1. Introduction
Ice clouds play a key role in the atmosphere of Earth, but in situ observations of ice clouds are difficult to make and thus are sparse. Remote sensing instruments have the potential to fill this gap because they can provide observations at high temporal and spatial resolutions. Among remote sensing instruments, ground-based vertically pointing Doppler cloud radars are the only instruments that can penetrate optically thick ice clouds and provide measurements of the fall velocity of hydrometeors. However, the observables of cloud radars are only indirectly linked to cloud and precipitation properties. The required transfer functions are not uniquely defined, resulting in substantial uncertainties in radar-based ice-cloud retrievals.
Different strategies have been used to increase the information content of radar observations: some studies suggested the use of radar–microwave radiometer combinations (Grecu and Olson 2008; Posselt and Mace 2014), radar–lidar combinations (Intrieri et al. 1993; Delanoë and Hogan 2008), or multiple radars operating at two (Hogan et al. 2000; Szyrmer and Zawadzki 2014) or three frequencies (Sekelsky et al. 1999; Kneifel et al. 2011). For single-instrument retrievals, some studies proposed to exploit not only equivalent radar reflectivity factor Ze but also mean Doppler velocity W (Matrosov et al. 2002; Szyrmer et al. 2012) and Doppler spectrum width σ (Mace et al. 2002; Deng and Mace 2006) of zenith-pointing radars. Also, the use of the full radar Doppler spectrum has been suggested (Verlinde et al. 2013).
Our research addresses the point of increasing the information content of observations from ice-cloud radars by studying whether additional information can be obtained from the higher-order moments and the slopes of the radar Doppler spectrum. For this, not only are Ze, W, and σ studied but also skewness γ and kurtosis κ. In addition to the higher-order moments, the left slope Sl and the right slope Sr of the radar Doppler maximum peak are also investigated. While σ, γ, and κ have the advantage that they are not influenced by radar calibration, they are strongly influenced by turbulence. Therefore, the retrieval implemented in this study provides not only microphysical properties—such as particle-size distribution, mass–size relation, and cross section–area relation—but also kinematic variables, such as vertical air motion and turbulent spectral broadening.
Lack of information about the mass–size relation of hydrometeors is one major reason for the uncertainty of most radar retrievals. A few studies solve this problem by including particle mass (or density) into the retrieval (e.g., Posselt and Mace 2014; Szyrmer and Zawadzki 2014). A retrieval providing microphysical parameters including the mass–size relation as well as kinematic properties, together with the corresponding uncertainties, has, to the authors’ knowledge, not yet been developed. The same is true for ice-cloud retrievals using higher-order moments even though there are studies using γ and κ for detection of drizzle onset (e.g., Kollias et al. 2011) and to locate supercooled liquid water (e.g., Luke et al. 2010). Maahn et al. (2015) found higher-order radar moments to be useful for evaluating ice-cloud parameterizations using observations from the Indirect and Semi-Direct Aerosol Campaign (ISDAC; McFarquhar et al. 2011), which took place in 2008 in the environs of the Atmospheric Radiation Measurement Program North Slope of Alaska (NSA) site in Barrow, Alaska. The dataset and the parameterizations recommended by Maahn et al. (2015) are the basis for this study.
To study the information content of higher-order moments and slopes qualitatively, response functions to ice-cloud parameters are assessed. For a quantitative analysis, the number of independent information pieces (total degrees of freedom for signal Df) are estimated using a retrieval that is based on optimal-estimation theory (Rodgers 2000). Optimal estimation is widely used in atmospheric remote sensing, in particular for passive and active microwave applications (e.g., Löhnert et al. 2004; Steinke et al. 2014).
The radar simulator of the Passive and Active Microwave Radiative Transfer (PAMTRA) model (Maahn 2015) is utilized as the forward operator. As input, both real and synthetic radar observations from the ISDAC campaign are used. The retrieval output is composed of variables that describe the particle-size distribution N(D), the mass–size relation m(D), and the cross section–area relation A(D) and kinematic variables that are related to turbulence and wind. For this, the a priori dataset and the parameterizations recommended by Maahn et al. (2015) are used. They used maximum particle dimension as the size descriptor, which is adopted here for consistency. Optimal estimation provides not only a retrieved state but also its uncertainty, assuming a Gaussian uncertainty distribution. Therefore, the analysis of retrieval results is not limited to bias but also covers uncertainties. The retrieval results are further analyzed in terms of the use of additional radar frequencies, the impact of calibration offset, measurement uncertainty, and prior knowledge of kinematic variables.
The datasets used, the forward model, and the optimal-estimation retrieval algorithm are described in section 2. The response functions of radar moments and slopes to microphysical parameters are discussed in section 3. In section 4, the retrieval results are investigated, and the impact of modifications to the retrieval is analyzed in section 5. The findings are discussed and concluding remarks are given in section 6.
2. Setup
In this section, the datasets and methods used in this study are introduced. This includes the definition of the state vector x and the measurement vector y and how they are connected by optimal estimation.
a. Dataset
All data used in this study were obtained during the ISDAC campaign (McFarquhar et al. 2011; Jackson et al. 2012), which took place in 2008 at the NSA site in Barrow and covered mostly stratocumulus ice and mixed-phase clouds. The ISDAC dataset consists of in situ cloud measurements by the Convair aircraft (McFarquhar and Jackson 2012) as well as ground-based spectral radar observations by the millimeter-wavelength cloud radar (MMCR) operating at a frequency of 34.86 GHz (Moran et al. 1998), which are available on request (Bharadwaj and Johnson 2003). Supercooled liquid water (SCLW) can be visible in the radar Doppler spectrum at high concentrations (Luke et al. 2010), which could bias the analysis. Therefore, only aircraft observations in the vicinity of the NSA site that contain less than 0.01 g m−3 of SCLW as measured by the “King” probe are used for this study. Even though such low SCLW values are, in the presence of cloud ice, usually not visible in the radar spectrum, rimed ice particles are probably still included in the dataset. Most of the observed clouds were precipitating, but aircraft observations were usually above cloud base. In the following, the quantities derived from the ISDAC dataset are briefly introduced. See Maahn et al. (2015) for a detailed discussion about why the following methods are used to parameterize the ISDAC dataset.
For kinematic effects, vertical air motion w (resulting in a shift of the Doppler peak) and kinematic broadening σk of the Doppler peak are considered. As discussed in Maahn et al. (2015), σk depends mainly on the horizontal wind u, the turbulence expressed by the eddy dissipation rate ε, the radar integration time, and the radar beamwidth. The u and w are obtained from aircraft measurements, whereby w is offset corrected as suggested by Gultepe et al. (1990). Even though the Convair aircraft was equipped with a Tropospheric Airborne Meteorological Data Report (TAMDAR) instrument (Moninger et al. 2010) that was capable of measuring turbulence, during ISDAC ε was mostly below the detection threshold of 10−3 m2 s−3 of the TAMDAR instrument. As a consequence, the TAMDAR instrument only provides an upper bound of ε; for ISDAC, Maahn et al. (2015) estimated ε to be 10−6 m2 s−3 on the basis of the comparison of in situ aircraft and ground-based radar observations.
b. Forward model
To simulate the full radar spectrum s and the corresponding moments and slopes on the basis of the provided in situ hydrometeor profiles, the PAMTRA model is used (Maahn 2015). Different from the method in Maahn et al. (2015), the self-similar Rayleigh–Gans (SSRG) approximation (Hogan and Westbrook 2014) is used here instead of the soft spheroid 
In this study, the PAMTRA forward simulator is configured in accordance with the technical specifications of the MMCR that was operating during ISDAC. It operated at a frequency of 34.86 GHz with Doppler velocity resolution of 4.1 cm, a full-width beamwidth of 0.31°, and a noise level of −33.25 dBz [as defined by Smith (2010)] at 1-km height (Moran et al. 1998; Bharadwaj and Johnson 2003). As attenuation is expected to be negligible for snow and ice at Ka band (Matrosov 2007), attenuation is not considered here.
c. Radar observables
d. Optimal estimation
The optimal-estimation code developed for this study is written in Python. It has been released as open source and is available online (https://github.com/maahn/pyOptimalEstimation).
e. State vector x
To simulate the radar Doppler spectrum, the PAMTRA forward operator requires information about particle mass m(D), particle cross-sectional area A(D), and particle number N(D) as a function of particle maximum dimension D. In addition, kinematic broadening σk and vertical air motion w are required to account for kinematic effects.
The state vector consists of the quantities introduced in section 2a (see also Table 1 for an overview). To estimate the prior information, 
A priori values and uncertainties of the state vector x and of derived quantities. Here, PSD is particle size distribution.
Optimal estimation requires that the uncertainties of xa (and y) be described by a Gaussian distribution with variance as provided in 
(a) Correlation matrix of state vector x used for all profiles, and (b) an example correlation matrix of the measurement vector y for σ = 0.08 m s−1.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
In addition to these quantities, PAMTRA output depends also but weakly on pressure p and temperature T, which are required for estimation of the particle fall velocity and the scattering properties. In this study, they are obtained from the ISDAC dataset; in a real-world application they can be estimated from radiosondes or model data. Therefore, they are treated as known parameters in the retrieval.
f. Measurement vector y
In what follows, different configurations of the measurement vectors are evaluated. In all cases, the measurement vector is composed of one or several quantities taken from the moments and slopes of the Doppler spectrum: equivalent radar reflectivity factor Ze, mean Doppler velocity W, Doppler spectrum width σ, skewness γ, and kurtosis κ as well as the left slope Sl and right slope Sr.
g. Measurement uncertainty
The measurement uncertainty and forward-model error are combined in the covariance matrix 
Measurement uncertainties are usually assumed to be constant, but they depend strongly on the signal-to-noise ratio (SNR) and on the number of bins of the Doppler spectrum that are covered by the peak (NB). In general, observations with highest SNR and largest NB have the smallest 
The measurement uncertainties corresponding to the square root of the diagonal of 
Measurement uncertainty (dashed) and combined measurement uncertainty and forward-model error (solid) for the quantities of y as a function of σ.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
h. Forward-model error
In addition to the measurement uncertainty, 
In addition to biases, there are random errors such as the following:
There is discretization error of measured N(D) and A(D). The impact of discretization of N(D) and A(D) was investigated in Maahn et al. (2015). For N(D), no significant impact was found when describing N(D) by a normalized gamma distribution [Eq. (2)], but for A(D) it was found that the approximation of A(D) as a power law leads to a differences in A of up to a factor of 2, which in turn cause biases in γ and κ. These differences are accounted for when running PAMTRA 1000 times. In this way,
There is also error in the fall velocity relation υ(D). In our forward model PAMTRA, the uncertainty in A(D) described above relates directly to an uncertainty in υ(D) of 11%. This is because A(D) is used in PAMTRA solely for estimation of υ(D). Therefore, we decided not to consider an additional uncertainty for υ(D) although Heymsfield and Westbrook (2010) reported an uncertainty of 25% for υ(D). We expect, however, in practice an error for υ(D) of smaller than 25% because of the large ensemble of simultaneously observed particles.
Third, there are errors in the estimation of backscattering cross section σbsc. In this study, a fixed value of 0.6 is used for the aspect ratio AR. Because the two other quantities, which according to SSRG theory determine the ensemble scattering properties, N(D) and m(D) are not fixed but are included in x, we can assume that the forward-model error for estimating σbsc is determined by the assumption of a fixed AR. Maahn et al. (2015) found that AR is greater than 0.5 on the basis of ISDAC in situ data, and literature values for AR vary between 0.6 (Hogan et al. 2012) and 0.7 (Tyynelä et al. 2011). Therefore, we draw AR from a normal distribution with mean 0.6 and standard deviation 0.1 when running PAMTRA 1000 times to estimate
Using the same data and forward-operator set in Maahn et al. (2015) and in this study allows for a consistent handling of the assumptions and uncertainties associated with the ISDAC dataset. The forward-model error combined with the measurement uncertainty is presented in Fig. 2. Inclusion of the forward-model error leads to clearly enhanced uncertainties for all elements of y. Only for small σ, the forward-model error does not increase the combined error for Ze, Sl, and Sr that is related to the dominating measurement uncertainty for small SNR. The measurement uncertainty is even slightly larger than the combined error for two data points, which we attribute to the averaging of 
To investigate the possibility that the discussion of forward-model and/or measurement errors is incomplete or that uncertainties have been underestimated, we analyze in section 5c the retrieval performance under the assumption that 
3. Response functions
The response functions of the forward model that show the impact of the parameters on the moments and slopes of the radar Doppler spectrum are now investigated. Note that this section follows Maahn (2015), with minor modifications.
The response functions are presented in Fig. 3. Because kinematic broadening σk can lead to peaks that are more Gaussian shaped and can potentially reduce the response functions of other elements of x, the response functions are estimated for three different values of σk. Because σk depends on the horizontal wind and on ε, this is implemented by assuming a fixed horizontal wind of 10 m s−1 and three different values of ε (1 × 10−6, 1.25 × 10−4, and 1 × 10−3 m2 s−3) covering the range of ε values found by Shupe et al. (2008) for the MPACE campaign in Barrow. Each response function is estimated 1000 times to assess its measurement uncertainty, that is, to investigate whether the moments and slopes are stable with respect to radar noise. The resulting uncertainty range indicated by the 10th and 90th percentiles is shown in addition to the median value.
Impact of the parameters of the state vector (columns) on moments and slopes of the radar Doppler spectrum (rows) for values of the eddy dissipation rate of 10−6 (yellow), 1.25 × 10−4 (green), and 10−3 (blue) m2 s−3. The uncertainties of moments and slopes (10th and 90th percentiles) caused by radar noise are indicated as colored areas around the median values’ lines. The black vertical line denotes the mean value of the parameters in the ISDAC dataset.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
Reflectivity Ze (Fig. 3a) depends strongly on Dm and on 
Mean Doppler velocity W (Fig. 3b) depends on both mass–size and area–size coefficients (α, b, c, and d) through the fall-velocity parameterization, with the response of W to c and d being particularly strong. For N(D), W depends only on Dm and not on 
In contrast to Ze and W, Doppler spectrum width σ (Fig. 3c) depends strongly on the assumed turbulence level, which causes an offset of σ but has little influence on the response functions of the various microphysical parameters. Apart from the coefficients of the area–size and mass–size relations, with a particularly strong response to b, σ also depends on the shape parameter μ* but does not depend on Dm and 
Skewness γ (Figs. 3d) is influenced by all of the parameters but 
Kurtosis κ (Fig. 3e) is mainly determined by the shape parameter μ* and the coefficients of the area–size and mass–size relations, leading to a response of up to 3. This is larger than the uncertainty of κ (~0.4). The response to b is, however, saturated for values of greater than 1.9. Similar to γ, the response of κ is reduced for the medium turbulence level. For the highest turbulence level, κ—like a Gaussian distribution—generally has a value of 3; the response functions are greatly reduced and are smaller than the uncertainties.
The left slope Sl (Fig. 3f) responds to all parameters except w, but only for μ*, b, c, and d is the response clearly greater than the uncertainty of up to 20 dB s m−1. In comparison with γ and κ, the uncertainty appears larger because the slopes are obtained from the Doppler spectrum in logarithmic scale, which increases the impact of radar noise. The moments, on the contrary, are estimated from the spectrum in linear scale. Similar to γ and κ, the response functions are reduced for the medium and high turbulence intervals and are often within or close to the noise estimate. The uncertainty estimate is, however, reduced for higher turbulence, which is probably related to the smoothing effect of turbulence. Hence, Sl might be even more exploitable by the retrieval in higher turbulence conditions despite the smaller response functions.
The response functions of the right slope Sr (Fig. 3g), which is mainly determined by the fastest-falling particles, are similar to Sl with respect to noise. For Dm, b, and c, the responses are largest and are significantly larger than the uncertainty.
4. Retrieval results
To investigate the benefit of adding higher-order moments and slopes to a radar retrieval, an optimal-estimation-based retrieval is applied to synthetic radar observations based on ISDAC in situ profiles using the parameterizations introduced in section 2a. The training dataset for a priori estimation is based on Maahn et al. (2015), who used only aircraft observations that were closer than 10 km to Barrow (Table 1). For separation of the training and the validation datasets, the retrieval is applied to 1371 synthetic radar observations obtained at distances from 40 to 10 km from Barrow. Using a maximum distance of 40 km results in almost equal sizes for the training and validation datasets.
The percentage of the 1371 ISDAC profiles that converge successfully to a solution P is presented in Fig. 4. The values of P vary between 80% and 100%. A profile does not converge if either the convergence criterion [Eq. (16)] is not met within 30 iteration steps or the retrieval iterates to a state that either is below radar sensitivity or is physically inconsistent (e.g., ice crystals with a density that is greater than that of pure ice) and that cannot be forward modeled. Addition of higher-order moments or slopes and omission of lower-order moments reduces P, which is most likely related to the greater noise in higher-order moments and slopes as well as their less linear response to parameters describing microphysical properties.
Mean (solid) and 10th and 90th percentile (shaded area) of total degrees of freedom for signal Df obtained for the 1371 synthetic observations. The different lines are for different, cumulative sets of measurement variables; e.g., the yellow line shows Df of a retrieval only with Ze (first point), for a retrieval with Ze and W (point marked with a 1 in a gray box), and so forth. The first point of the light-green line is for a configuration only with W, the second point is for W and σ, and so forth. For the configurations marked with a 1 in a box, the retrieval solutions are investigated in detail. The dashed lines denote the percentage of retrievals of the different configurations that converged successfully.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
a. Total degrees of freedom for signal
The total degrees of freedom for signal Df that can be obtained from radar observations are presented in Fig. 4. As expected, a retrieval exploiting only reflectivity can achieve only Df = 1.0. Addition of W (using only Ze and W is called LM in the following) adds 1.0 to Df. If σ is also included, Df reaches values around 2.8 depending on the profile. When adding higher-order moments γ and κ to the retrieval, the median of Df increases only by 0.1–0.2 per moment. For some profiles, however, γ and κ can contribute a multiple as can be seen from the spread. This is probably due to the reduced response functions of the higher-order moments for stronger turbulence: higher-order moments contain exploitable information only, if the radar peak is not Gaussian shaped. This is similar for the slopes. For some profiles, Sl and Sr do not contribute any information to the retrieval, but on average they add 0.1 and 0.7, respectively, to Df. The greater additional information content of Sr in comparison with Sl is probably related to the sensitivity of Sr to large, fast particles. Since fall velocity depends on particle mass and cross-sectional area, this reveals microphysical properties of these particles that cannot be captured by other moments. The Sl, in contrast, describes the slowly falling particles whose fall velocities converge to zero with decreasing size. Therefore, the fall velocity does not reveal microphysical properties for these particles. When using all moments and slopes (AM) together, Df is 3.9 ± 0.9.
When Ze is removed from the retrieval, Df is reduced by approximately 1 as long as Sl is not added to y. Then, Df is only reduced by 0.5. Therefore, it can be concluded that Sl can partly replace Ze because they share partly redundant information. This is different than for Sr, which is particularly sensitive to large particles with large fall velocities, and therefore addition of Sr always increases Df by 0.5–0.7.
A retrieval consisting only of σ to Sl (HM) can achieve a value of Df around 2.8 ± 0.7, which is approximately 1.1 less than when using AM. Configurations without Ze and W have in general a high spread of up to 0.7 around the mean value for Df.
Using a retrieval that is based only on a single higher-order moment or slope shows that Df values of 0.3–1.0 are reached. This means that the reason for the small increase in Df when adding γ or κ to a lower-order-moments retrieval is not that γ and κ do not contain any information but rather that this information is partly redundant with that included in the lower-order moments, particularly in W and σ.
b. Using additional frequencies
Here, the benefit of adding additional frequencies to the measurement vector is investigated. For this, the impact of a dual-frequency (Ka and W bands) combination as well as a triple-frequency (Ku, Ka, and W band) configuration is studied. For simplicity, beamwidth, Nyquist range, number of FFT points, and radar noise characteristics of all frequencies were configured with respect to the MMCR specifications. This might underestimate the obtained Df because kinematic broadening σk can be directly retrieved from σ if radars with different beam widths are used (Maahn and Kollias 2012). When applied to real W-band observations, attenuation has to be accounted for even though attenuation for ice and snow is low in the W band (Nemarich et al. 1988).
The impact on Df of using two or three frequencies is presented in Fig. 5 for three configurations: LM (lower moments Ze and W), HM (σ, γ, κ, Sl, and Sr), and AM (all moments and slopes). It can be seen that the use of additional frequencies increases Df up to 5.4. Using AM in a single-frequency retrieval leads to a Df value of 3.9, which is 1.2 (1.0) more than when using a dual-frequency (triple frequency) retrieval with LM with multiple frequencies. Note that this additional information, however, has to be used to retrieve σk as well, which is not required for the LM configuration. Also when using a dual- or triple-frequency configuration, the inclusion of higher-order moments and slopes into the retrieval is beneficial because Df rises faster with the number of frequencies for AM than for LM. If only HM are used, the retrieved Df is between the configurations with AM and LM.
Mean (solid) and 10th and 90th percentile (shaded area) of total degrees of freedom for signal Df obtained for the three retrieval configurations LM (gray), HM (green), and AM (yellow) as a function of observations used: real single-frequency observations (R in a box) as well as synthetic single-, dual-, and triple-frequency observations (1, 2, and 3, respectively, in the boxes). The dashed lines indicate the percentage of converged profiles.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
The increase of Df by adding a second frequency is greater than when adding a third one. This is especially true for the LM configuration for which only a little additional information can be obtained for most cases when using a third frequency. This is in contrast to the findings of, for example, Leinonen and Moisseev (2015) and Kneifel et al. (2015), who found information in the triple-frequency signatures of reflectivity for aggregation and riming. This is most likely because our dataset contains mostly cloud ice and only a few precipitating snow particles because the aircraft was probing above cloud base most of the time. Also heavily rimed particles are expected to be infrequent in the dataset because data that contain larger amounts of supercooled water are filtered out. It is also unclear whether the SSRG approximation is able to reproduce triple-frequency signatures or whether the coefficients found by Hogan and Westbrook (2014) have to be altered to represent heavily aggregated and/or rimed particles better [see also Kneifel et al. (2016)].
c. Application to real radar data
So far, retrieval performance was only investigated by using synthetic radar observations. For the Ka band, real radar observations from the MMCR in Barrow are available during ISDAC. To make sure that the real dataset contains ice clouds with microphysical properties comparable to the synthetic dataset, only the radar observations that are closest in time and space to ISDAC in situ observations of the training dataset are analyzed. The MMCR dataset is offset corrected as proposed by Protat et al. (2011) and is filtered for values of liquid water path of greater than 0.1 kg m−2, which is the detection threshold of the used microwave radiometer retrieval (Cadeddu 1993). For a cloud with 2.5-km thickness, 0.1 kg m−2 of liquid corresponds to approximately −32 dBz. Therefore, the impact of SCLW on the radar moments and slopes can be neglected. The radar dataset contains 1354 profiles, and Df is estimated for these profiles and the three retrieval configurations of LM, HM, and AM (Fig. 5). It can be seen that Df is as large as for the synthetic observations. This result shows that retrievals using HM or AM are feasible in practice and that the forward model works sufficiently well. Only the percentage of converged profiles P is reduced by 11% and 4% for HM and AM, respectively. This is most likely related to measurement effects that were not considered in the measurement uncertainty such as nonuniform beam filling but can also be related to forward-model errors.
d. Individual degrees of freedom for signal
The individual degrees of freedom for signal df for the elements of x can be estimated from the diagonal elements of 
Box-and-whisker plots showing individual degrees of freedom for signal df that can be achieved for the elements of x. The boxes indicate the 25th and 75th percentiles, and the whiskers show the 5th and 95th percentiles. Note that the color indicates the set of moments and slopes (gray: LM, yellow: AM, and green: HM) and the symbol indicates the median for real (label R for Ka band) and synthetic radar observations (label 1 for Ka band; label 2 for Ka and W band; label 3 for Ku, Ka, and W band). The blue × indicates the impact of previous knowledge of w and σk on the retrieval; the magenta × is for quadrupling 
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
As expected, no information about σk can be obtained from the LM. Moreover, df is low for 
The spread of df is strongly increased when using HM and/or multiple frequencies. This shows the noisiness of HM and that they do not work for all conditions equally well. Another reason is likely that the retrieval might distribute the information content among the variables differently depending on the profile. For example, for κ the response function of b saturates for values larger than 1.9, which might reduce df of b for profiles with b > 1.9.
The impact of using additional frequencies is particularly large for 
Using only HM gives mostly results between LM and AM, but the spread is largest, indicating the variability of how much information can be obtained from HM among the individual profiles. Applying the retrieval to real observations gives results very similar to the synthetic single-frequency configuration. Only between the highly correlated pair c and d is the information distributed differently among the variables when using HM or AM.
e. Retrieval uncertainty
The relative uncertainty of the retrieval solution is shown together with the relative a priori uncertainties in Fig. 7 for the individual elements of x. Note that the uncertainty of the variables treated in logarithmic scale by the retrieval (Dm, 
Relative uncertainty of the solution for the elements of x. Colors and symbols are as in Fig. 6. The thick horizontal line indicates the a priori uncertainty.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
Similar to df, the impact of using AM and adding more frequencies to the retrieval can be clearly seen. The uncertainty of some variables depends more on the number of frequencies (Dm, 
When using multiple frequencies, but in particular when using HM, the spread of uncertainties among retrieval solutions is very large. This highlights, similar to df, that the advantage of using higher-order moments and slopes but also of multiple frequencies depends on the profile. Similar to df, the results for the real observations are on average very similar to the results of the synthetic single-frequency observations.
When using only HM, the median of the relative uncertainty is between LM and AM for α, b, c, d, and σk. For Dm, 
f. Retrieval bias
For the synthetic observations, it is also possible to investigate whether the retrieval solution is in agreement with the aircraft in situ profile that was used to simulate the observations (Fig. 8). In general, the bias between truth and retrieved state is below 5%, with the exception of c and w. This result is related to the fact that these quantities typically have values close to zero, which leads to large relative biases even though the absolute biases are small. The spread of parameters describing N(D) is slightly increased for single-frequency retrievals but reduced for multifrequency retrievals.
Solution with respect to truth, in percent. Colors and symbols are as in Fig. 6. The thick horizontal line indicates a bias of zero.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
g. Derived values
Relative uncertainty of the solution for the derived variables IWC, Ntot, and reff. Colors and symbols are as in Fig. 6. The thick horizontal line indicates the a priori uncertainty.
Citation: Journal of Applied Meteorology and Climatology 56, 2; 10.1175/JAMC-D-16-0020.1
5. Retrieval modifications
a. Prior knowledge of kinematics
A potential improvement of the retrieval would be the inclusion of continuity in time and space because microphysical and, in particular, kinematic variables do not change drastically between two range gates and/or time steps. In practice, this could be achieved by updating the a priori knowledge depending on neighboring/previous observations and the correlation length of the corresponding parameters (analogous to a Kalman filter). Because only the potential of this modification is evaluated, here a different approach is chosen: we investigate how the retrieval changes if w and σk are excluded from the retrieval and assumed to be known from another source (two-step retrieval). Uncertainties of externally retrieved w and σk are expressed by 
Exclusion of the kinematic variables from the retrieval reduces the total Df by 0.6–1.3 depending on configuration when using AM and HM (not shown). This indicates that the advantage of knowing w and σk can only be partly exploited by a retrieval when using HM or AM. The impact of known kinematic variables on the median of df and the relative uncertainty are presented in Figs. 6 and 7, respectively. When using LM, a significant increase of df can be found only for 
b. Radar calibration
Radar calibration is still an urgent topic, and comparisons with satellite-based radar revealed large offsets (Protat et al. 2011). Of the radar moments, only Ze is affected by radar calibration (exceptions are possible for very low SNR ratios). The slopes are only a relative measure between the noise level and the top of the peak, and they should not be affected by calibration either. However, correct determination of the noise level is crucial for estimating the slopes correctly from the simulated Doppler spectrum. Because the absolute estimation of the noise level depends on the calibration as well, the slopes are indirectly affected by calibration offsets, a fact that is also considered in the following. To investigate the impact of a potential radar miscalibration on retrieval bias, the retrieval was applied to synthetic observations assuming a relatively large calibration offset of ±3 dB. A comparison with the retrieval configured without calibration offset revealed only a small increase of the bias of 1%–5% for most variables (not shown). Only for w and α, the change in bias is found to be up to 10% for some retrieval configurations. For all variables, the median bias is still within the 25th–75th-percentile range of the spread among the ISDAC profiles.
c. Increased forward-model error
To investigate the impact of a doubled forward-model and measurement error, 
6. Discussion and outlook
The potential of adding higher-order moments of the radar Doppler spectrum (Doppler spectrum width σ, skewness γ, and kurtosis κ) as well as the left and right slope of the Doppler spectrum (Sl and Sr, respectively) to a retrieval targeted to Arctic ice clouds with low to medium turbulence conditions was investigated with a vertically pointing radar. An optimal-estimation-based retrieval was set up that retrieves quantities describing microphysical properties such as N(D) (
The key findings of this study can be summarized as follows:
Higher-order moments and slopes respond to quantities describing microphysical and kinematic properties even though they are more noisy than lower-order moments.
For single-frequency observations, the use of lower- and higher-order moments as well as the slopes (AM) can double the information content in comparison with the use of lower-order moments (LM). When using more than one frequency, the increase in information content is larger for AM than for LM. The information that higher-order moments and slopes can contribute to a retrieval varies from profile to profile.
When using only higher-order moments and the slopes (HM), retrieval performance is between those of LM and AM.
The use of AM reduces the uncertainty of all state quantities with respect to the prior knowledge. The use of multiple frequencies can reduce the posterior uncertainties for all quantities even further. Best results can be obtained when using AM and a triple-frequency configuration.
Application to real MMCR observations shows only minor differences with synthetic radar observations. This result indicates the general feasibility of the approach.
Prior knowledge about kinematic quantities enhances retrieval results—in particular, for m(D)—while an increased measurement/forward-model error decreases retrieval results. Calibration offset was found to have only a secondary impact on the results. Doubling of the combined measurement uncertainty and forward-model error leads to enhanced retrieval uncertainties but has no impact on the general conclusions.
Together with the methods to estimate the a priori dataset developed in Maahn et al. (2015), this study introduces a generalized retrieval framework that can also be transferred to periods other than ISDAC and/or locations other than Barrow. The question of whether this is also true for the a priori dataset itself has yet to be answered. For this, the representativeness of the used ISDAC dataset has to be investigated. This could be achieved by, for example, comparing the prior information of microphysical properties with other studies (e.g., Wood et al. 2014). Because the aircraft was mostly probing above cloud base, it also has to be estimated as to whether the prior can be successfully applied to snowfall consisting of larger particles. As of now, the applicability of the retrieval method is limited to ice clouds consisting of aggregates. To overcome this limitation, the applicability of SSRG theory to heavily rimed and pristine particles has to be investigated. Moreover, a key assumption in the retrieval is that knowledge of D, m, and aspect ratio is sufficient to estimate the ensemble backscattering properties (Hogan and Westbrook 2014). If this assumption is invalid (e.g., because of an additional dependence on particle type), the retrieval results might be biased to a degree that exceeds the reported uncertainties. The same is true for the estimation of fall velocity, which was assumed to depend only on D, m, and A (Heymsfield and Westbrook 2010).
Most Arctic clouds are mixed phase (Battaglia and Delanoë 2013), and the retrieval includes no limitations with respect to lightly rimed particles. However, the retrieval is not designed for mixed-phase clouds for which the SCLW is visible in the radar Doppler spectrum. Therefore, a cloud-classification algorithm (e.g., Shupe 2007) has to be used when applying the retrieval in mixed-phase clouds. An extension of the retrieval to mixed-phase clouds is desirable but requires the addition of multiple, additional variables to the state vector. Moreover, high amounts of liquid can create bimodal peaks or even two separate peaks in the radar Doppler spectrum (Luke et al. 2010). For these peaks, new moments or other quantities have to be developed to express the bimodality properly. As an alternative, the full Doppler spectrum could be used instead of radar moments and slopes for the observation vector y [as was done for rain by Tridon and Battaglia (2015)]. Using the full spectrum would also allow one to investigate whether there is additional information included in Doppler spectra of ice clouds that cannot be obtained from moments and slopes. Another possibility would be the use of radar attenuation, which is stronger for liquid water (Gosset and Sauvageot 1992), or the inclusion of other remote sensing instruments such as microwave radiometer and lidar that are sensitive to cloud water.
Even though the statistical agreement of retrieval results of real and synthetic radar observations indicates that the retrieval gives consistent results for real observations as well, its output has to be evaluated. This is challenging because collocated, independent observations of microphysical properties do not exist. Possible options would be comparison with models such as numerical weather models or large-eddy simulations (e.g., Solomon et al. 2014), comparison with surface observations (e.g., Wood et al. 2014), or radiative closure studies (e.g., Shupe et al. 2015).
A Monte Carlo–based retrieval approach could be used as an alternative to optimal estimation. Even though it is much more expensive, it could be used for selected profiles to map the solution space and to investigate whether optimal estimation converges to a global minimum or to a local minimum (Posselt and Mace 2014).
Estimating the scattering properties from database-supported discrete dipole approximation calculations (e.g., Liu 2008; Tyynelä et al. 2011) is potentially more accurate than the self-similar Rayleigh–Gans approximation and could increase the quality of the retrieval. The problem has yet to be solved, however, as to how particle distributions that follow arbitrary mass–size and area–size relations (as it is required by retrievals) can be obtained from a database that contains only certain particle types with certain discrete mass–size and area–size relations. One possibility to solve this issue would be to select different particles from the database and mix them such that the mixture matches the required microphysical bulk properties.
Acknowledgments
The authors thank Robert C. Jackson, Stefan Kneifel, Pavlos Kollias, Edward P. Luke, and Greg M. McFarquhar for providing data and fruitful discussions. This study was carried out within the project ADMIRARI II supported by the German Research Association (DFG) under research Grant LO901/5-1 and was also supported by Energy Transition and Climate Change (ET-CC) under DFG Grant ZUK 81. Author M. Maahn was also supported by Grant GSGS-2015B-F03 of the Graduate School of Geosciences of the University of Cologne. Data were obtained from the Atmospheric Radiation Measurement Program sponsored by the U.S. Department of Energy Office of Science Office of Biological and Environmental Research (BER) Climate and Environmental Sciences Division. Computing time was gratefully allocated by the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the University of Cologne. We thank the editor and the three anonymous reviewers for their helpful comments.
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