation 1, vector p is determined using the CTH derived from the WALES lidar. This vector ends in point 30 P . The vector x ends in point X. Now we use the x-component of the differential vector between P and X which indicates Figure 2. Illustration of the alignment of the SWIR camera which looks 2.6°towards the HALO nose. OSWIR denotes the origin of the SWIR coordinate system and OWALES denotes the origin of the WALES coorindate system. WALES looks directly downwards. The SWIR camera will see a cloud by an time offset of ∆t cth earlier than WALES when the cloud is sufficiently far away. This time offset depends also on the CTH. The higher the cloud, the smaller the time offset.
the horizontal distance between the two points. The temporal offset can then be calculated by dividing this x-component by the aircraft velocity.
Secondly, relative angular offsets between the instruments were identified by correlating the SWIR radiance measured in the nadir pixels with the BSR (BackScattering Ratio) at 532 nm measured by the WALES lidar for different rotation angles.

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Having found the best correlation, we can rotate the SWIR camera system with the rotation matrix R where α, β and γ describe the rotation around the z, y and x axis in the SWIR coordinate system, respectively. α denotes the yaw angle offset, β denotes the pitch angle offset and γ denotes the roll angle offset. What is not taken into account is the bending of the whole aircraft fuselage, but normally this is a small effect within such a short aircraft as HALO. Therefore, this 10 can be neglected.

Radiative Transfer Calculations
The radiance arriving at the observer for different wavelengths is modeled using either a plane-parallel or a three-dimensional atmosphere. In case of a plane-parallel atmosphere, the one-dimensional radiative transfer solver DISORT (Discrete Ordinates Radiative Transfer Program for a Multi-Layered Plane-Parallel Medium, Stamnes et al. (1988)) is used. In case of a three- As libRadtran input we used the closest dropsonde profile and extended it above the aircraft with the profile of the tropical atmosphere as defined by (Anderson et al., 1986). A background aerosol with a surface visibility of 50 km was defined and the proper sun-earth distance is used. The reflectance of the ocean is calculated according to the Bidirectional Reflectance 5 Distribution Function (BRDF) following Cox and Munk (1954) using the wind velocity of the nearest dropsonde. Mie scattering (Mie, 1908) is used for clouds. The REPTRAN band parameterization (Gasteiger et al. (2014), Buehler et al. (2010)) with medium resolution (5 cm −1 ) is used for molecular absorption.  Typical cumulus clouds form due to rising warm and moist air parcels that cool with height adiabatically.
where T (0) is the ambient temperature and τ d (0) is the dew point temperature at sea surface level (e.g., Wallace and Hobbs, 2006). The measured humidity profile of the dropsonde is not used since the dropsonde seldomly flies through a cloud.

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According to Gutleben et al. (2019), the along-track cloud top height z cth is determined using a threshold of 20 in vertical profiles of backscatter ratio. The Cloud Geometrical Thickness z cgt can then be calculated using 3.2 Microphysics 10 Cloud microphysics is described by the Liquid Water Content (LWC), the effective radius r eff , the cloud optical thickness τ , the thermodynamic phase I p , the Cloud Droplet Number Concentration N and the cloud droplet number density N (a).
The effective radius r eff of cloud droplets was defined by Hansen and Travis (1974) as an extinction-weighted mean of the size distribution of the droplets, i.e., the ratio of the third to second moment of the size distribution which is often expressed in µm. z c is the height above the CBH, N (a) is the cloud droplet number density and a is the radius of the cloud droplets.
The cloud optical thickness τ c can be expressed as the the integrated Extinction Efficiency Q ext between the heights z c1 and 20 z c2 in a plane-parallel atmosphere approximation. Hansen and Travis (1974), for example, expressed the optical thickness as The Liquid Water Content is the amount of liquid water per unit volume of air and often expressed in g m −3 . The LWP (Liquid Water Path) is defined as the integral of the Liquid Water Content over the vertical height of the cloud z c , from cloud bottom z cbh to cloud top z cth and given in g m −2 . It can be expressed as with the density of liquid water ρ lw .
The thermodynamic phase can be derived using the approach of Jäkel et al. (2013) where L 1700 is the radiance at 1700 nm and L 1553 is the radiance at 1553 nm. A positive slope I p shows that ice clouds 5 are observed and a negative slope I p means that water clouds are observed. This approach is possible since electromagnetic radiation in the wavelength range between 1553 and 1700 nm is absorbed stronger by ice than by liquid water (e.g. Pilewskie and Twomey, 1987).

Adiabatic Theory
Liquid Water Content and effective radius r ef f are both a function of cloud height z c . It is interesting to know how both develop 10 with height z c . Air rises due to thermal updrafts or lifting and starts to condensate when the rate of condensation is larger than the rate of evaporation. The air gets saturated and water condenses on condensation nuclei. Droplet growth by condensation happens when the radii are below 18 µm and no ice exists (Rogers, 1976). This droplet growth can be described by an adiabatic model. Similar adiabatic models have already been used by, for example, Brenguier et al. (2000).

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The adiabatic Liquid Water Content is the maximum amount of liquid water that can be contained in a specific volume.
Thus, we can express it with the density of condensed water ρ H2O using where m lw is the mass of liquid water and V air is the specific volume of air. The partial pressure of the condensing water e con must be the difference between the partial pressure of the water vapor e at a given height and the saturation pressure e sat 20 where z c is the height above the cloud bottom and T c (z c ) the cloud temperature profile. This difference increases with height after the Lifted Condensation Level is reached (see Figure 4).
Using the ideal gas law, the adiabatic Liquid Water Content (Equation 10) becomes 25 where R v is the specific gas constant for water vapor. Furthermore, the ideal gas law states inside the cloud where ρ v is the density of water vapor. The density of water vapor ρ v can be expressed with the specific humidity q. With the help of the ideal gas law for the density of air ρ air , the partial pressure of water vapor e(T c (z c ) can be described as 30 Figure 4. Schematic showing the partial pressure of water vapor e (light blue line) and the saturation pressure esat (dark blue line) decreasing with height. Condensation starts when the Lifted Condensation Level is reached, i.e., when saturation pressure is equal the partial pressure of water vapor. The gray area shows the maximum condensed water econ. Droplets will grow as long as the partial pressure is larger than the saturation pressure. This schematic is initialized with 25°C surface temperature and a surface partial pressure for water vapor of 10 hPa.
with the air pressure p air (T c (z c )). Inserting this Equation into Equation 12 results in: The change of saturation pressure with cloud height e sat (T c (z c )), the change of temperature inside the cloud T c (z c ) and the change of air pressure with cloud height p air (T c (z c )) can be calculated using the Clausius-Clapeyron equation, the moist adiabatic lapse rate and the barometric formula, respectively (see Appendix A).

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The theoretical adiabatic Liquid Water Content value is usually not measured but a lower one since, for example, entrainment processes play a role. Morton et al. (1956) defines entrainment as the incorporation of ambient air into a localized circulation.
This mixing of environmental air into organized air clusters leads to the evaporation of cloud droplets and also can reduce the cloud droplet radii. Therefore, we get a measured Liquid Water Content LW C m which is less or equal than the theoretically 10 calculated adiabatic Liquid Water Content LW C ad The parameter a ∈ [0, 1] is the degree of entrainment and shows how strong the measured Liquid Water Content differs from the theoretical value, i.e., how strong the entrainment is. The left part of Figure 5 shows some Liquid Water Content profiles for different entrainment factors a according to Equations 16 and 15. The closer the factor a to 1, the weaker the entrainment.  Martin et al. (1994) showed that, for a measured cloud droplet size distribution, the effective and mean volume radius can be connected by an empirical and averaged factor k (k ≈ 0.80 ± 0.07 for maritime clouds). Thus, it can be shown that the Cloud Droplet Number Concentration for maritime clouds is where ρ w is the density of liquid water, N is in units cm −3 and r is in units µm (e.g. Reid et al., 1999). In case of entrainment 5 we know that the measured LW C will be different from the theoretical LW C ad . Therefore, we must consider Equation 16.

Thus, Equation 17 yields
Moreover, we can reconstruct the effective radius profile using Equation 18. The effective radius profile becomes 10 where N en is given in units cm −3 and the radius r eff in units µm. In this case, we assume that the Cloud Droplet Number Concentration N en remains constant within the cloud. The right part of Figure 5 shows three effective radius profiles for three different N with identical initial temperature and pressure values at cloud base. The dashed and dotted orange lines show the profiles for an entrainment factor a of 0.8 and 0.6, respectively.
Optical thickness, effective radius and thermodynamic phase can be retrieved for the full specMACS field-of-view. Quantities such as CTH and LWP can only be determined for the nadir direction directly below the aircraft where lidar and radiometer measurements exist. One possible approach to spread the information of nadir measurements to the wider specMACS field-ofview is described by Barker et al. (2011).
5 Figure 6 shows the basic idea of Barker et al. (2011). The spectra between nadir columns and off-nadir columns are compared with each other within about 25 km distance to the observer. If the spectra match, the nadir-measured remote sensing information of another instrument is transferred to the off-nadir column. The nadir pixel will then be called donor and the off-nadir pixel will be called recipient. This mapping is possible since the spectrum contains information about cloud macro-and microphysics. For example, the spectrum at two different cloud locations with the same geometrical thicknesses but with two different LWPs will look different. Figure 7 shows the shortwave infrared spectrum of specMACS for different LWP values at a constant CTH (blue) and for different CTHs at constant Liquid Water Path values (dashed, red). We see, for example, between 1400 and 1800 nm 15 that the radiance decreases with increasing LWP values because the cloud droplet effective radii usually become larger which means that the reflection becomes weaker in this wavelength range. Moreover, the radiance increases with increasing CTH where i is the temporal axis and j the spatial axis of the specMACS, m is the temporal axis and 0 the spatial location of the nadir pixel, λ the wavelength and L the radiance. The donor pixel D is the pixel with the smallest distance d: The Euclidean distance alone does not include physics. In the shortwave infrared region, radiances at smaller wavelengths 10 are higher than radiances at longer wavelengths because the solar radiation is more powerful. Subsequently, radiances at shorter wavelengths contribute more to the distance and radiances at longer wavelengths contribute less to the distance. Therefore, we introduce a weight ω which we define as where L toa,λ is the radiance at the top of the atmosphere calculated by using Kurucz (1992) in libRadtran. This weight 15 decreases the contribution to d of radiances at shorter wavelengths and it increases the contribution at longer wavelengths.
We got initial good results with this weight. We also, for example, increased the weight at radiances where water vapor absorption is enhanced, but did not see any significant influence on the result. We also tested the algorithm following the suggestion of Barker et al. (2011) to reconstruct the nadir cross-section and got similar results (not shown).

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The data amount is, with about 120 GB each flight, not small. In Figure 7 not only differences between the spectra are visible, but also similarities. Thus, it should be possible to describe most of the spectra with fewer parameters which can be done with Principal Component Analysis (PCA, see e.g. Jolliffe (2002)). We apply the PCA on the measured radiance L

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where P is the PCA transformation, L is the radiance in the PCA notation, L is the measured radiance, c the PCA components and k is the mean value of the measured radiance. Furthermore, we yield for x: Inserting the result into Equation 23 yields: 20 18 PCA components explain more than 99.7 % of the variability of the original signal. We tested the algorithm with 50 components without any visible changes in the results. Due to the use of the PCA, we save around a factor of ten in memory.
Having found the according donors, the CTH of the WALES lidar, the reflectivity of the HAMP radar and the LWP of the HAMP radiometer are mapped on the wider specMACS field-of-view. Only donors having a maximum distance of about 25 km from the observer are used. Barker et al. (2011) found that the usual distance between the recipient and the donors are less than 30 km and mostly below 5 km. We assume that meteorological conditions are mostly similar within that distance. Over the cloud-free ocean areas, the donor pixels are influenced by the sun reflection and cloud shadows. Therefore, we see 10 over the ocean in some cases a signal. Knowing the velocity v, the height z plane , as well as the roll, yaw and pitch angles of the aircraft, the height of the cloud z cth and the viewing angles of all camera pixels, we can transform the CTH points p b seen from the specMACS SWIR camera into the three-dimensional reconstruction following four steps:

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Firstly, we put the origin of the local reference system in the middle of one cloud scene (see also Figure 9). Secondly, we calculate the CTH points p b and transform them from the specMACS SWIR camera coordinate system into the local reference system with 15 m box resolution following α is the viewing angle of the SWIR camera pixel, O SWIR the origin of the specMACS SWIR camera which is in the focal point Thirdly, we average all points p s within a circle of 5 m radius. Thereby, and for performance reasons, a K-D Tree (Bentley, 1975) was used to create a space-partitioning data structure to find for each CTH point p b all points p s within a circle of 5 m radius (circle on plain in Figure 9). Single gaps can occur when the cloud blocks the specMACS SWIR view to pixel lying behind. These gaps are filled with the average height of the surrounding pixel. 5 Finally, we use the theoretical determined CBH z cbh of the nearest dropsonde and fill the columns up to the WALES retrieved CTH z cth with microphysical information as described in the following.

Microphysics
Lookup tables are calculated for the optical thickness and the effective radius for different atmospheric parameters and solar positions using a plane-parallel approach. The radiance at 750.2 nm is used to derive the optical thickness and the radiance 10 at 2160 nm is used to derive the effective radius similar to Nakajima and King (1990). Comparing the measured values to the simulated values allows to determine the optical thickness and effective radius. The thermodynamic phase is calculated  Figure 10 shows on the left part the effective radius and on the right part the corresponding optical thickness with an applied cloudmask. Both quantities have been derived with specMACS alone. The effective radius over the ocean is significantly higher than at cloud areas and increases to values above 25 µm. Inside cloud areas, the effective radius has values between 6 and 17 µm. On cloud edges, as well as at cloud shadow areas, the effective radii values increase strongly. We also see a wave pattern inside the ocean caused by sun reflection. The optical thickness increases in cloud center to values of about 35. On cloud borders, the optical thicknesses is below 5.   The Liquid Water Content Profile is derived with the adiabatic equations (Equation 15) using dropsond data, the CBH, the CTH and the LWP. We adapt this profile with the factor a so that the theoretical LWP matches the measured Liquid Water Path

Filtering
The following filtering method is very strict and eliminates about 90 % of the available data but has to be done to find the pixels in a cloud domain where the effective radius retrieval of Nakajima and King (1990) works well with a high certainty. With the  2010)). Hence, we can only retrieve the effective radius for shadow-free areas, optical thick clouds, ice free areas and cloud parts with less three-dimensional effects. 15 We made the assumption that the effective radius usually grows with height (see Subsection 3.3). When the retrieved effective radii vary within the same cloud height significantly, the values are probably affected by shadows, three-dimensional photon losses or gains, ice effects or, in the worst case, by entrainment (the mixing of ambient clear sky air into the cloud). One way to define the quality of the retrieved effective radii for the whole effective radius profile is the standard deviation averaged over all contributing cloud layers which we define as where i is the number of effective radii each layer, j the number of discrete height layers, r eff the effective radii andr eff the mean effective radii of all effective radii r reffi at height layer j. In an idealized case and without mixing effects this mean standard deviation would be very close to zero. 25 First, we apply the cloudmask as described in Appendix B. Then, we filter areas that contain shadows using ray tracing. Figure 12 shows an example of the shadow mask. The area of a ray will be identified as shadow if it intersects with another cloud surface or when it is close to one. In addition, we exclude all ice parts of the cloud. We then calculate the mean standard deviation for increasing optical thickness thresholds and for increasing geometrical distance thresholds from the shadow areas. After an applied optical thickness threshold of above roughly 8, the mean effective standard deviation does not decrease so much anymore. Thus, we removed all parts of the clouds having a smaller optical thickness than 8. Marshak et al. (2006) also showed in their Figure 5 that optical thicknesses smaller than 10 are connected to strong fluctuations. Thus, an optical thickness threshold of 8 was applied and the mean standard deviation could be reduced from about 4.8 to below 3 µm. 5 Moreover, we found that the mean effective standard deviation does not decrease strongly for more than 45 m geometrical distance from shadow areas. Thus, a geometrical distance threshold of 45 m was applied. The mean standard deviation in this case could be reduced from about 4.8 to about 2.5 µm. Combining both filters, the mean standard deviation could be reduced in most cases to values below about 2 µm.

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Applying these thresholds on our data, we hypothesize that the areas left over are mostly not influenced by three-dimensional effects, shadow, ice, or surface reflectivity. Figure 13 shows the retrieved effective radius profile and the theoretical one. The increase of the effective radius with cloud height (see also Figure 13) is only visible after this described filtering method.   and east is towards the right side x [m]. We see that only for the largest clouds and central areas some effective radius values are available after the filtering method. Strong effective radius fluctuations cannot be seen in these areas.

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For these filtered effective radius areas, the Cloud Droplet Number Concentration N is calculated and averaged. It is 38 ± 15 cm −3 in this cloud scene. Moreover, the best theoretical effective radius is fitted onto the retrieved and filtered effective radius data and used for all clouds within this cloud scene.

Three-Dimensional Reconstruction
A reconstructed cloud scene with about 35 % cloud cover is shown in Figure 15. This cloud scene shows reconstructed cumu- We see single spikes on the cloud edges with high Cloud Geometrical Thicknesses. We see also that the effective radius is identical in every cloud. Entrainment effects and mixing processes, especially on the cloud borders, are, in the effective radius 15 profile, not considered. Also the constant CBH is visible. The Liquid Water Content is not identical in every cloud since it is derived via the LWP of the HAMP.

Consistency Check
The quality of the reconstructed three-dimensional cloud fields will be examined in the following. Since in-situ measurements for a validation against real data are not available, theoretical experiments with simulations are performed. Thereby, the radiative transfer software MYSTIC included in libRadtran is used (see Section 2.7). Based on reconstructions introduced above, the radiance field observable by specMACS is simulated. The resulting synthetic measurement is then compared to the true 5 measurement. Thus, general deficiences as well as limitations of our method should become visible. We use simulations at two wavelengths, 750.2 and 1553.5 nm. 750.2 nm is a wavelength in the visible spectrum and is mainly sensitive to the optical thickness. 1553.5 nm is a wavelength in the shortwave infrared spectrum and has additional sensitivity to the effective radius.
The simulated radiance, the measured radiance and the corresponding histograms for 750.2 nm are shown in the left part of Figure 16. The right part, shows the same for the radiance at 1553.5 nm.

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The two histograms in Figure 16 show The shadows on the sea surface at both wavelengths show that CTHs and CBHs are reconstructed well. However, the extent of the shadows is larger for the simulation and more pronounced. This is mainly a consequence of missing variability of the 20 CBH measurement which is replaced by a constant Convective Condensation Level representing the lowermost possible cloud height.
At 750.2 nm, the reflectivity is sensitive to the optical thickness. In general, the simulated radiances (b) at the cloud centers of the smaller clouds agree well with the measured radiances (a). However, in the cloud centers of the two biggest clouds 25 close to the edge of the specMACS field-of-view, differences of up to roughly 65 W · sr −1 · m −2 can be seen. Furthermore, cloud edges appear brighter in the simulation. The averaged reflected radiance of the cloudy areas is 109.86 W · sr −1 · m −2 in the simulation and 98.57 W · sr −1 · m −2 in the measurement. This is a difference of about 11.4 % with higher values in the simulation. The radiance distributions at 750.2 nm are shown in the histograms in Figure 16c. 30 At 1553 nm, the forward simulation shows additional sensitivity to liquid water absorption, and thus, to droplet effective radius. Looking into details we find additional differences compared to the comparison at 750.2 nm. For example, the level of brightness in the simulation is, in general, elevated which is especially obvious in cloud centers. Moreover, cloudy areas appear larger because cloud edges are brighter. The averaged reflected radiance of cloudy areas is 20.91 W · sr −1 · m −2 in The LWP is scaled so that the cloud optical thickness remains constant.
the simulation and 18.58 W · sr −1 · m −2 in the measurement. This is a difference of about 12.5 % with higher values in the simulation.
The differences of the radiances at 750.2 nm between measurement and simulation has to be investigated further. At 750.2 nm the radiance distribution of the simulation in Figure 16c shows a higher occurrence of median radiance values 5 (at about 80 W · sr −1 · m −2 to 150 W · sr −1 · m −2 ). In addition, the simulated peak values are smaller than the measured ones.
Both differences are possibly caused by the rather wide 5°field-of-view of the HAMP radiometer. The wide field-of-view tends to average out large LWP values to lower values which will result in a lower reflectivity and small LWP values become larger which result in a higher reflectivity. This is shown in Figure 17. (a) shows the measured radiance at 750.2 nm, (b) shows the measured radiance averaged to the HAMP resolution, (c) shows the simulated radiance and (d) shows the value distribution. We see that the largest measured radiance values become smaller and the smallest measured radiance values become larger when we reduce the measured radiance to the HAMP resolution. Also the peak simulated radiances agree well with the measured radiances reduced to the HAMP footprint. shows the measured specMACS radiance at 750.2 nm reduced to the HAMP resolution and (c) shows the simulated radiance values. The blue color in the distribution (d) represents the measured radiance at 750.2 nm, the orange color represents the measurements reduced to the HAMP resolution and the green color is the distribution of the simulated values of the reconstructed cloud.
The resolution problem explains most of the differences but not all. Further errors at 750.2 nm might be caused by a lack of matching donor pixels providing the necessary large LWP value. Figure 18 shows the deviation d ijm as introduced in Section 4 which also could be used as a quality filter for the matches. Especially at the center areas of the biggest clouds the deviation is high (d ijm > 0.010 W sr −1 m −2 ) indicating probably erroneous donor-recipient matches. For the rest of the cloud, the deviation is small (d ijm < 0.010 W sr −1 m −2 ) indicating fitting donor-recipient matches.

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The differences of the radiances at 1553.5 nm between measurement and simulation has to be inspected further. Source of the additional brightness at 1553.5 nm could either be a general overestimation of optical thickness or an underestimation of effective radius by our reconstruction method. An overestimation of optical thickness is not likely because this would mean that the brightness of the simulated radiance values at the visible wavelength must be larger than the measured radiance values.
This is not the case. An underestimation of the reconstructed effective radius would be possible. While shadow affected cloud areas and cloud edges are removed by the filtering method (tendency to strongly overestimate effective radius), bright illuminated cloud parts remain in our data (tendency to underestimate effective radius). These three-dimensional illumination effects would lead to smaller effective radii Platnick (2011), Marshak et al. (2006)).

5
In order to put the quality of our reconstruction with its remaining limitations into perspective, the additional simulations can be investigated (dashed lines in Figure 16). The simulated base case shows signs of an underestimation of effective radius because the simulated radiance values are higher than the measured radiance values. Consequently, the test with larger effective radii (dashed blue line in Figure 16f) should provide a better match. 20 % larger effective radii decreases the difference between 10 the measured and simulated base case from 12.5 % to 4.1 % with higher values in the simulation. A reduction of cloud effective radius by 20 % (dashed cyan line) causes a stronger overestimation in the simulation with a difference of about 19 % to the measurement. Table 1 concludes the additional investigations for ±5 • and ±10 • limited specMACS field-of-views. Thus, we conclude that the effective radius uncertainty is close to roughly 20 %.  Table 1 presents the additional investigations for ±5 • and ±10 • limited specMACS field-of-views. Thus, LWP and optical thickness uncertainty is most likely within the range of 20 %. This accuracy is not larger than the general accuracy of the microwave radiometer derived underlying LWP values (Jacob et al., 2019a). We presented an approach to combine different active and passive remote sensing instruments on-board the HALO research flights during the NARVAL-II campaign. This approach makes it possible to reconstruct, with a box-size resolution of 15 m, three-dimensional cloud macro-and microphysics of liquid trade-wind cumuli.
The three-dimensional extent of clouds is provided by a combination of Cloud Top Height from WALES lidar measurements, 10 Cloud Bottom Height from dropsonde air-parcel analysis and horizontal cloud mask from spectral imagery. Lidar information from the track below the aircraft is spread on the wider imager swath following the spectral re-sampling approach. In the same way, Liquid Water Path information provided by the microwave radiometer HAMP is also spread on the specMACS swath. This is the starting point for a reconstruction of three-dimensional cloud microphysics.

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Cloud optical thickness and effective radius retrieved via passive spectral remote sensing are added to provide a consistent microphysical representation of the cloud. Microphysical profiles within the clouds are determined using the HAMP measurements and the retrieved effective radius as constraints for a microphysical sub-adiabatic model. A strict filtering method eliminates effective radius values which are most likely affected by three-dimensional radiative transfer effects. Both shadow areas within clouds and cloud areas having small optical thicknesses close to cloud edges are removed from further data anal-20 ysis as they are prone to retrieval errors (as also noted by Marshak et al. (2006)). Based on the remaining values, a most likely averaged effective radius profile and an averaged cloud droplet number concentration can be derived for each cloud domain.
In-situ cloud data were not available for the discussed NARVAL-II campaign cases. Therefore, we performed a consistency check using simulations. From these we estimate that the accuracy of liquid water and effective radius values of our cloud reconstruction is in the range of about 20 % of the absolute values. The LWP errors are most likely random and scene dependent.

5
The effective radius shows a bias towards smaller values. This bias visible in the near-infrared wavelength tests can probably be explained by remaining three-dimensional illumination effects not considered in the plane-parallel effective radius retrieval (also compare, e.g., Zhang and Platnick, 2011).
The cloud droplet number concentration of randomly selected cloud scenes on 19th August corresponding to a flight path of 10 about 100 km over the Atlantic Ocean vary between 27 ± 7 and 47 ± 16 cm −3 using the standard deviation as error. Assuming an error of 10 % in the effective radius, the cloud droplet numbers are about 27 ± 11 and 47 ± 21 cm −3 . These values are consistent to the annual average close to Barbados as shown in Figure 1 by Roelofs et al. (2006) or to previous studies such as Brenguier et al. (2000). They are also in the range of similar cloud scenes retrieved by Wolf et al. (2019) using non-imaging solar reflectivity observations. Nonetheless, we are confident that our strict filtering leads to a strong reduction of uncertainty.

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Using the horizontal coverage of our specMACS data, especially information on cloud horizontal extent and cloud surface orientation, we are able to reduce the most important impact of tree-dimensional radiation transport.
It has to be kept in mind that we use a simple sub-adiabatic microphysical model assuming that both the cloud droplet number concentration and the effective radius profile remain constant in every cloud part. Entrainment effects which might 20 explain detailed differences are not considered. Visible differences at cloud edges might be caused by using only one constant Cloud Bottom Height for the whole cloud domain. In reality, the Cloud Bottom Height likely rises towards the cloud edge due to mixing. Unfortunately, available cloud radar data was not suited to determine a more realistic cloud bottom height due to the limited sensitivity to the small droplets. A combination of this retrieval with stereoscopic retrieved cloud top structure, as recently presented by Kölling et al. (2019) using specMACS, could lead to better results. A more detailed and more direct 25 reconstruction of the upper cloud border at cloud edges would become possible.
In subsequent work the radiative diabatic contributions to the local energy budgets in these reconstructed cloud fields will be analysed as well as their influence on potential vorticity and, thus, on dynamics. This method described in this paper will also be useful for the EarthCARE satellite mission since similar instruments will be used (Illingworth et al., 2015) as well as 30 the upcoming NARVAL-II follow-up EUREC4A campaign (Bony et al., 2017).
Data availability. The specMACS data are available at https://macsserver.physik.uni-muenchen.de (accessed at: 18th November 2019) after requesting a personal account. The radar data of the NARVAL-II flight on 12th August 2016 used in this paper is described by Konow et al. (2019) and the dataset was published by Konow et al. (2018). The Liquid Water Path data of the NARVAL-II flight on 19th August 2016 is described by Jacob et al. (2019a)  Appendix A: Adiabatic Theory -Standard Equations (a) The decrease of saturation pressure with height e sat (T c (z c )) can be calculated using an approximated Clausius-Clapeyron 10 Equation which is valid for temperatures −35 • C < T c < 35 • C and accurate up to 0.3 % in this range (Bolton, 1980): The decrease of temperature inside the cloud T c (z c ) can be calculated using the moist adiabatic lapse rate Γ f which is defined as (e.g., Vallis, 2019, p.234) 15 with the Mixing Ratio of liquid water r, the specific heat c p , the latent heat of vaporization L, the specific gas constant for dry air R dry and the specific gas constant for moist air R v . The Mixing Ratio of liquid water can be approximated by r ≈ 0.622 · e sat (T (z cbh ))/(p air (z cbh )) because e sat p air . Moreover, we approximate T c (z c ) with T (z cbh ) for shallow cumulus clouds since the vertical extent, as found in this study, is mostly less than 1000 m. Thus, the temperature differences inside the clouds are usually less than 5 K. Thus, we can express the decrease of temperature inside the cloud with height as: The decrease of the air pressure with height p air (T c (z c )) can be calculated using the Barometric formula with M dry as the molar mass of the Earth's air, g the gravitational acceleration and R 0 the universal gas constant: p air (T c (z c )) = p air (z cbh ) · exp − M dry · g R 0 · T c (z c ) · z c (A4) 25 The pressure p air (z cbh ) at the cloud base is used.
clouds, solar radiation will be reflected by the clouds at higher altitudes, and thus, both the passed distance and the water vapor absorption will be smaller. Figure B1 shows examples of the simulated spectra of the SWIR camera. The cloud mask was generated according to the following steps: First, one reference spectrum with atmospheric molecular absorption and one reference spectrum without atmospheric molecular absorption are simulated using the DISORT solver 10 included in libRadtran. These spectra are then convolved with the spectral response function of the specMACS SWIR camera.
Second, the transmittance T ref,λ of the reference atmosphere is determined by dividing the simulated spectrum with absorption by the simulated spectrum without absorption. Finally, the product of the spectrum without absorption but with transmittance is fitted to each measurement following L meas,λ = a · L noabs,λ · (T ref,λ ) x .
15 L meas,λ denotes the measured radiance at wavelength λ and L noabs,λ denotes the simulated radiance without absorption. The parameter a scales with the brightness of the measured spectrum and the parameter x scales with the transmittance and is, therefore, a measure of absorption. Since the transmittance is an exponential function depending on the propagated distance of the light and the absorption coefficient, the parameter x is an exponent. A threshold of the parameter x is defined by visual inspection to distinguish cloudy from clear sky measurements which is then adapted dynamically depending on the viewing zenith angle of the camera, the solar zenith angle, and the column integrated water vapor density of ECMWF ERA-Interim reanalysis data (Dee et al., 2011). As clouds reflect large amounts of solar radiation they appear as bright features in the measurements. This makes it also useful to define a threshold of the parameter a which has been done by a visual inspection. Additional, the signal to noise ratio, providing information about 5 the reliability of individual measurements, is used to determine the cloud mask. Also all clouds smaller than 3x3 are removed because it was seen that in cases with sun-glint accompanied by either high aerosol concentrations or too high water vapor concentrations in ERA-Interim data, the cloud mask fails by detecting widely scattered and very small clouds.
Author contributions. LH was in charge of the presented method, development of the three-dimensional cloud macro-and microphysics retrieval and the manuscript. FG developed and provided the cloudmask. MG derived the used WALES cloud top heights. TK developed the shadow mask, the effective radius retrieval and also provided valuable input during the development and verification of the method. BM and TZ prepared the field campaigns, provided valuable input during the development of the method and contributed to the final version of this paper.
Competing interests. The authors declare that they have no conflict of interest.
Acknowledgements. The authors thank the German Science Foundation (DFG) for supporting the HALO NARVAL-II and NAWDEX cam-