We have developed an algorithm that retrieves the size, number concentration and density of falling snow from multifrequency radar observations. This work builds on previous studies that have indicated that three-frequency radars can provide information on snow density, potentially improving the accuracy of snow parameter estimates. The algorithm is based on a Bayesian framework, using lookup tables mapping the measurement space to the state space, which allows fast and robust retrieval. In the forward model, we calculate the radar reflectivities using recently published snow scattering databases. We demonstrate the algorithm using multifrequency airborne radar observations from the OLYMPEX–RADEX field campaign, comparing the retrieval results to hydrometeor identification using ground-based polarimetric radar and also to collocated in situ observations made using another aircraft. Using these data, we examine how the availability of multiple frequencies affects the retrieval accuracy, and we test the sensitivity of the algorithm to the prior assumptions. The results suggest that multifrequency radars are substantially better than single-frequency radars at retrieving snow microphysical properties. Meanwhile, triple-frequency radars can retrieve wider ranges of snow density than dual-frequency radars and better locate regions of high-density snow such as graupel, although these benefits are relatively modest compared to the difference in retrieval performance between dual- and single-frequency radars. We also examine the sensitivity of the retrieval results to the fixed a priori assumptions in the algorithm, showing that the multifrequency method can reliably retrieve snowflake size, while the retrieved number concentration and density are affected significantly by the assumptions.

Atmospheric ice formation and growth processes have a major impact on the
Earth's radiative balance and on the hydrological cycle. Ice clouds and
snowfall occur nearly everywhere, as ice processes occur at high altitudes
even in areas where freezing temperatures at the surface are rare

Observational data are needed to evaluate the representation of ice and snow
in models. While direct measurements of ice particle properties can be made
in situ, such measurements only produce limited samples and are difficult
and expensive to make, especially when surface observations are not possible
and aircraft-based measurements are needed. Remote-sensing instruments are
able to sample far larger volumes. Radars, in particular, can make
range-resolved measurements and thus map the vertical structure of the ice
cloud–precipitation column. However, the interpretation of radar signatures
of ice particles is subject to uncertainties because the microwave scattering
properties of icy hydrometeors depend on their size, shape and structure.
These are extremely variable, as deposition growth alone results in diverse
and often complicated shapes, and further growth through aggregation and
riming adds to the complexity

Multifrequency radars have emerged as a potential tool for ice microphysics
investigations. It has been recognized for a while that snowflake size can be
constrained with collocated measurements at two different frequencies

Studies on the triple-frequency signatures of snow have, so far, been mostly
limited to numerical and theoretical investigations, as well as empirical
studies that demonstrated the plausibility of the concept. Only very
recently have databases of snow scattering properties covering a wide range of
snow growth processes

In this paper, we introduce a method for retrieving certain microphysical
properties of snow – namely, the number concentration, size and density –
from multifrequency radar observations. The algorithm is based on a Bayesian
framework and uses radar cross sections from detailed snowflake models that
cover a wide range of sizes and densities. In Sect.

The objective of a radar retrieval algorithm for snowfall is to provide the
best estimate of the microphysical properties of the snowflakes based on the
received radar signals. The unattenuated equivalent radar reflectivity factor

The attenuation of the radar signal must be accounted for in radar-only
retrieval algorithms. The attenuated reflectivity at distance

It was shown as early as

We use a technique similar to

Attenuation also results from atmospheric gases and from supercooled liquid
water. The gaseous attenuation was calculated and corrected for with the
ITU-R P.676-11 model

In order to manage the dimensionality of the problem, the microphysical
properties of the snowflakes must be parameterized. We utilize two common
assumptions for this. First, we assume that the particle size distribution
(PSD) follows the exponential distribution

The forward model in an inversion algorithm is responsible for calculating
the measurements that correspond to a given state vector – in our case, the
radar reflectivity at each wavelength given the microphysical parameters. The
simulation of radar reflectivity from snowflakes whose diameters are
comparable to or larger than the wavelength is known to require calculations
that account for the internal structure of the snowflake

While there have been considerable recent advances on the problem of modeling
snowflakes produced by different ice processes and calculating their
scattering properties, the abundance of available snowflake models leads to
another question: which set of snowflakes should be used by the forward model
in a particular situation? We use an approach that does not force us to
select any one dataset. Instead, the scattering properties are given as a
function of mass and size:

With a method to calculate the cross sections as a function of

A radar retrieval algorithm needs to invert Eqs. (

The retrieval problem is commonly stated as finding a state vector

In our multifrequency radar retrieval algorithm, the most straightforward
way to formulate the measurement vector would be to use each of the three
radar reflectivities. However, earlier studies

The measurement vector must be accompanied by an error estimate, which should
include not only the radar instrument error but also the error due to the
forward model assumptions. In our case, the latter includes the errors due to
the assumptions of an exponential size distribution, a fixed
mass–dimensional exponent

In atmospheric remote sensing, the inversion problem is often solved using
optimal estimation

Despite the shortcomings of OE, a Bayesian approach was still desirable in
order to constrain the retrieved microphysical parameters. We found that the
retrieval can be performed in a robust way through a global calculation of
the expected value of the state

Using Eq. (

Error estimates for the retrieved values can be computed using the same
technique. The error covariance matrix of the state given an observation,

The method described above allows the state and its covariance to be retrieved robustly and very quickly, with only a table lookup and an interpolation needed for each measurement. This comes at the cost of a relatively expensive initialization of the tables before the retrieval is started. However, with our parameters for the discretization, this only took about 1 min on a modern laptop computer with no parallelization, so it does not present a major computational burden.

The results of the retrieval are the parameters of Eq. (

When discussing the snowflake size,

The quantities in Eqs. (

The main source of data that we use to demonstrate the triple-frequency
retrieval is from the Airborne Third Generation Precipitation Radar

We investigated the ability of the triple-frequency algorithm to identify
snowfall processes qualitatively by comparing the results to collocated
ground-based dual-polarization radar observations. These observations were
made by the NASA S-Band Dual-Polarimetric Radar (NPOL), which was deployed

During the OLYMPEX campaign, the University of North Dakota Citation aircraft
often flew in the same area as the NASA DC-8. Typically, the Citation flew at
lower altitudes than the DC-8, and consequently there are many data points
where the Citation measurements are collocated with the APR-3. A total of 16 cases from
OLYMPEX were analyzed. The APR-3 gate closest to the Citation is found using
a

The ground-based observations of snowfall microphysics used to derive the a
priori distribution were gathered at the Hyytiälä Forestry Field Station
(

We also used balloon sounding data to support the analysis of the case studies. These data were derived from publicly available operational soundings launched daily at 00:00 and 12:00 UTC from Quillayute, Washington, near the area where the radar measurements took place.

Bayesian retrievals depend on the availability of a priori data. We based our
a priori values on two sources of in situ data: the Citation dataset from
OLYMPEX and the ground-based measurements from BAECC. Both of these datasets
can be used to derive the

For the purposes of demonstrating the algorithm, we based the a priori
distribution used in this study on a combination of the two datasets, taking
an equal number of samples from each for a total

The analysis resulted in means of

We assume that the a priori distribution is multivariate normal. Given the
limited scope of the datasets used to derive the prior distribution in this
study, we cannot rigorously test this assumption, but the choice is motivated
by probabilistic arguments that the normal distribution is the most natural
choice for an unknown distribution

The paths of the flights used in Sect.

The first of the two cases that we examined together with NPOL data took
place on 3 December 2015. The APR-3 flight leg started at 16:17:23 UTC over
the Olympic Mountains, from where the DC-8 flew towards the coast, passing
directly over the NPOL site. A map of the flight path is shown in
Fig.

Data from the 3 December 2015 case described in
Sect.

The retrievals from the case are shown in Fig.

The transition from ice crystals to aggregates is also detected at around

Another interesting feature found in this case is denoted by the red boxes in
Fig.

On 4 December 2015, precipitation originated mostly from postfrontal
convection following the passage of the front on the previous day

As Fig.

The large convective plume found by APR-3 in this case is marked with a red
box in Fig.

On the left side of NPOL, another graupel-containing region is denoted by an
orange box. This region is also accompanied by an NPOL detection of graupel
in the vicinity. The time separation in this region was longer, between

Our retrieval and the NPOL HID also seem to be in reasonably good agreement
regarding the transition from ice crystals to aggregates. Both indicate the
presence of ice crystals (i.e., small, relatively dense hydrometeors) at
higher altitudes and aggregates at lower altitudes (below approximately

As described in Sect.

To filter out outliers and poor collocations, we applied two filters. First,
to ensure an acceptably accurate collocation between the two measurements,
the time separation between them was required to be less than

Scatter plots of in situ measured (horizontal axis) and retrieved
(vertical axis) microphysical values from the collocated Citation–APR-3
dataset. The columns correspond to different microphysical parameters: from
left to right, the intercept parameter

The comparisons of the retrievals against the in situ values are shown on the
top row of Fig.

The retrieved IWCs

In the assessment of a multifrequency algorithm, one interesting question
is what are the benefits of introducing additional frequencies? To evaluate
this, we reran the analysis of Sect.

The scatter plots of the in situ and retrieved microphysical parameters are
shown in Fig.

Another way to evaluate the sensitivity to the number of frequencies is to
examine the a posteriori errors reported by the algorithm itself. These
errors, derived from the 4 December 2015 case, are shown in
Fig.

The average posterior retrieval errors of the logarithms of
microphysical variables with different combinations of radar frequencies. The
data from the 4 December 2015 case (Sect.

The errors in the single-frequency retrievals are all similar; the W band
seems to have somewhat smaller errors for

We have additionally created plots of the microphysical parameters shown in
Fig.

In order to examine the sensitivity of the results of the retrieval algorithm
to the prior assumptions, we ran the case of 4 December 2015 with shifted
prior means. We changed the mean of each variable in the state vector

The root-mean-square changes in the microphysical parameters in
response to changes in the prior. The change in the prior is indicated on the
left side of each row. The data are from the 4 December 2015 case
(Sect.

The effects on other variables from adjusting the prior of one variable are
not straightforward to interpret. These are connected in a complicated way
due to the significant a priori correlations among the different variables,
as well as the necessity of explaining the observed reflectivities with other
parameters when one of them is shifted. The dependencies are clearly not
linear. The shifts in the prior also interact with the limits of the
scattering database, which further complicates the interpretation. The IWC is the most sensitive to the prior of

In Figs. S22–S28, we repeat this analysis with the reduced frequencies. These clearly show the increasing dependence on the prior assumptions with fewer available frequencies. Again, the difference between triple and dual frequency is fairly modest, while the single-frequency retrievals shift much more in response to changes in the prior.

The most significant fixed parameter in the retrieval is the exponent

The root-mean-square changes in the microphysical parameters in
response to changes in the mass–dimensional exponent

The changes in the retrieval results for different values of

In this study, we described and evaluated an algorithm for snow microphysical retrievals using multifrequency radar measurements. The probabilistic method is based on direct application of Bayes' theorem using lookup tables. We examined the capabilities and limitations of the retrieval algorithm using data from the OLYMPEX–RADEX measurement campaign, comparing the results to ground-based radar measurements from the NASA NPOL radar and to in situ measurements from the UND Citation aircraft, both of which were collocated with the APR-3 measurements. We also examined the sensitivity of the algorithm to various assumptions used in its formulation.

The results indicate that, at least for the retrieval approach presented here, triple-frequency radar retrievals provide modest benefits over dual-frequency retrievals of snowfall properties. The probabilistic error estimates from the triple-frequency retrievals are generally only slightly smaller than those from dual-frequency retrievals, but closer examination of the retrieved values shows that the triple-frequency approach produces more detailed retrievals with higher degrees of variability than the dual-frequency retrievals. The triple-frequency method can also determine particle size throughout the range of snowflake sizes studied here, avoiding problems with some of the dual-frequency methods with sizing either small or large particles. Multifrequency retrievals significantly outperform those using only one frequency, and none of the three dual-frequency configurations studied (Ka–W-, Ku–Ka- and Ku–W-bands) appear to be decisively better than the others, although the Ka–W band combination was found to have more sensitivity to the snowflake density than the Ku–Ka- or Ku–W-band combinations. Similarly, we found the relative performances of Ku-, Ka- and W-band single-frequency retrievals to be approximately equal. Thus, information content analysis appears to suggest that multifrequency radars are preferable to single-frequency radars in snowfall retrievals, but it does not provide much insight into the exact choice of frequencies; this choice should probably be more dependent on other factors such as achievable sensitivity and resolution, the importance of attenuation, and cost.

The triple-frequency technique appears to be useful at identifying graupel,
that is, ice particles that are heavily rimed and thus considerably denser
than most aggregate snowflakes, providing a sufficient signal for the
triple-frequency retrieval to detect. This was confirmed in this study with
the comparison to polarimetric observations with the NPOL ground-based radar.
Globally, graupel occurs in relatively rare events that represent only a
small fraction of snow cases, and consequently graupel events do not impact
the statistics much. However, graupel (and hail, which is even
denser) can have a substantial societal impact where it occurs, and thus
detecting it can be valuable even though it only occurs in a small percentage
of icy precipitation. Detecting graupel plumes, together with accurate
snowflake size determination elsewhere in a precipitating region, can also
shed light on the processes involved in the formation of graupel. These
plumes are usually small in their horizontal extent, of the order of

Despite the improvements in retrieval precision in multifrequency
retrievals, the retrieved results are still dependent on the assumptions
regarding the a priori distribution of the retrieved microphysical
parameters, as well as the mass–dimensional exponent

The findings of this study concern the retrieval accuracy of multifrequency
radars and do not address their other potential benefits. For instance,
multifrequency radars can utilize lower-frequency channels (e.g., Ku band) to
penetrate deeper into precipitation, particularly heavy rain that can
attenuate higher frequencies (e.g., W band) heavily enough to block detection
altogether. Conversely, higher-frequency radars can generally be made more
sensitive, allowing detection in regions below the sensitivity thresholds of
low-frequency bands. These benefits should be considered together with the
retrieval performance when decisions about instrument specifications are
made; see, e.g.,

This work builds on earlier experimental and modeling results that suggested
that triple-frequency radars can be used to constrain snowflake habits and
examines this capability in practice with a prototype retrieval algorithm.
Based on the experience gained in this study, we can identify two
requirements for future research that need to be fulfilled in order to use
such an algorithm in an operational setting. First, the snowflake scattering
database, while more extensive than those previously available, is still
limited in its scope, and its coverage of snowflake sizes, densities and
habits should be expanded in order to support the forward model in all
scenarios. Second, the a priori distributions used in the retrievals in this
study are based on relatively few data points. An abundance of in situ data
from ice clouds and snowfall currently exists as a result of many ground- and
aircraft-based field campaigns; analyses of the data from these are needed to
support retrieval algorithm development by providing representative a priori
distributions of snowfall properties. The substantial cross
correlations found in this study among the snow microphysical properties
(Eq.

The APR-3 data files can be downloaded from the OLYMPEX
data repository at

Consider a scalar

The supplement related to this article is available online at:

JL planned the study, formulated and implemented the retrieval algorithm, and performed the analysis presented in this paper. He also led the preparation of this article, with contributions from all authors. MDL advised on the algorithm formulation and data analysis. ST and OOS calibrated and quality controlled the APR-3 data and provided support for using them; ST also participated in the collection of the APR-3 data during OLYMPEX–RADEX. BD processed the NPOL data, provided advice on their use and participated in the NPOL operations during OLYMPEX. RJC and JAF analyzed and processed the Citation data, collocated them with APR-3, and advised on the comparisons to the retrievals. AvL and DM coordinated the collection of the BAECC in situ data and processed them for use in this study.

The authors declare that they have no conflicts of interest.

We thank the two anonymous reviewers for their constructive comments. The research of Jussi Leinonen, Matthew D. Lebsock, Simone Tanelli and Ousmane O. Sy was carried out at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under contract with NASA. The work of Jussi Leinonen and Matthew D. Lebsock was supported by the NASA Aerosol-Cloud-Ecosystem and CloudSat missions under RTOP WBS 103930/6.1 and 103428/8.A.1.6, respectively. Jussi Leinonen was partly funded under subcontract 1559252 from JPL to UCLA. Simone Tanelli and Ousmane O. Sy acknowledge support from the GPM GV program and the ACE Science Working Group funding for the acquisition and initial processing of APR-3 data, and support from the Earth Science U.S. Participating Investigator program for the detailed analysis of W-band Doppler data. Funding for the research of Brenda Dolan, Randy J. Chase and Joseph A. Finlon was provided by NASA Precipitation Measurement Missions grants NNX16AI11G (BD) and NNX16AD80G (Randy J. Chase and Joseph A. Finlon) under Ramesh Kakar. Dmitri Moisseev acknowledges the funding received through ERA-PLANET, trans-national project iCUPE (grant agreement 689443), funded under the EU Horizon 2020 Framework Programme, and the Academy of Finland (grant nos. 307331 and 305175). The research work of Annakaisa von Lerber was funded by EU's Horizon 2020 research and innovation program (EC-HORIZON2020-PR700099-ANYWHERE). Edited by: Mark Kulie Reviewed by: two anonymous referees