The scattering of electromagnetic waves on spherical particles is fully described by Mie theory . In general, for radiation traveling along the positive z-axis the scattered electric field vector Esc=(El,Er)scT is related to the incoming electric field vector Ein=(El,Er)inT by the scattering amplitude matrix e.g.,Sect. 3.2 1ElErsc=eik(R-z)-ikRS2S3S4S1ElErin. The state of the polarization of the electromagnetic radiation is conveniently described in terms of the Stokes vector 2I=IQUV=ElEl*+ErEr*ElEl*-ErEr*ElEr*+ErEl*iElEr*-ErEl* and the incoming Iin and outgoing Isc Stokes vectors are related by the 4×4 scattering phase matrix P 3Isc=σsca4πR2PIin, where σsca is the scattering cross-section. P can be calculated from the amplitude scattering matrix, which is diagonal for spherical particles (S3=S4=0) due to symmetry, such that P is fully described by four elements, namely P11,P12,P33 and P34.

The existing polarimetric retrievals of the CDSD use that the polarized reflectance Rp=-πQμsI0 is proportional to the measured component Q of the scattered Stokes vector in the scattering plane when U≈0 and that in turn, Rp∝Q∝P12 e.g.,. Here, μs is the cosine of the solar zenith angle and I0 is the incoming extraterrestrial solar irradiance. P12 is for spherical particles calculated from the elements S1 and S2 of the amplitude scattering matrix as 4P12=2πσscak2|S2|2-|S1|2.

Following , the average polarized phase function P‾12(θ) in terms of the scattering angle θ for an arbitrary normalized droplet size distribution n(r) can be written as 5P‾12(θ)=∫0∞σsca(r)P12(r,θ)n(r)dr∫0∞σsca(r)n(r)dr. In case of the polarimetric retrievals, n(r) is often assumed to have the form of a monomodal gamma distribution which, as defined by , can be written as 6n(r)=n0r(1-3veff)/veffe-r/(reffveff). Hereby, reff is the area-weighted mean or effective radius 7reff=∫0∞rπr2n(r)dr∫0∞πr2n(r)dr and veff is the dimensionless effective variance 8veff=∫0∞(r-reff)2πr2n(r)drreff2∫0∞πr2n(r)dr. As first introduced by and following the notation of , a general fit function of the form 9Qfit=A×P12[reff,veff](θ)+B×cos⁡2(θ)+C is used to determine the best fitting effective radius and variance for the observed polarized radiance under the assumption of a single mode gamma distribution. The parameters A,B and C are fitting parameters.

As already observed by from the application of the Rainbow Fourier Transform, the DSD can have a pronounced multimodal shape and decomposed the retrieved DSD into several single-mode gamma distributions. Assuming now that the DSD can be written as a superposition of M single-mode distributions n(r)=∑i=1Mni(r), we can rewrite Eq. () as 10P‾12(θ)=∑i=1M∫0∞σsca(r)ni(r)dr∫0∞σsca(r)n(r)dr×∫0∞σsca(r)P12(r,θ)ni(r)dr∫0∞σsca(r)ni(r)dr≡∑i=1Mβext,iβext,totP‾12i≡∑i=1Mfβ,iP‾12i. Hence, the observed scattered polarized radiation can be expressed in terms of a superposition of the polarized radiation scattered by the single distributions. The fraction that the individual size distribution contributes to the total signal fβ,i equals hereby the fraction of the extinction coefficient βext,i to the total extinction coefficient βext,tot when absorption is negligible. Note that this holds in general for any arbitrary shape of the individual size distributions ni(r). However, assuming that the individual distributions obey the shape of single-mode gamma distributions, we introduce the fit function in the presence of multiple modes in accordance to Eq. () as 11Qfit=∑i=1MAi×P12i[reff,i,veff,i](θ)+B×cos⁡2(θ)+C.

Comparing Eqs. () and () suggests that the fit factors Ai∝fβ,i, but they incorporate for the conversion between the phase matrix elements to the measured radiance Q, as A does in Eq. (). For example, we do not convert the measured Q to the polarized reflectance. Further, the measured polarization can be reduced due to ice or aerosol particles within or above the cloud target.

This allows to define a “signal fraction” fA as the normalized fitted amplitude of each mode 12fA,i=AiAtot,Atot=∑i=1MAi where Ai refers to the fit parameter of Eq. (). Atot is hereby the normalization factor which ensures that ∑i=1MfA,i=1. Provided that the multiplicative scaling factor within Ai is the same for all modes, we get 13fA,i=AiAtot≈fβ,i=βext,iβext,tot=τiτtot, where fβ,i is the same factor as in Eq. () and τi and τtot denote the optical thickness of mode i and the total optical thickness, respectively. The assumption of a common multiplicative scaling factor is motivated by the saturation of the polarized cloud signal at sufficiently large liquid optical thicknesses, typically around 3-5 . Then, the scaling factor may become less sensitive to further increases in optical thickness and is mainly determined by the conversion from polarized radiance (Q) to reflectance and external factors, such as depolarizing aerosol or ice particles above or within the clouds, respectively and the normalization of the phase functions. Further, we assume here that we observe a homogeneously mixed target, such that βext,i/βext,tot=τi/τtot holds.

Moreover, the signal fraction can be used to calculate a fraction with respect to the liquid water content fLWC and the droplet number concentration fN. Using that the extinction efficiency Qext≈2 for particles that are much larger than the wavelength , the definition of the reff after Eq. (), the extinction coefficient 14βext=∫0∞Qext(r)πr2n(r)dr and the liquid water content 15LWC=43πρ∫0∞r3n(r)dr=43πρNr‾v3, where N is the droplet number concentration and ρ is the water density, the extinction coefficient can be approximated as

16βext≈3LWC2reff. Further, Eq. () can be written in terms of the effective radius as 17LWC=43πkreff3ρN, where we used that k=(1-veff)(1-2veff) connects the mean volume radius r‾v3=kreff3 to the effective variance of the gamma distribution. Hence, we can derive the liquid water content fraction as 18fLWC,i=LWCiLWC=fA,i×reff,i∑i=03fA,i×reff,i and the fraction of the droplet number concentration as 19fN,i=NiNtot=fA,i/(ki×reff,i2)∑i=03fA,i/(ki×reff,i2).

While Mie theory provides an exact solution of the Maxwell equations for the scattering of light by spherical particles, it is represented by an infinite series of Legendre Polynomials. Thus, for practical applications, the series has to be truncated when a reasonable accuracy has been reached. In general, the Mie solution depends on the ratio between the radius and the wavelength λ of the scattered light, which is in most applications expressed in terms of the size parameter x defined as 20x=2πrλ. The series of Legendre Polynomials converges approximately when Nl≳x Sect. 3.4, p. 102, where Nl is the number of calculated terms. Hence, calculations become more expensive the larger the size parameter.

For our application, we have to extend the existing phase function table generated by to effective radii of rain droplet size distributions (RDSD). Therefore, we assume that RDSDs can be described in terms of gamma distributions to be consistent with the existing phase function table. A widely used assumption for the rain droplet size distribution is the Marshall-Palmer distribution Sect. 2.1.6, which follows an exponential relationship between the droplet number concentration ND and the droplet diameter D=2r: 21ND(D)=N0e-ΛD=N0e-2Λr further found N0 to be constant with N0=8×104m-3cm-1 and the slope parameter Λ=41R-0.21 to vary with the rainfall rate R in mmh-1. Since this is a special case of the modified gamma distribution in Eq. () with veff=1/3 and Λ=1/(2reffveff), the phase function table was not only extended in terms of the effective radius, but also by two further effective variances veff=1/3 and veff=0.4. Multiple studies, such as and have shown that the use of the gamma distribution in a more general form can be used to describe RDSDs fairly well and further justifies our assumption of underlying gamma distributions for larger rain droplet size spectra.

For our extended phase function table, we performed Mie calculations using the MIEV0 Code by , which is implemented into the Mie tool that comes with the library for radiative transfer libRadtran;. Since we want to use the phase function table for a retrieval based on measurements of the polarization resolving RGB cameras of specMACS, the phase function calculations were performed for the wavelength range between 380 and 710nm and integrated over the spectral response function of the cameras . Our extended phase function table includes now 143 effective radii from 1 µm to including 1000 µm with reff,i+1=reff,i×1.05 for the same 16 effective variances between 0.01 and 0.325 as used by and the two new effective variances 1/3 and 0.4.

Figure  shows the phase matrix components P11 and P12 in the scattering angle range between 120° and 180° integrated for the green channel of the RGB polarization cameras of specMACS for an effective variance of veff=0.1 and different effective radii between 1 and 1000 µm (every 10th radius of the extended phase function table). The gray shaded area in both panels marks the region of the cloud- (and rainbow) between 135° and 165° scattering angles which has been used for the polarimetric retrievals of CDSDs e.g, . It can be seen, that in particular the polarimetric phase functions P12 split up in that regime for the different effective radii and the minimum moves towards lower scattering angles for larger effective radii. However, it also becomes clear that there is less sensitivity for larger effective radii. We will investigate this in more detail in Sect. . Next to the cloud- and rainbow, we can also see the second order rainbow as a secondary maximum in P11 and a secondary minimum in P12 between 120° and 130°, which becomes more pronounced for larger effective radii. In between the two rainbows, Alexander's dark band is clearly visible for larger radii. Around 180°, the backscatter glory can be identified, which has also been used for remote sensing of cloud droplet size distributions .

Figure  shows the dependence of the polarized phase function P12 on the effective variance for four different effective radii of 1.00 µm (Fig. a), 9.91 µm (Fig. b), 98.13 µm (Fig. c), and 1000 µm (Fig. d) in the cloud-/rainbow range between 135 and 165°. While the phase functions split up well for 1.00, 9.91, and 98.13 µm, hardly any difference can be seen for 1000 µm. For 9.91 and 98.13 µm, the polarized phase functions show next to the mostly aligned major minimum so-called supernumerary bows for the smaller effective variances. In contrast, the phase functions for 1.00 µm show significant differences between the effective variances as well, however, the minimum shifts clearly towards larger scattering angles for smaller effective variances. For veff=0.01, the major minimum is not even visible in the range of the cloudbow. Since shifts in the minimum occur also due to changes in the effective radius, there might be less sensitivity in the fit also for very small radii. Due to the indifference of P12 for different effective variances at 1000 µm, it can be expected that large uncertainties in veff might occur at those values. As mentioned before, this will be investigated further in Sect. .

The motivation for the implementation of the multimode retrieval can be seen in Fig. . It illustrates the effect on the polarized phase function when two modes coexist in the droplet size distribution for a first mode with reff,1=9.91 µm and a second one with reff,2=98.13 µm. Different fractions that the individual modes contribute to the overall signal were chosen with f1∈{1.0,0.8,0.5,0.3,0.0} and f2=1-f1. The existence of a second mode in the DSD changes the polarized phase function significantly, and hence would bias a retrieval that assumes a single mode gamma distribution.

The retrieval algorithm is based on the aggregated angular radiance measurements in the scattering angle range between 135° and 165° described by and uses the single mode results as a starting point. The results from the single mode fit contain several quality measures which evaluate the quality of the fit : The root mean square error RMSE, Eq. 7 in 22RMSE=1N∑i=0n(Qfit(θi)-Qmeas(θi))2 and the quality index QI, Eq. 8 in 23QI2=A2P122-〈P12〉2RMSE2. As in Eq. (), θ denotes the scattering angle and A is the fit parameter that scales the polarized phase functions to the measured amplitude of the cloudbow. Cloud targets of the single mode fit results are not considered for further analysis, when RMSE>2.5 and QI<4 . Those targets, which do not fulfill the quality measures of the single mode fit are then used as an input for the multimode retrieval. However, targets where A<0.001 and QI<1.5 are not considered, since the signals do not contain a lot of information, e.g. due to ice, shadows or ocean influence.

While the single mode fit function from Eq. () has five free parameters for the fit (A, B, C, reff, and veff), every additional mode in the fit function after Eq. () adds another three free parameters to the parameter space, because the background B and C is fitted only once. Hence, it would be ambiguous to search the whole lookup table (LUT) for each mode and we decided on a strategy, which subsequently adds modes to the fit and searches the LUT in three different stages. First, an independent one-mode search is performed within the multimode retrieval by performing a coarse search through the LUT, which keeps every sixth effective radius and effective variance values. Since the fit is much more sensitive to reff than veff, we look for the best fitting effective radii for the central effective variance of the kept values first. The results for the five best effective radius fits are then combined with all possible effective variances to find again the five best candidates. Then, the fit is refined in a secondary coarse search around these best results from the first coarse search, which takes every second reff and veff in a window that includes the two neighbors of the coarsened LUT in both dimensions (reff and veff) around the found result into account. Again, the five best candidates from the second stage fit are kept, to perform a final search, which takes all candidates within a window of size two (±2 neighbors) around those results. Hereby, the result of the first mode found can be interpreted as the dominant mode generating the signal and does not resemble an average over the whole multimodal distribution, as already observed by .

The result of the first mode serves as a starting point for sequentially adding up to a predefined maximum number of modes nm. For the new mode, the fit strategy described above is repeated, while the results for the reff and veff of the single best fit from the preceding step are used as centers for local search windows. Hence, when adding a second mode, the possible reff and veff values of the coarsened LUT are combined with the neighboring values of the LUT of the best reff and veff of the first mode. For each combination, the fit parameters Ai, B, and C are refitted to minimize the objective function (here: RMSE), and the best result is kept to perform the secondary coarse search and the fine search with the newly added mode.

Modes are added until either the predefined maximum number of modes is reached or the newly obtained fit satisfies the RMSE threshold (currently 1.2) for all fitted color channels. A newly added mode is accepted only when the objective function does not worsen and the fitted mode amplitudes are positive for all fitted color channels. If the new fit worsens the objective for any of the color channels, the preceding best valid fit is retained.

Because different combinations of phase functions can produce similar angular radiance curves, the multimode inverse problem may admit several comparably good solutions. Hence, it should be mentioned that the result might describe a local minimum rather than the global minimum which could lead to errors in the resulting droplet size distributions. Further, the resulting solution is conditional on the LUT, search-window strategy, objective function, and positivity constraints. However, first applications of the current fit strategy show promising results, which are presented in Sect. .

Afterwards, the quality of the results of the multimode fit can still be ensured by applying a threshold filter as for the single mode retrieval. Next to the RMSE, we define the quality index QI for a multimodal fit analog to the quality index for the single mode fit from Eq. () as 24QI2:=Var∑i=1MAiP12iRMSE2=∑i=1MAiP12i2-∑i=1MAiP12i2RMSE2. Similar to the goodness thresholds for the single mode fit, we filter targets when RMSE<2.5 and QI>4. The individual steps of the retrieval algorithm and the fit strategy are summarized in Fig. .

Figure  shows that the phase functions for larger droplets do not split up as much as the phase functions for the well tested smaller droplets in the cloudbow region. Hence, the first question to be answered is the sensitivity of the fit to larger droplets. To this end, we added random measurement noise equivalent to the noise in real measurements of the specMACS polarization cameras to the phase functions in the LUT and generated a data set which includes several targets for each (reff,veff) pair. The fit routine described in Sect.  was applied to the measurement resembling data set for a maximum number of modes nm=1 allowing to test the sensitivity of a single mode fit dependent on the actual effective radius and variance. The results are shown in Fig. . In Fig. a, a larger spread of the derived effective radius can be seen both towards small effective radii smaller than about 2 µm and as expected also towards effective radii larger than about 200 µm. In Fig. b, the comparison between the actual input effective variance and the derived effective variance is plotted, showing a spread for all values. While the mean follows the 1:1 line well, a slight overestimation can be seen for small effective variances and an underestimation for larger effective variances, which is due to the end of the LUT. Further, it can be seen that the retrieved effective variances accumulate both in the smallest (veff=0.01) and the largest (veff=0.4) effective variance bins, which is mainly due to insensitivities of the phase functions for large effective radii as seen in Fig. .

In Fig. c and d, the absolute (black, left axis) and the relative (red, right axis) mean biases are shown with respect to the input effective radius and variance respectively. The mean bias of the effective radius remains approximately zero for effective radii up to 200 µm which is consistent with the findings from Fig. a and also holds for the relative mean bias, although the relative mean bias is increased for reff<2 µm as seen before. For radii between 200 µm and about 700 µm, the mean bias increases to about +20 µm around 500 µm, before it falls off to negative values down to about -175 µm at 1000 µm. The overestimation for reff<2 µm and the underestimation for reff≳700 µm might also be related to the fact that they are close to the borders of the LUT, such that potential errors are always positive or respectively negative. Altogether, it can be concluded that the effective radius can be determined with an accuracy better than 10% between about 1.5 µm and 800 µm and hence, fulfills the requirement determined by for the reff of liquid droplet size distributions to monitor the long-term aerosol and cloud forcing from space. For the bias of the effective variance shown in Fig. d, a steady decrease towards larger veff can be seen. While the effective variance is overestimated for veff≲0.15, it is underestimated for values above 0.15. The overestimation at small effective variances results in a relative bias of up to about 2, with a fast decrease down to about 0.2 at veff=0.05. At the largest effective variance of veff=0.4 an underestimation of about -0.07 or -20% is found. Again, it should be kept in mind that overestimates at small effective variances and underestimates at large veff might be caused by reaching the borders of the LUT. Compared to the accuracy requirements determined by of 0.05 or 50%, the requirements are met for all effective variances, either because of their low absolute or relative bias.

The same is shown for the standard deviation in Fig. e and f. As denoted before, the standard deviation of the derived effective radius is large for very small effective radii (reff≪2 µm), which is mostly due to single outliers. While it drops to a minimum of about 0.1 µm at reff≈3 µm, it steadily increases for larger reff up to nearly 200 µm for reff≳500 µm. However, this means that the relative standard deviation stays below about 0.4 for all reff except for the very small values close to about 1 µm. The standard deviation of the derived effective variance generally increases for larger input effective variances from about 0.06 to about 0.10 for veff≳0.3. However, the relative standard deviation steadily decreases for larger veff with values below 0.5 for veff≳0.15.

performed an extensive study on the accuracy of the single mode polarimetric retrieval applied to measurements of the specMACS instrument using 3-D radiative transfer simulations of a field of shallow cumulus clouds. They found an uncertainty in the effective radius of (-0.17±1.30) µm, which corresponds to a relative mean bias of less than 2% when compared to the average radius of 10.49 µm expected from the model data. While the values found were averaged for all effective radii between 1 µm and about 40 µm (the values included in the existing LUT for the single mode retrieval), they compare well to the values found from Fig. c. With regard to the effective variance, found an uncertainty of (0.02±0.05), which corresponds to a relative mean bias of 15% at an average expected effective variance of 0.14. Note that the simulations did not include any veff<0.1 as this was the effective variance assumed for each grid box of the model simulations. In contrast to the results from Fig. d, found rather an overestimation in the effective variance. This might be explained by the idealized model that was used for this study, which includes typical measurement uncertainties with respect to the measured value of Q, and hence only indirectly includes errors that might occur for example due to cloud movements or uncertainties in the cloud top height.

Figure  illustrates the sensitivity of the single mode fit on both the effective radius and variance of the input. In Fig. a, the mean bias of the retrieved effective radius is shown with red values denoting an overestimation while blue regions show underestimates. Each pair of input (reff,veff) was calculated 12 times. As expected, larger uncertainties are found for larger input reff with both overestimates and underestimates occurring. However, overestimates tend to dominate in the beginning while underestimates take over the larger the effective radius, which might be due to the end of the LUT coming closer as discussed before. This shift from over- to underestimates tends to occur later for smaller effective variances, while the effective radii are mostly underestimated for the smallest effective variance of 0.01. Finally, the overestimation of the effective radius for small effective radii as seen already in Fig. a is also visible, with the largest overestimates occurring for small effective variances. Further slight overestimates can be seen for larger variances, while there is hardly any overestimation for medium effective variances between about 0.07 and 0.16. Figure c shows the relative mean bias for each pair of (reff,veff). As before, it is mostly well below 10%, particularly for 2µm≲reff≲200 µm and veff≲0.1. The corresponding relative standard deviation is shown in Fig. e. As stated above, most of the values are well below 0.5 except for very small effective radii smaller than 2 µm and low effective variances as well as single outliers at larger effective radii (≳200 µm).

The analog plots for the effective variance are shown in Fig. b, d and f. While large effective variances tend to be underestimated especially at very small effective radii (≲2 µm) and for large effective radii ≳200 µm, the opposite holds for small effective variances ≲0.1. Parts of this might be explained by the end of the LUT, although the determination of the effective variance works well in between except for slight overestimates for 0.15≲veff≲0.25. This is reflected by the relative mean bias in Fig. d, with overestimates above 100% occurring only for veff<0.1 and reff<3 µm or reff>200 µm. In between, slightly red colors seem to dominate except for veff≳0.27. This might explain the findings of , that the effective variance is overestimated while we found on average rather underestimates for veff≳0.12 from Fig. d, which seems to be dominated by the underestimates at very small and very large effective radii. The relative standard deviation in Fig. f shows a similar picture with large values occurring particularly at small and large effective radii. Hereby, values larger than 1 are found at small effective variances smaller than about 0.15 and for reff≲3 µm and reff≳200 µm. Hence, for those ranges the determination of the effective variance becomes highly uncertain, which is as expected from Fig. .

In conclusion, an accurate determination of the effective radius is still possible even for larger effective radii where the phase functions become less distinct, in particular with respect to the relative mean bias and standard deviation. For the effective variance larger uncertainties were found towards larger effective radii, and particularly small effective variances might be hard to determine.

As shown in the previous section, the effective radius of the multimode retrieval shows some systematic patterns for the derived modes, which hints towards a stable retrieval algorithm. To further evaluate the results, we compare the results for four individual targets in Figs. – to the original result of the single mode fit and the RFT. All four figures show the location of the target in panel (a), and the measured signal, the resulting multimode fit as well as the single mode fit result and RFT fit result in panel (b). The RFT fit result is computed from the result of the mode decomposition after as 25Qfit,RFT=ARFT∑i=1Mζi×P12i[reff,i,veff,i](θ)+BRFT×cos⁡2(θ)+CRTF, where ζi is the percentage scale of each mode derived from the mode decomposition and the parameters ARFT, BRFT and CRFT are derived from a least-squares fit to the measurements, which makes the results of the RFT and the multimode retrieval comparable as best as possible. Panels (c) and (d) show the resulting so-called area size distributions of the single mode, and the multimode fit results as well as the distributions derived by the RFT and the corresponding mode decomposition. The area size distribution is hereby defined by as 26na(r)=r2n(r)∫0∞r2n(r)dr,

where n(r) is the droplet size distribution as in Eq. (). As the area distribution defines another gamma distribution which can be described in the same form as in Eq. (), the “effective radius” and “effective variance” of the area distribution reff,a and respectively veff,a can be derived from the effective radius and variance of the droplet size distribution as 27reff,a=reff(1+2veff)andveff,a=veff1+2veff. The corresponding inverse transformation is given by 28reff=reff,a(1-2veff,a)andveff=veff,a1-2veff,a, allowing to calculate the one distribution from the parameters of the other. Panels (c) and (d) differ from each other as we used a logarithmic scale in panel (d) to eventually show the multimodal structure more clearly, as larger cloud droplets are suspect to much lower cloud droplet number concentrations.

In Fig. , the measured polarized radiance (Fig. b) and the results from the multimode fit (red), the RFT (blue) and the single mode fit (green) are shown for a cloud target located within the region close to the observed rainbow, where at least two modes were found in the shown figures in Sect. . This example shows a double cloudbow as obtained from the combination of phase functions shown in Fig. . Furthermore, we can observe two supernumerary minima, which hints to at least one mode with a narrow DSD, which is also what we find both from the multimode fit and the RFT: The multimode fit sees two modes, a signal dominating one with reff,1=185 µm and a second one with reff,2=7.39 µm. While the first mode is broad (veff,1=0.4), the small droplet mode has a narrow distribution with veff,2=0.01 explaining the supernumerary bows. From Fig. b, it can be seen that the found combination explains the measured signal very well with a RMSE of 1.17. The mode decomposition of the RFT on the other hand, finds a first mode that is with reff,1=7.65 µm and veff,1=0.03 very similar to the small droplet mode found from the multimode fit. It further finds a second mode with even smaller droplets and reff,2=4.22 µm and veff,2=0.05. Figure c hereby reveals, that both retrieval results find a distribution with a narrow peak around 7 µm, where both distributions are well aligned. In contrast, the multimode retrieval does not retrieve the secondary peak towards the left of the main peak of the RFT distribution, while Fig. d shows that the underlying broader distribution with its maximum around 150 µm is missed by the RFT. This is due to the fact that the RFT in its current implementation only retrieves radii up to 100 µm in the area size distribution, which means that the effective radii retrieved from the mode decomposition are in general lower than 100 µm. Moreover, the mode decomposition technique finds peaks in the area size distribution, but since droplet number concentrations are usually much smaller for larger droplets as can be seen from the distribution retrieved from the multimode retrieval in Fig. c, peaks at larger droplets might also be missed in general by the method. From Fig. c and d, we can also see one issue, that we usually observe when applying the RFT to specMACS measurements: the derived area size distribution shows negative values which are not removed by the corrections described by making the results at least partly physically unrealistic.

The results of the cloud target in Fig.  also show a clear multimodal structure as the target in Fig. . In Fig. b, we can clearly see that the signal is well explained by the new multimodal fit (red), while the single mode fit does not capture the measured signal properly. Here, the multimode fit finds two modes with reff,1=152 µm and reff,2=7.76 µm. Again, the first mode is broad with an effective variance of veff,1=0.4, while this time, also a broader distribution of the second mode is found with veff,2=0.25. From the single mode retrieval, a radius of reff,max=40.77 µm is found, which is consistent with the maximum radius of the LUT. The RFT finds four modes from the mode decomposition with effective radii reff,1=4.12 µm, reff,2=6.63 µm, reff,3=12.15 µm and reff,4=8.47 µm. This results in a fit calculated after Eq. () that seems to capture the lower effective radius mode well, while it does not explain the signal originating from the large droplets. Here, Fig. d makes clear that the RFT misses droplets larger than 100 µm where currently the definition of the RFT ends.

The example shown in Fig.  shows the capability of the multimode retrieval to explain the measured signal with three modes: One with reff,1=14.64 µm and an effective variance of 0.30, a second one with an effective radius of 60.24 µm and an effective variance of 0.4 and one with reff,3=5.00 µm and veff,3=0.1. Altogether, the measured signal is described with a RMSE of 0.51 and a high quality index of 23.29 is found. In comparison, the mode decomposition from the RFT finds two modes, a first mode with reff,1=5.36 µm and veff,1=0.18, which is on the same order of the third mode found by the multimode and a second mode with reff,2=16.87 µm and veff,2=0.09. The resulting DSDs in Fig. c and d show that the DSDs of the RFT and the multimode retrieval are very similar. The major peak around 5 µm is both found by the RFT and the multimode retrieval and both distributions show a tail towards larger droplets, although the small peak around 11 µm is not as pronounced in the multimode retrieval as in the RFT. As before, the signal computed by the mode decomposition result does not describe the actual measured signal completely, which is most likely due to the missing contribution from the mode around 60 µm. This mode derived by the multimode retrieval is not encountered by the mode decomposition of the RFT, since droplet numbers are too small compared to the smaller droplets. In this example, the single mode fit does not hit the maximum effective radius of the LUT, but finds an effective radius of 15.16 µm. This corresponds to a combined effective radius which lies in between the 14.64 µm found by the multimode retrieval and the 16.87 µm of the RFT.

Figure  shows an example target, where the RFT captures the supernumerary bows of the measured signal very well, while the multimode retrieval fits the main minimum and the second minimum but not the supernumerary bows. Hereby, the mode decomposition of the RFT finds three modes, each with a rather small distribution: A first mode with reff,1=4.75 µm and veff,1=0.06, a second one with reff,2=16.77 µm and veff,2=0.02, and a third one with reff,3=7.81 µm and veff,3=0.02 are found. This explains that the supernumerary bows are fitted well by the RFT, while on the other hand, the multimode fit finds three modes as well with reff,1=84.77 µm and veff,1=0.08, a second one with reff,2=15.37 µm and veff,2=0.15, and a third one with reff,3=4.12 µm and veff,3=0.23. While the effective radii of both second modes are similar, the effective variance of the multimode fit is larger. This holds in general for all three modes and might explain why the multimode fit does not capture the supernumerary bows. Again, the RFT mode decomposition is subject to smaller effective radii and does not find the larger mode with an effective radius of about 84 µm, which explains why the fit does not capture the major minimum. Furthermore, the single mode retrieval is again biased towards larger radii and finds the maximum effective radius of the LUT with reff,max=40.77 µm and a broad distribution (veff=0.32). Figure c and d show, that the resulting size distributions differ between the different methods, although both the multimode distribution and the distribution from the RFT show three maxima, but they are shifted. One explanation for the deviation of the methods might be the noise of the measured data in Fig. b, which could be related to errors in the tracking of the target and hence, errors in the retrievals.