Retrieval of snowﬂake microphysical properties from ...olympex.atmos.washington.edu/publications/2018/AMT...In this paper, we introduce a method for retrieving certain microphysical

Atmos. Meas. Tech., 11, 5471–5488, 2018https://doi.org/10.5194/amt-11-5471-2018© Author(s) 2018. This work is distributed underthe Creative Commons Attribution 4.0 License.

Retrieval of snowflake microphysical properties frommultifrequency radar observationsJussi Leinonen1,2, Matthew D. Lebsock1, Simone Tanelli1, Ousmane O. Sy1, Brenda Dolan3, Randy J. Chase4,Joseph A. Finlon4, Annakaisa von Lerber5, and Dmitri Moisseev5,61Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA2Joint Institute for Earth System Science and Engineering, University of California, Los Angeles, California, USA3Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado, USA4Department of Atmospheric Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA5Radar Science, Finnish Meteorological Institute, Helsinki, Finland6Institute for Atmospheric and Earth System Research/Physics, Faculty of Science, University of Helsinki, Helsinki, Finland

Correspondence: Jussi Leinonen ([email protected])

Received: 14 March 2018 – Discussion started: 27 March 2018Revised: 29 August 2018 – Accepted: 12 September 2018 – Published: 5 October 2018

Abstract. We have developed an algorithm that retrieves thesize, number concentration and density of falling snow frommultifrequency radar observations. This work builds on pre-vious studies that have indicated that three-frequency radarscan provide information on snow density, potentially improv-ing the accuracy of snow parameter estimates. The algorithmis based on a Bayesian framework, using lookup tables map-ping the measurement space to the state space, which allowsfast and robust retrieval. In the forward model, we calculatethe radar reflectivities using recently published snow scat-tering databases. We demonstrate the algorithm using multi-frequency airborne radar observations from the OLYMPEX–RADEX field campaign, comparing the retrieval results tohydrometeor identification using ground-based polarimetricradar and also to collocated in situ observations made us-ing another aircraft. Using these data, we examine how theavailability of multiple frequencies affects the retrieval ac-curacy, and we test the sensitivity of the algorithm to theprior assumptions. The results suggest that multifrequencyradars are substantially better than single-frequency radars atretrieving snow microphysical properties. Meanwhile, triple-frequency radars can retrieve wider ranges of snow densitythan dual-frequency radars and better locate regions of high-density snow such as graupel, although these benefits are rel-atively modest compared to the difference in retrieval perfor-mance between dual- and single-frequency radars. We alsoexamine the sensitivity of the retrieval results to the fixed a

priori assumptions in the algorithm, showing that the multi-frequency method can reliably retrieve snowflake size, whilethe retrieved number concentration and density are affectedsignificantly by the assumptions.

1 Introduction

Atmospheric ice formation and growth processes have a ma-jor impact on the Earth’s radiative balance and on the hy-drological cycle. Ice clouds and snowfall occur nearly every-where, as ice processes occur at high altitudes even in ar-eas where freezing temperatures at the surface are rare (Fieldand Heymsfield, 2015; Mülmenstädt et al., 2015). Ice cloudshave also long been a challenge for weather and climatemodels (Waliser et al., 2009). Improving the microphysicsschemes, which describe nucleation of small ice crystals andtheir transformation into precipitation-sized particles, is alsocurrently an active area of model development in which con-ceptually new schemes have been recently introduced (Har-rington et al., 2013; Morrison and Milbrandt, 2015).

Observational data are needed to evaluate the represen-tation of ice and snow in models. While direct measure-ments of ice particle properties can be made in situ, suchmeasurements only produce limited samples and are diffi-cult and expensive to make, especially when surface ob-servations are not possible and aircraft-based measurements

Published by Copernicus Publications on behalf of the European Geosciences Union.

5472 J. Leinonen et al.: Retrieval of snowflake microphysical properties

are needed. Remote-sensing instruments are able to samplefar larger volumes. Radars, in particular, can make range-resolved measurements and thus map the vertical structureof the ice cloud–precipitation column. However, the inter-pretation of radar signatures of ice particles is subject touncertainties because the microwave scattering properties oficy hydrometeors depend on their size, shape and structure.These are extremely variable, as deposition growth alone re-sults in diverse and often complicated shapes, and furthergrowth through aggregation and riming adds to the complex-ity (Pruppacher and Klett, 1997; Lamb and Verlinde, 2011).

Multifrequency radars have emerged as a potential tool forice microphysics investigations. It has been recognized fora while that snowflake size can be constrained with collo-cated measurements at two different frequencies (Matrosov,1993, 1998; Hogan et al., 2000; Liao et al., 2005). More re-cently, several studies have shown, using detailed numeri-cal scattering simulations and empirical evidence, that triple-frequency measurements provide information on both thesize and density of icy hydrometeors (Kneifel et al., 2011,2015; Leinonen et al., 2012a; Kulie et al., 2014; Stein et al.,2015; Leinonen and Moisseev, 2015; Leinonen and Szyrmer,2015; Gergely et al., 2017; Yin et al., 2017). The availabilityof this information has been expected to enable more accu-rate quantitative estimation of ice water content (IWC) andsnowfall rate and to provide a method to remotely distinguishand characterize icy hydrometeor growth processes.

Studies on the triple-frequency signatures of snow have, sofar, been mostly limited to numerical and theoretical inves-tigations, as well as empirical studies that demonstrated theplausibility of the concept. Only very recently have databasesof snow scattering properties covering a wide range of snowgrowth processes (e.g., Leinonen and Szyrmer, 2015; Kuoet al., 2016; Lu et al., 2016) become available, enabling thedevelopment of a versatile radar forward model that can pro-duce the radar signatures of many types of snowflakes. This,together with the expanded availability of collocated triple-frequency measurement datasets from field campaigns, hasprovided the prerequisites for the development of a practicalsnowfall retrieval algorithm for triple-frequency radars.

In this paper, we introduce a method for retrieving certainmicrophysical properties of snow – namely, the number con-centration, size and density – from multifrequency radar ob-servations. The algorithm is based on a Bayesian frameworkand uses radar cross sections from detailed snowflake mod-els that cover a wide range of sizes and densities. In Sect. 2,we describe the algorithm formulation. Section 3 describesthe datasets used for demonstrating and evaluating the algo-rithm, and Sect. 4 describes how the a priori distributionsused in the retrieval were derived. In Sect. 5, we investigatecase studies of airborne radar data from the Olympic Moun-tain Experiment–ACE Radar Definition Experiment 2015(OLYMPEX–RADEX’15) coordinated by NASA and com-pare the retrieval results to ground-based polarimetric radarobservations. Section 6 describes comparisons to collocated

in situ measurements. Section 7 presents statistical analysesof the sensitivity of the algorithm to the number of frequen-cies available and to the a priori assumptions. Finally, we dis-cuss the implications of the results and summarize the studyin Sect. 8.

2 Algorithm

2.1 Physical basis

The objective of a radar retrieval algorithm for snowfall is toprovide the best estimate of the microphysical properties ofthe snowflakes based on the received radar signals. The unat-tenuated equivalent radar reflectivity factor Ze for a givenwavelength λ is

Ze =λ4

π5|Kw|2

∞∫0

σbsc(D)N(D)dD, (1)

where σbsc(D) is the backscattering cross section as a func-tion of the maximum diameter D, N(D) is the particle sizedistribution and Kw is the dielectric factor defined as Kw =(n2w−1)/(n

2w+2), where nw is the complex refractive index

of liquid water assumed at a reference temperature and fre-quency.

The attenuation of the radar signal must be accounted forin radar-only retrieval algorithms. The attenuated reflectivityat distance r from the radar is given by

Z′e(r)= Ze(r)exp

−2 r∫0

∞∫0

σext(D,r′)N(D,r ′)dD dr ′

, (2)where σext is the extinction cross section. The resulting re-flectivity is usually expressed in logarithmic units of decibelsrelative to Z (dBZ), defined by

Z′dB = 10log10Z′eZ0, (3)

where Z0 = 1mm6 m−3. The attenuated reflectivity can bewritten as

Z′dB(r)= 10log10Ze(r)

Z0−

r∫0

AdB(r′)dr ′, (4)

where AdB is the two-way specific attenuation, that is, theattenuation in decibels per unit length.

It was shown as early as Hitschfeld and Bordan (1954)that weather radar attenuation correction is subject to mathe-matical instabilities that can lead to small errors multiplyingin a positive feedback loop. Namely, overestimation of at-tenuation in one radar range bin leads to overcompensationin all subsequent bins away from the radar, causing overes-timation of the precipitation signal, which in turn leads to

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J. Leinonen et al.: Retrieval of snowflake microphysical properties 5473

further overestimation of the attenuation. In multifrequencyradars, the lower-frequency signals are generally attenuatedless. In the case of snowfall, the W-band signal can be sig-nificantly attenuated, the Ka band much less so, and the Kuband is practically unattenuated by the snowflakes. Thus, theKu-band radar reflectivity can be used to correct the Ka- andW-band signals in a stable manner.

We use a technique similar to Kulie et al. (2014) for at-tenuation correction: we draw samples from the a priori dis-tribution (described in Sect. 4), calculate both the Ku-bandreflectivity and the specific attenuation at the Ka or W bandfor each sample, and fit a function between the reflectivityand the attenuation. We found that a linear function betweenZdB and ln AdB fits the relationship well. We validated thisapproach by computing attenuation afterwards from the re-trieved microphysical values; the root-mean-square (RMS)difference in the total attenuation, calculated over all binsin the case shown in Sect. 5.2, is only 0.27dB, so this ap-proximate approach to attenuation correction seems to workadequately.

Attenuation also results from atmospheric gases and fromsupercooled liquid water. The gaseous attenuation was cal-culated and corrected for with the ITU-R P.676-11 model(ITU, 2016), using radio sounding data for the temperature,pressure and humidity required by the model. The gaseousattenuation varies spatially since it is dependent on water va-por, but the error introduced by this is likely small given thatthe maximum two-way gaseous attenuation in the cases an-alyzed in this study is only 1.1dB at 94GHz (W band), andmuch less at the lower frequencies. However, supercooledliquid water found in mixed-phase clouds can cause signifi-cant radar attenuation. However, the radar echo of the super-cooled water is very weak because of the small size of thedrops, making it practically impossible, using radar signalsalone, to detect supercooled water coexisting with ice. Thus,we do not correct for attenuation caused by supercooled wa-ter, while acknowledging its role as a potential error source.

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

N(D)=N0 exp(−3D), (5)

where N0 and 3 are the intercept and slope parameters, re-spectively. Although gamma distributions, and other formsthat introduce additional parameters, are sometimes used,the exponential distribution has been found to describesnowflake size distributions well (Sekhon and Srivastava,1970; Heymsfield et al., 2008). We also found it to be a goodmatch to the in situ airborne size distribution measurementsused in this study (see Sect. 6). Therefore, we find it prefer-able over more complicated alternatives. Second, we assumethat the mass of snowflakes is given as a function of the di-

ameter as

m(D)= αDβ (6)

as has been commonly done in microphysics literature (e.gPruppacher and Klett, 1997). In the following section, weexplain how these assumptions are used to compute the radarreflectivities.

2.2 Forward model

The forward model in an inversion algorithm is responsi-ble for calculating the measurements that correspond to agiven state vector – in our case, the radar reflectivity at eachwavelength given the microphysical parameters. The simu-lation of radar reflectivity from snowflakes whose diametersare comparable to or larger than the wavelength is knownto require calculations that account for the internal struc-ture of the snowflake (Petty and Huang, 2010; Botta et al.,2011; Tyynelä et al., 2011). Recently, such calculations havebeen used for a wide variety of model snowflakes in orderto establish databases of scattering properties. We chose acombination of two such datasets as the basis of our for-ward model: the rimed snowflakes of Leinonen and Szyrmer(2015) and the OpenSSP database of Kuo et al. (2016). Thedataset of Leinonen and Szyrmer (2015) covers a wide rangeof snowflake densities, but due to the relatively coarse reso-lution of the volume elements, it mostly contains moderate-and large-sized snowflakes. The Kuo et al. (2016) dataset wasused to augment the set of snowflakes used by the forwardmodel at small sizes, D < 1mm.

While there have been considerable recent advances onthe problem of modeling snowflakes produced by differentice processes and calculating their scattering properties, theabundance of available snowflake models leads to anotherquestion: which set of snowflakes should be used by the for-ward model in a particular situation? We use an approachthat does not force us to select any one dataset. Instead, thescattering properties are given as a function of mass andsize: σ(D,m), where σ can be one of σbsc, σsca or σabs, thelast two being the scattering and absorption cross sections,respectively, with σext = σsca+ σabs. The function σ(D,m)is constructed by organizing all model snowflakes from thecombined scattering database into bins by D and m; we use128× 128 logarithmically spaced bins to cover the range ofdiameters and masses found in the dataset. For each bin, wecompute the average of σnorm ≡ σ/mγ , where γ = 2 for thebackscattering and scattering cross sections, and γ = 1 forthe absorption cross section. The reason for the normaliza-tion by m2 or m is that in the Rayleigh scattering regime(D� λ) σbsc and σsca are proportional to m2, and σabs isproportional to m (Bohren and Huffman, 1983). It followsthat the normalized cross sections are roughly constant atthe small-particle limit. To smoothen the binned values, thesamples used in the averaging are weighted using a Gaussianfunction of the distance from the bin center, with a standard

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deviation of 0.15 for both lnD and lnm. A continuous func-tion of the form

ln σnorm(ln D, ln m) (7)

is then formed by interpolation among the bin centers. Notall bins have snowflakes in them; for those we are unableto do the averaging and instead assign the scattering prop-erties to zero. This means that the limits of the coverage ofthe snowflake database in the (D,m) space are effectively as-sumed to be the limits of the natural variability in snowflakes.While this is not exactly true, the combined database doescover a wide range of microphysical processes. The assump-tion that the cross section goes to zero (as opposed to, forinstance, extrapolating it) outside the coverage area also ef-fectively truncates the integrals in Eqs. (1) and (2).

With a method to calculate the cross sections as a functionof D and m, it is relatively straightforward to compute radarreflectivities from the microphysical inputs. As can be seenfrom the previous section, the input parameters for the for-ward model are N0, 3, α and β. We start with a fixed set of1024 logarithmically spaced integration points that span theinterval [Dmin,Dmax]. The parameters α and β are used tofind the corresponding masses using Eq. (6). The cross sec-tion for each integration point is then found from the lookuptable using interpolation. The cross sections are multipliedwith the size distribution determined by N0 and3, which al-lows us to compute the integral in Eq. (1) with fixed-pointnumerical integration.

2.3 Retrieval

A radar retrieval algorithm needs to invert Eqs. (1) and (2)such that an input ofZ′e at one or more wavelengths yields theproperties of N(D) and m(D). The inversion is unavoidablyinexact, as the wide variety of snowflake number concentra-tions, size distributions and densities leads to a variability toolarge to constrain with a few radar reflectivities. The retrievalmust be performed in a probabilistic sense, deriving the mostlikely solution from the possible alternatives, using the priorinformation about snowflake properties as a constraint.

The retrieval problem is commonly stated as finding a statevector x that explains a given measurement vector y. The for-mulation of the state vector depends on which variables arechosen for retrieval and which ones are simply assumed. Inour experimentation with different combinations, we foundthat the most stable solution was to retrieve N0, 3 and α.The β parameter was fixed at 2.1. While β varies in na-ture, many experimental and modeling studies (e.g., Mitchellet al., 1990; Pruppacher and Klett, 1997; Westbrook et al.,2004; Leinonen and Moisseev, 2015; Delanoë et al., 2014;Erfani and Mitchell, 2017; Moisseev et al., 2017; Mascioet al., 2017; Mascio and Mace, 2017) have found exponentsnear this value for various types of snowflakes; we will ex-amine the sensitivity of the results to this assumption inSect. 7.3. We retrieve the logarithm of each microphysical

parameter because the dynamic ranges of the retrieved valuesare large and because using the logarithmic values makes theforward model more linear; this was examined analyticallyfor the simpler case of cloud water retrieval by Leinonenet al. (2016). The state vector then becomes

x = [ln N0 ln 3 ln α]T . (8)

In our multifrequency radar retrieval algorithm, the moststraightforward way to formulate the measurement vectorwould be to use each of the three radar reflectivities. How-ever, earlier studies (e.g., Kneifel et al., 2011; Leinonenand Szyrmer, 2015) have shown that combinations of dual-wavelength ratios (DWRs), such as simultaneous measure-ments of Ka–W-band and Ku–Ka-band DWRs, contain infor-mation about the size and density of the snowflakes. Follow-ing this concept, we form the measurement vector with theKu-band reflectivity and the Ka–W-band and Ku–Ka-bandDWRs. The measurement vector is then

y =[ZdB,Ku DWRKa/W DWRKu/Ka

]T. (9)

The choice of the Ku-band reflectivity is somewhat arbitrary,as any of the three bands could be used, but the Ku band doesbenefit from that band being the least attenuated of the three.In studies in which we omit one of the radar bands, insteadoperating with a dual-frequency radar, y consists of the re-flectivity from the lowest-frequency radar and the DWR. Forsingle-frequency retrievals, y simply contains ZdB at the sin-gle band.

The measurement vector must be accompanied by an er-ror estimate, which should include not only the radar instru-ment error but also the error due to the forward model as-sumptions. In our case, the latter includes the errors due tothe assumptions of an exponential size distribution, a fixedmass–dimensional exponent β and the orientation distribu-tions assumed in the scattering databases. The extent of theseerrors is difficult to quantify, but their effect should be simi-lar on each collocated radar frequency: for example, the radarcross section will increase with increasing particle size for allfrequencies, and thus the errors in radar reflectivity at differ-ent frequencies will partially cancel out when computing theDWRs. This suggests that the DWRs can be assumed to havesmaller errors than the absolute value of the reflectivity. Ac-cordingly, we assign 3dB of error standard deviation for theabsolute value of the radar reflectivity and 1dB for each ofthe DWRs.

In atmospheric remote sensing, the inversion problem isoften solved using optimal estimation (OE; Rodgers, 2000).This is a Bayesian method that assumes that x and y arejointly distributed according to the multivariate normal dis-tribution and which is solved using optimization methods.We found this technique to be problematic for our retrieval,partly due to the limited and discrete nature of the snowflakescattering database used in the forward model. The optimiza-tion in OE often converged to local minima, especially near



the extreme values supported by the snowflake database, in-troducing sudden changes to the retrieved values.

Despite the shortcomings of OE, a Bayesian approach wasstill desirable in order to constrain the retrieved microphysi-cal parameters. We found that the retrieval can be performedin a robust way through a global calculation of the expectedvalue of the state x given a measurement y. This is given by

E[x|y] =∫

x p(x|y)dx =1

p(y)

∫x p(y|x)p(x)dx, (10)

where p(y) is the marginal probability of y, p(y|x) is theconditional probability of a measurement y given a state xand p(x) is the a priori probability of x, described in detail inSect. 4. This approach is slightly different from the commonstrategy of finding the most likely solution given the priorand the measurement: That method aims to find the mode ofthe conditional distribution; ours determines the mean.

Using Eq. (10), we can construct a lookup table that mapsdiscrete values of y to the corresponding expected valuesE[x|y]. Multilinear interpolation is used to estimate E[x|y]for values of y that fall between the discrete values usedin the table. The errors associated with the discretizationcan be reduced to be arbitrarily small by making the inter-vals between the values finer. In the studies presented here,the lookup table for E[x|y] ranged between 0 and 35dBZfor ZdB,Ku, between −2 and 14dB for DWRKa/W, and be-tween −2 and 9dB for DWRKu/Ka, with 0.25dB discretiza-tion for each dimension. The integral in Eq. (10) was com-puted by evaluating the integrand at approximately 10000discrete points, which were distributed uniformly across afinite search space spanning (xi − 3σi ,xi + 3σi) along eachvariable, where xi is the prior mean of the ith variable in x,and σi is its prior standard deviation. Making the discretiza-tion finer than this did not seem to change the retrieval resultssignificantly in our case, although we encourage those usingthis approach for other problems to establish the appropriatediscretization for their problem.

Error estimates for the retrieved values can be computedusing the same technique. The error covariance matrix of thestate given an observation, Sx|y , can be computed as

Sx|y = E[x⊗ x|y] −E[x|y]⊗E[x|y], (11)

where “⊗” is the outer product. E[x⊗x|y] can be evaluatedusing a lookup table and interpolation in the same manner asexplained for E[x|y] above.

The method described above allows the state and its co-variance to be retrieved robustly and very quickly, with onlya table lookup and an interpolation needed for each measure-ment. This comes at the cost of a relatively expensive ini-tialization of the tables before the retrieval is started. How-ever, with our parameters for the discretization, this only tookabout 1 min on a modern laptop computer with no paral-lelization, so it does not present a major computational bur-den.

2.4 Derived variables

The results of the retrieval are the parameters of Eq. (8), butfor further analysis of the results, it is useful to derive othervariables that are important for microphysics or more intu-itively understood by end users. Perhaps most importantly,the IWC (denoted by Wice), which expresses the ice mass ina unit volume of air, is given by

Wice =

∞∫0

m(D)N(D)dD. (12)

Consistently with the calculation of the scattering properties,we set m(D)= 0 in the integral (and other integrals in thissection) where no snowflake samples are available for the(D,m) combination. If this truncation is not used, the as-sumptions of Eqs. (5) and (6) give Wice in the simple form

Wice =N0α3−β−10(β + 1), (13)

where 0 is the gamma function.When discussing the snowflake size, 3−1 gives the aver-

age diameter for the untruncated exponential size distribu-tion, but it is often clearer and more convenient to state thediameter that contributes most to the IWC. This is the mass-weighted mean diameter

Dm =

∫∞

0 Dm(D)N(D)dD∫∞

0 m(D)N(D)dD. (14)

Similarly, the total number concentration of snowflakes maybe a more meaningful quantity than N0. This is given simplyby

NT =

∞∫0

N(D)dD. (15)

Also, the density of the snowflakes depends on the diameter,but a bulk density for the snowflake ensemble can be com-puted by dividing the IWC by the volume spanned by theenclosing spheres of the snowflakes in a unit volume:

ρbulk =Wice∫

∞

0π6D

3N(D)dD. (16)

We use this definition for simplicity; a somewhat higher den-sity would be obtained by using the volume of the enclosingspheroid or ellipsoid in the integral in the denominator, butthe shape of this ellipsoid is in general dependent on D andm, which would complicate the calculation.

The quantities in Eqs. (12)–(16) are nonlinear functionsof the state x, and consequently estimating their errors isnot completely straightforward. Since our algorithm returnsa probability distribution function (PDF) for x, we can obtainstatistically valid error estimates by computing the standarddeviation of a quantity over the PDF. This can be estimatedquickly with Gauss–Hermite quadratures; see Appendix A.



3 Data

The main source of data that we use to demonstrate thetriple-frequency retrieval is from the Airborne Third Genera-tion Precipitation Radar (APR-3; Sadowy et al., 2003) flownonboard the NASA DC-8 aircraft during the OLYMPEX–RADEX experiment, which took place around the OlympicMountains of Washington State, USA, in late 2015 (Houzeet al., 2017). The RADEX involvement in this field cam-paign was intended specifically to assess the the capabili-ties of multifrequency radar observations for satellite remotesensing of precipitation processes. APR-3 acquired simul-taneous measurements at three frequencies: 13.4GHz (Kuband), 35.6GHz (Ka band) and 94.9GHz (W band). APR-3is a scanning polarimetric cloud-profiling radar with Dopplercapability. With a vertical resolution of 30m, it provideshigh-resolution 3-D measurements of clouds and precipita-tion. OLYMPEX was the first time it was deployed in itstriple-frequency configuration.

We investigated the ability of the triple-frequency algo-rithm to identify snowfall processes qualitatively by compar-ing the results to collocated ground-based dual-polarizationradar observations. These observations were made by theNASA S-Band Dual-Polarimetric Radar (NPOL), which wasdeployed 2km from the coast at 47.277◦N, 124.211◦W,157m above mean sea level (m.s.l.). The NPOL scanningstrategy interleaved planned position indicator scans (PPIs)with a series of high-resolution range-height indicator (RHI)sector scans to the west over the ocean and to the east over theQuinault River valley (Houze et al., 2017). During OLYM-PEX, the NASA DC-8 aircraft frequently flew directly alongNPOL RHI azimuths, making it relatively straightforward tocollocate with the nadir-pointing scans from APR-3. We col-located NPOL data to the APR-3 radar coordinates using thePython ARM Radar Toolkit (Helmus and Collis, 2016) byfirst identifying RHI scans whose time and direction coin-cided with the APR-3 overpass, then copying data from thenearest NPOL bin to each APR-3 bin. We used two variablesfrom NPOL: the radar reflectivity and the hydrometeor iden-tification (HID) product (Dolan and Rutledge, 2009). The lat-ter uses fuzzy logic to assign the most likely hydrometeorclass to each radar bin based on temperature and the radarreflectivity and polarimetric parameters. We use this productto provide independent estimates of the type of icy hydrom-eteors and compare them to the microphysical properties re-trieved by our algorithm.

During the OLYMPEX campaign, the University of NorthDakota Citation aircraft often flew in the same area as theNASA DC-8. Typically, the Citation flew at lower altitudesthan the DC-8, and consequently there are many data pointswhere the Citation measurements are collocated with theAPR-3. A total of 16 cases from OLYMPEX were analyzed.The APR-3 gate closest to the Citation is found using a k-dimensional-tree search algorithm. The Citation measuredthe PSD using the 2D-S (Stereo) Probe (Lawson et al., 2006)

in the range of 225µm≤D < 1mm and the High-VolumeParticle Spectrometer (1mm≤D ≤ 3.25cm). To eliminateshattered artifacts created from ice crystals colliding withthe probe housing, anti-shattering tips are used in conjunc-tion with the University of Illinois Oklahoma Optical ArrayProbe Processing Software (Jackson et al., 2014). In additionto the optical array probes, the Citation also carried a Nev-zorov probe (Korolev et al., 1998) to measure bulk total watercontent.

The ground-based observations of snowfall microphysicsused to derive the a priori distribution were gathered at theHyytiälä Forestry Field Station (61.845◦N, 24.287◦E, 150mabove mean sea level) of the University of Helsinki, Finland,during the Biogenic Aerosols – Effects on Clouds and Cli-mate (BAECC) campaign (Petäjä et al., 2016) and the fol-lowing winter of 2014–2015. The weather conditions dur-ing BAECC and the following winter were mostly mild, andmost of the snowfall observations were collected at temper-atures above −4 ◦C. Both aggregation and riming occurredfrequently during the measurement period (Moisseev et al.,2017). The PSDs were measured with a video disdrometer,the Snowflake Video Imager (SVI; Newman et al., 2009), asa function of the disk-equivalent diameter (the diameter of adisk with the projected area of the particle image). The meanPSD was calculated for every 5 min period. The resolutionof the SVI is 0.1mm, although in practice, the smallest disk-equivalent diameter used in the computations was approx-imately 0.2mm. The PSD was divided into 120 bins with abin size of 0.2mm; the highest bin is for diameters larger than26.0mm. A linear scaling factor between the disk-equivalentdiameter and the maximum diameter was determined by an-alyzing SVI images of snowflakes from each case and uti-lized to give the PSD as a function of maximum diameter.The mass retrievals were obtained by combining SVI obser-vations with a collocated precipitation gauge. Based on theparticle fall velocity and shape measurements provided bythe SVI, the masses of individual falling snow particles wereestimated with hydrodynamic theory (Mitchell and Heyms-field, 2005; von Lerber et al., 2017). The mass–dimensionalrelation in the form of Eq. (6) was determined for every 5minwith mass as a function of maximum diameter and with a lin-ear regression fit in the log scale.

We also used balloon sounding data to support the anal-ysis of the case studies. These data were derived from pub-licly available operational soundings launched daily at 00:00and 12:00 UTC from Quillayute, Washington, near the areawhere the radar measurements took place.

4 A priori assumptions

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-basedmeasurements from BAECC. Both of these datasets can be



used to derive theN0,3 and α parameters. For both datasets,N0 and 3 can be derived from the binned PSDs. The α pa-rameter can be derived by fitting a curve defined by Eq. (6)to the mass as a function of diameter; this is included inthe BAECC data, in which the mass was derived from thesnowflake fall velocity (von Lerber et al., 2017). In calculat-ing α from the BAECC dataset, we fixed β to 2.1, consistentwith the assumptions in the retrieval algorithm. For the Ci-tation data, mass is not directly available as a function ofdiameter, but Wice is estimated with the Nevzorov probe andthus α can be roughly estimated using Eq. (13).

For the purposes of demonstrating the algorithm, we basedthe a priori distribution used in this study on a combinationof the two datasets, taking an equal number of samples fromeach for a total N ≈ 6000. We recognize that this is an im-perfect solution, and a further analysis using these and otherdatasets should be conducted to establish a priori distribu-tions suitable for remote-sensing retrievals of snowfall undervarious atmospheric conditions. Doing this rigorously willlikely require an entire study of its own.

The analysis resulted in means of ln N0 = 15.4, ln 3=7.50, and ln α =−2.30 and standard deviations ofStd[ln N0] = 1.67, Std[ln 3] = 0.52 and Std[ln α] = 0.69.Because the two datasets cannot be expected to cover theentire natural distribution of these parameters, basing the apriori distribution on them would likely result in an overlyrestrictive prior. To compensate for this, we increase the stan-dard deviations given above by a factor of 1.5, acknowledg-ing that this choice is somewhat arbitrary. The correlationmatrix of x derived from the datasets is

Ca =

1 0.46 −0.070.46 1 0.54−0.07 0.54 1

, (17)from which the a priori covariance matrix can be computedas

Sa = DCaD, (18)

where D is a diagonal matrix with the standard deviationsof x on the diagonal. The resulting distribution, used as theprior in all retrievals in this study, is then given by the meanxa and covariance Sa:

xa =[

15.4 7.50 −2.30]T, (19)

Sa =

6.28 0.90 −0.180.90 0.61 0.44−0.18 0.44 1.07

. (20)In Sect. 7.2 we examine the sensitivity of the results to thechoice of prior.

We assume that the a priori distribution is multivariatenormal. Given the limited scope of the datasets used to de-rive the prior distribution in this study, we cannot rigorouslytest this assumption, but the choice is motivated by proba-bilistic arguments that the normal distribution is the most

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Figure 1. The paths of the flights used in Sect. 5.1 (a) and 5.2 (b).The darker sections of the paths show the flight data used in thisstudy (the rest of the measurements were discarded for the lack ofuseful data). The time stamps (UTC) denote the beginning and endof each flight and the beginning and end of the data that were used.The gray background shows the outline of the Olympic Peninsula,with Vancouver Island to the north.

natural choice for an unknown distribution (Jaynes, 2003).Global distributions for microphysical quantities have alsooften been found to be lognormal (e.g., Kedem and Chiu,1987; Leinonen et al., 2012b), meaning that the distributionsof their logarithms (we use the logarithmic values in the statevector) are normal. Thus a multivariate normal distributionis a reasonable assumption for this study, although largerdatasets should be analyzed in this manner in order to deriveappropriate global priors.

5 Case studies and comparison to NPOL

5.1 3 December 2015

The first of the two cases that we examined together withNPOL data took place on 3 December 2015. The APR-3flight leg started at 16:17:23 UTC over the Olympic Moun-tains, from where the DC-8 flew towards the coast, passingdirectly over the NPOL site. A map of the flight path is shownin Fig. 1a. The case consisted primarily of prefrontal strati-form precipitation; see Houze et al. (2015a) for details. Weonly used data from regions above the melting layer, whichwe identified just below 3km in altitude based on the theradar bright band; this also agrees with the 0 ◦C isothermof 2.85km in the 12:00 UTC balloon sounding from nearbyQuillayute, Washington.

The retrievals from the case are shown in Fig. 2a–e. Onthe left side of Fig. 2a–c, an orange box delineates a col-umn in which Dm increases significantly with decreasing al-titude, accompanied by a rapid decrease in ρbulk. Together,these changes point to the onset of aggregation, which re-sults in rapid growth of snowflakes accompanied by a de-crease in density as single ice crystals stick together to formaggregates, whose density decreases as a function of size.The transition can also be seen in Fig. 2e, in which orangedots denote the data points from the orange box in Fig. 2a–c.



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Figure 2. Data from the 3 December 2015 case described in Sect. 5.1. (a) The mass-weighted mean diameter Dm (Eq. 14). (b) The bulkdensity ρbulk (Eq. 16). (c) The ice water content (Eq. 12). (d) The NPOL hydrometeor identification. (e) A scatter plot ofDm and ρbulk from(a) and (b), with the red and orange points identifying the data inside the boxes of corresponding colors shown in those panels. (f) The radarreflectivity observed by NPOL.

The transition from ice crystals to aggregates is also de-tected at around 5 km in altitude, 20–45km on the distancescale, by both the triple-frequency retrieval, which shows asudden increase in Dm (Fig. 2a), and by NPOL, which iden-tifies a change in the hydrometeor type at roughly the samealtitude. According to NPOL HID, the hydrometeors abovethis altitude consist mostly of a mixture of ice crystals andaggregates, while the hydrometeors below it are identified asaggregates. While the altitude at which aggregation initiatesappears to be similar between NPOL and our retrieval, smalldiscrepancies are to be expected because the APR-3 obser-vations are not perfectly simultaneous with the NPOL scan.The time difference ranges from 4min at the beginning ofthe observations shown in Fig. 2 to 14min at the end. Fur-ther evidence for aggregation is provided by sounding data,which indicate a temperature between−15 and−12 ◦C in thelayer at 5.0–5.5km in altitude, a common temperature rangefor the onset of aggregation driven by dendritic growth ofice crystals at these temperatures (Bailey and Hallett, 2009;Lamb and Verlinde, 2011).

Another interesting feature found in this case is denotedby the red boxes in Fig. 2a–c. In this region, the retrieved mi-crophysical variables indicate moderately sized snowflakes

with relatively high ρ, which suggests that rimed snowflakesoccur in the area. The data points located within this box areshown in red in the scatter plot of Fig. 2e, which confirmsthese attributes. It is interesting to note that the red regionand the bottom of the orange region have similar IWCs, butthe sizes and densities are very different. NPOL also detectssome graupel in this region, which suggests that the three-frequency retrieval detects snowflake riming and graupel for-mation. In the following case, we further explore this capa-bility.

5.2 4 December 2015

On 4 December 2015, precipitation originated mostly frompostfrontal convection following the passage of the fronton the previous day (Houze et al., 2015b). The DC-8 fol-lowed a flight path similar to in the previous case (Fig. 1b);APR-3 data collection for the dataset shown here started at14:53:21 UTC. We collocated two NPOL RHIs to APR-3 co-ordinates, one pointing toward land and the other toward theocean. For the ocean-pointing scan, we selected an RHI thatis offset by 4◦ from the optimal collocation with APR-3 in or-der to better capture a convective plume that was observed by



APR-3 but had moved before being scanned by NPOL 2 minlater. This shifted the location of the scan by only 500 m at thedistance of the plume. The sounding data and the radar brightband both placed the melting level at around 1.3km, lowerthan on the previous day. We again only used data points lo-cated above the melting layer.

The large convective plume found by APR-3 in this case ismarked with a red box in Fig. 3a–c. As with Fig. 2, the datapoints from this box are denoted with red dots in Fig. 3e. Inthis case, the data points from the plume are particularly dis-tinct from the rest of the joint distribution of Dm and ρbulk,indicating moderately large particles with high density, char-acteristic of graupel. NPOL also indicates a similarly sizedplume of graupel in this region. The time separation of thescans in the region to the right of NPOL in Fig. 3 is only2min, so it seems likely that the same plume was capturedby both radars. The spatial shift between the plumes observedby APR-3 and NPOL appears to be 1–2km; this is consistentwith the 13ms−1 wind speed measured by the sounding at3km in altitude, which translates to a 1.5km distance over2min.

On the left side of NPOL, another graupel-containing re-gion is denoted by an orange box. This region is also ac-companied by an NPOL detection of graupel in the vicinity.The time separation in this region was longer, between 4 and8min, so the plume had more time to move away from thevertical cross section before being observed by APR-3. Re-gardless, the two radars agree on location of the plume towithin 2km and on its height to within 0.5km.

Our retrieval and the NPOL HID also seem to be in rea-sonably good agreement regarding the transition from icecrystals to aggregates. Both indicate the presence of ice crys-tals (i.e., small, relatively dense hydrometeors) at higher alti-tudes and aggregates at lower altitudes (below approximately4km), with the transition point varying considerably withinthis case. Both products also identify the presence of smallerparticles at 2–3km in altitude in the region, around 20km onthe horizontal scale.

6 Comparison to in situ data

As described in Sect. 3, the UND Citation aircraft gatheredparticle probe measurements simultaneously with the NASADC-8 radar observations during OLYMPEX. This resulted ina set of collocated radar and in situ data. The retrieval algo-rithm was run using the collocated and attenuation-correctedradar reflectivity values. The retrieved microphysical quanti-ties were then compared to those measured in situ. The avail-ability of variables from the in situ dataset is somewhat lim-ited: while the number and projected sizes of the ice particlescan be measured quite accurately using the imaging probes,the two-dimensional nature of the imager limits the accuracyof the maximum dimension as this must be estimated from aprojection of the particle. The snowflake masses are also dif-

ficult to determine. The bulk IWC can be estimated with theNevzorov probe, but its inlet is only 8mm in diameter, whichcauses it to underestimate IWC when the maximum particlesize exceeds approximately 4mm (Korolev et al., 2013). Un-fortunately, the cases with large snowflakes are where onewould expect the largest benefits from multifrequency meth-ods because of the stronger resonance effects involved inscattering. Thus, this limitation of the Nevzorov probe some-what diminishes its value in validating the retrievals. Whilethe Citation measurements do not give the masses of individ-ual particles, α can be estimated from Eq. (13) if the IWCgiven by the Nevzorov probe is assumed to be correct.

To filter out outliers and poor collocations, we applied twofilters. First, to ensure an acceptably accurate collocation be-tween the two measurements, the time separation betweenthem was required to be less than 2min. Second, for adequatesampling, the total number concentrationNT was required tobe more than 103 m−3. These criteria successfully removedmost outliers that we found in the unfiltered comparisons.

The comparisons of the retrievals against the in situ valuesare shown on the top row of Fig. 4 (the same analysis runwith reduced frequencies, shown on the other rows of Fig. 4,is discussed in Sect. 7.1). The figures show that retrievalsof the slope parameter 3 compare considerably better to thein situ values than do the retrievals of the intercept param-eter N0, which in turn are better than those of the mass–dimensional factor α. The 3 parameters agree well through-out the range of values (for ln3, root-mean-square error isRMSE[ln3] = 0.41, bias is bias[ln3] = 0.023 and correla-tion is Cor[ln3] = 0.70), showing that particle sizing canbe carried out reliably using the multifrequency retrieval. N0is also quite well matched (Cor[ln N0] = 0.56), but the rela-tive errors are much larger than for3 (RMSE[ln N0] = 3.01,bias[ln N0] = −0.73). The α parameters are poorly matchedbetween the two datasets, although the retrieval producessome variation in this parameter. In any case, one should beskeptical of the α comparison as the in situ values have beenderived from the Nevzorov probe data, which suffers fromthe abovementioned problems, and using Eq. (13), which isan approximation. Furthermore, fixing the β parameter mayfurther exacerbate the problem with estimating α.

The retrieved IWCs Wice correspond quite well to thein situ values (RMSE[lnWice] = 0.72, bias[lnWice] = 0.30,Cor[lnWice] = 0.67). Interestingly, the IWC, which is afunction of N0 and α, appears to be better retrieved than ei-ther of those parameters. Opposite errors in N0 and α, seenin their respective scatter plots, suggest that their retrieval er-rors compensate for each other in a way that allows Wice tobe constrained better than either N0 or α alone. This is alsosupported by the correlation matrix of the retrieval errors, forwhich the error correlation between ln N0 and ln α is −0.30on average. The red dots in Fig. 4 correspond to larger iceparticles, where the Nevzorov probe might be prone to un-derestimation. However, there does not appear to be a signifi-cant difference inWice between the small and large particles.



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Figure 3. As Fig. 2, except for the 4 December 2015 case described in Sect. 5.2. The location of NPOL on the flight track is marked by thearrow in panels (a)–(d) and (f).

However, the large particles stand out in the α scatter plot,where they are clearly the worst match between the in situand retrieved values.

7 Sensitivity analysis

7.1 Sensitivity to the number of frequencies

In the assessment of a multifrequency algorithm, one inter-esting question is what are the benefits of introducing addi-tional frequencies? To evaluate this, we reran the analysis ofSect. 6 with subsets of the frequencies used in the full anal-ysis. We examined all the possible combinations of availablebands, always using the lowest frequency for the absolute re-flectivity, combined with the DWRs that were available (oneDWR for dual-frequency retrievals and two DWRs for thetriple-frequency retrieval).

The scatter plots of the in situ and retrieved microphysicalparameters are shown in Fig. 4. These plots suggest that theresults of the triple-frequency retrieval are similar to those ofthe dual-frequency retrievals. However, the multifrequencyretrievals clearly outperform single-frequency retrievals. Thetriple- and dual-frequency scatter plots are visually similarfor all two- and three-frequency combinations for 3, and

to a lesser extent N0. The dual-frequency retrieval using theKa–W bands seems to be limited in its ability to determinethe size of large particles (small 3), presumably because thedual-frequency ratio saturates at large sizes, while the Ku–Ka-band retrieval suffers from a similar problem with smallparticles. The Ku–W-band retrieval and the triple-frequencyretrieval do not suffer from this problem. Meanwhile, thesingle-frequency retrievals all have poor sensitivity to N0.Ku- and Ka-band single-frequency retrievals have some sen-sitivity to 3 for small particles, while the W-band retrievalalso cannot discern this parameter particularly well. None ofthe retrievals perform adequately with α, although the multi-frequency retrievals, especially the triple-frequency retrieval,permit considerably more variation in the values of that pa-rameter: α is almost constant with the single-frequency re-trievals, while its relative standard deviation is about 60 % inthe triple-frequency results, indicating that the retrieval algo-rithm is confident enough in the signal to estimate α as some-thing other than the a priori mean. The results for α should beinterpreted skeptically because of the issues with the deriva-tion of α, as explained in Sect. 6. The single-frequency re-trievals appear to constrain Wice much better than they con-strain any of the individual microphysical parameters.



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Figure 4. 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 N0, the slopeparameter 3, the mass–dimensional prefactor α and the ice water content Wice. The rows correspond to different combinations of radarfrequencies and DWRs used to run the retrieval, as shown to the left of each row. The color denotes the size of the snowflakes: blue dotscorrespond to small particles (largest 25% of 3), orange to medium-sized particles and red to large particles (smallest 25% of 3). In eachplot, the black line is the 1 : 1 line. Note the logarithmic scales on the axes.



ln N0 ln ln ln Wice ln NT ln Dm ln bulk

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1.75 0.36 0.76 0.92 1.54 0.35 0.96

1.10 0.31 0.84 0.79 1.11 0.30 1.08

2.43 0.58 0.81 1.05 1.92 0.58 1.17

2.34 0.62 0.81 0.95 1.81 0.61 1.19

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Figure 5. The average posterior retrieval errors of the logarithmsof microphysical variables with different combinations of radar fre-quencies. The data from the 4 December 2015 case (Sect. 5.2) areused in this figure.

Another way to evaluate the sensitivity to the number offrequencies is to examine the a posteriori errors reported bythe algorithm itself. These errors, derived from the 4 Decem-ber 2015 case, are shown in Fig. 5 for the different frequencycombinations. According to the error estimate from the algo-rithm, the three-frequency retrieval seems to yield a modestbut fairly consistent improvement over the dual-frequency re-sults. These, like with the in situ data comparison, are clearlybetter than the single-frequency results for all parameters,although the differences for α, Wice and ρbulk are less pro-nounced.

The errors in the single-frequency retrievals are all sim-ilar; the W band seems to have somewhat smaller errorsfor Wice and N0 while the Ku band is slightly better withthe particle size. Notably, the a posteriori errors for thesingle-frequency retrievals are not much smaller than the apriori errors of Stda[ln N0] = 2.45, Stda[ln 3] = 0.83 andStda[ln α] = 1.13, which emphasizes the poor informationcontent in the single-frequency retrievals. Regardless, withWice the single-frequency retrievals perform nearly as well asthe multifrequency ones, consistent with what was shown inthe comparison to in situ values. None of the dual-frequencyoptions are significantly better than the others, either, al-though the Ku–Ka-band configuration underperforms theKa–W-band and Ku–W-band configurations in retrievals ofN0 and NT , and to a lesser extent Wice. The Ka–W- and Ku–W-band configurations are nearly equally good.

We have additionally created plots of the microphysicalparameters shown in Fig. 3 using each of the frequency com-binations found in Fig. 5. Due to the large number of plotsresulting from this analysis, these plots are not shown here,but can be found in Figs. S1–S21 of the Supplement accom-panying this article. A notable feature of these plots is thehigher level of detail and wider range of variation found in

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Figure 6. The root-mean-square changes in the microphysical pa-rameters in response to changes in the prior. The change in the prioris indicated on the left side of each row. The data are from the 4 De-cember 2015 case (Sect. 5.2).

the triple-frequency plots of Dm and especially ρbulk com-pared to the dual-frequency plots. The Ka–W band dual-frequency retrieval appears to capture the plume found by thetriple-frequency approach, albeit with a more subdued signal;the other two dual-frequency configurations miss the plumealtogether. Consistent with the results of other comparisonsshown in this section, the dual-frequency plots capture moredetail than the single-frequency plots. This is especially strik-ing for the plots of ρbulk, in which the single-frequency re-trievals appear to always give nearly the same density. Incontrast to Dm and ρbulk, Wice has only small differences,and similar levels of detail between the single-frequency andmultifrequency retrievals. This is again similar to the findingsin Fig. 4.

7.2 Sensitivity to prior assumptions

In order to examine the sensitivity of the results of the re-trieval algorithm to the prior assumptions, we ran the case of4 December 2015 with shifted prior means. We changed themean of each variable in the state vector x, one at a time, by±1 standard deviation of that variable. The results are shownin Fig. 6. The results are consistent with the retrievals in thesense that a shift in the prior of a variable causes a smallershift of the same sign in the a posteriori value of that variable.

The effects on other variables from adjusting the prior ofone variable are not straightforward to interpret. These areconnected in a complicated way due to the significant a pri-ori correlations among the different variables, as well as thenecessity of explaining the observed reflectivities with otherparameters when one of them is shifted. The dependenciesare clearly not linear. The shifts in the prior also interact withthe limits of the scattering database, which further compli-cates the interpretation. The IWC is the most sensitive tothe prior of ln N0. The results are the least sensitive to theprior assumption of ln 3, indicating that ln 3 is very well



ln N0 ln ln Wice ln NT ln Dm ln bulk

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Figure 7. The root-mean-square changes in the microphysical pa-rameters in response to changes in the mass–dimensional exponentβ. The standard assumption of this paper, β = 2.1, is used as thebaseline. The value of β is indicated on the left side of each row.The analysis is based on the 4 December 2015 case (Sect. 5.2).

constrained by the observations. Changes to the priors of ei-ther ln N0 or lnα induce considerably larger changes in theresults. Thus, the triple-frequency algorithm is clearly stillsomewhat dependent on the a priori assumptions, althoughthe changes in the posterior values are much smaller than thecorresponding changes in the prior, showing that the radarsignal constrains them quite effectively.

In Figs. S22–S28, we repeat this analysis with the re-duced frequencies. These clearly show the increasing depen-dence on the prior assumptions with fewer available frequen-cies. Again, the difference between triple and dual frequencyis fairly modest, while the single-frequency retrievals shiftmuch more in response to changes in the prior.

7.3 Sensitivity to mass–dimensional exponent

The most significant fixed parameter in the retrieval is the ex-ponent β of the mass–dimensional relationship (Eq. 6). Simi-lar to Sect. 7.2, we carried out an analysis of the sensitivity ofthe retrieval results to the choice of β. We used the value usu-ally adopted in this paper, β = 2.1, as the reference and com-pared the results obtained with β = 1.9, β = 2.3 and β = 2.5to the reference retrieval. The values were chosen based onexponents found in the literature for single crystals, aggre-gate snowflakes and rimed particles (e.g., Mitchell et al.,1990, their Tables 1 and 2); higher exponents such as thoseclose to 3.0 often found for graupel (Locatelli and Hobbs,1974; Heymsfield and Kajikawa, 1987) were not tested be-cause the distribution of particles in the scattering databasesdoes not support such high exponents well. The results areshown in Fig. 7. This figure is similar to Fig. 6, but we haveomitted the changes in the mass–dimensional prefactor α be-cause this parameter does not have a physical meaning inde-pendent of β.

The changes in the retrieval results for different values of βexhibit patterns similar to those resulting from the change inprior values: The parameters corresponding to number con-centration (N0 and NT ) and density (ρbulk) are the most sen-sitive to the assumptions. Meanwhile, parameters related to

particle size (3 and Dm) and, to a lesser extent, the IWCWice are less affected by changes in β. The changes in re-trieved parameters with changing β can be substantial, sug-gesting that a good estimate of β is important for quantita-tively correct retrievals. However, the changes are predictableand reasonable, which suggests that the algorithm is robustand can function with different values of β without majorproblems. A notable exception to the predictable behavior isthat of Wice, whose retrieved value increases in response toboth increase and decrease in β from 2.1.

8 Conclusions

In this study, we described and evaluated an algorithm forsnow microphysical retrievals using multifrequency radarmeasurements. The probabilistic method is based on directapplication of Bayes’ theorem using lookup tables. We exam-ined the capabilities and limitations of the retrieval algorithmusing data from the OLYMPEX–RADEX measurement cam-paign, comparing the results to ground-based radar measure-ments from the NASA NPOL radar and to in situ measure-ments from the UND Citation aircraft, both of which werecollocated with the APR-3 measurements. We also examinedthe sensitivity of the algorithm to various assumptions usedin its formulation.

The results indicate that, at least for the retrieval ap-proach presented here, triple-frequency radar retrievals pro-vide modest benefits over dual-frequency retrievals of snow-fall properties. The probabilistic error estimates from thetriple-frequency retrievals are generally only slightly smallerthan those from dual-frequency retrievals, but closer exami-nation of the retrieved values shows that the triple-frequencyapproach produces more detailed retrievals with higher de-grees of variability than the dual-frequency retrievals. Thetriple-frequency method can also determine particle sizethroughout the range of snowflake sizes studied here, avoid-ing problems with some of the dual-frequency methods withsizing either small or large particles. Multifrequency re-trievals significantly outperform those using only one fre-quency, and none of the three dual-frequency configura-tions studied (Ka–W-, Ku–Ka- and Ku–W-bands) appearto be decisively better than the others, although the Ka–Wband combination was found to have more sensitivity to thesnowflake density than the Ku–Ka- or Ku–W-band combina-tions. Similarly, we found the relative performances of Ku-,Ka- and W-band single-frequency retrievals to be approxi-mately equal. Thus, information content analysis appears tosuggest that multifrequency radars are preferable to single-frequency radars in snowfall retrievals, but it does not pro-vide much insight into the exact choice of frequencies; thischoice should probably be more dependent on other factorssuch as achievable sensitivity and resolution, the importanceof attenuation, and cost.



The triple-frequency technique appears to be useful atidentifying graupel, that is, ice particles that are heav-ily rimed and thus considerably denser than most aggre-gate snowflakes, providing a sufficient signal for the triple-frequency retrieval to detect. This was confirmed in this studywith the comparison to polarimetric observations with theNPOL ground-based radar. Globally, graupel occurs in rela-tively rare events that represent only a small fraction of snowcases, and consequently graupel events do not impact thestatistics much. However, graupel (and hail, which is evendenser) can have a substantial societal impact where it oc-curs, and thus detecting it can be valuable even though it onlyoccurs in a small percentage of icy precipitation. Detectinggraupel plumes, together with accurate snowflake size deter-mination elsewhere in a precipitating region, can also shedlight on the processes involved in the formation of graupel.These plumes are usually small in their horizontal extent, ofthe order of 1km, requiring a fairly high spatial resolution inthe radars used to detect them, which can be challenging toachieve if multifrequency radars are considered for satelliteapplications.

Despite the improvements in retrieval precision in mul-tifrequency retrievals, the retrieved results are still depen-dent on the assumptions regarding the a priori distribution ofthe retrieved microphysical parameters, as well as the mass–dimensional exponent β. Different retrieved parameters havewidely different sensitivities to the assumptions: the retrievedsnow particle size changes only modestly in response tochanges to the prior and to β, indicating that the size canbe retrieved robustly with the multifrequency method. Incontrast, the retrieved number concentration and density aremuch more sensitive to the assumptions and therefore po-tentially susceptible to retrieval errors caused by inaccurateprior data. Therefore, it is still vital to constrain the algorithmusing in situ measurements that provide not only the sizeand number concentration of snowflakes but also their mass–dimensional scaling parameters α and β. Later versions ofthe algorithm should include β as a retrievable parameter andincorporate it in the multivariate prior so that the retrieval er-rors originating from the uncertainty of β can be properlyquantified.

The findings of this study concern the retrieval accuracyof multifrequency radars and do not address their other po-tential benefits. For instance, multifrequency radars can uti-lize lower-frequency channels (e.g., Ku band) to penetratedeeper into precipitation, particularly heavy rain that can at-tenuate higher frequencies (e.g., W band) heavily enoughto block detection altogether. Conversely, higher-frequencyradars can generally be made more sensitive, allowing de-tection in regions below the sensitivity thresholds of low-frequency bands. These benefits should be considered to-gether with the retrieval performance when decisions aboutinstrument specifications are made; see, e.g., Leinonen et al.(2015) for a quantitative assessment of retrieval capabilitiesof a potential spaceborne triple-frequency radar.

This work builds on earlier experimental and modeling re-sults that suggested that triple-frequency radars can be usedto constrain snowflake habits and examines this capabilityin practice with a prototype retrieval algorithm. Based onthe experience gained in this study, we can identify two re-quirements for future research that need to be fulfilled in or-der to use such an algorithm in an operational setting. First,the snowflake scattering database, while more extensive thanthose previously available, is still limited in its scope, andits coverage of snowflake sizes, densities and habits shouldbe expanded in order to support the forward model in all sce-narios. Second, the a priori distributions used in the retrievalsin this study are based on relatively few data points. An abun-dance of in situ data from ice clouds and snowfall currentlyexists as a result of many ground- and aircraft-based fieldcampaigns; analyses of the data from these are needed tosupport retrieval algorithm development by providing rep-resentative a priori distributions of snowfall properties. Thesubstantial cross correlations found in this study among thesnow microphysical properties (Eq. 17) emphasize the needfor a multivariate analysis of these datasets.

Data availability. The APR-3 data files can be down-loaded from the OLYMPEX data repository athttps://doi.org/10.5067/GPMGV/OLYMPEX/APR3/DATA201(Durden and Tanelli, 2018), and the NPOL data fromhttps://doi.org/10.5067/GPMGV/OLYMPEX/NPOL/DATA301(Wolff et al., 2017). The Citation data are available athttps://github.com/dopplerchase/Chase_et_al_2018 (last ac-cess: 1 October 2018), maintained by Randy J. Chase (email:[email protected]). The BAECC campaign data are availableat https://github.com/dmoisseev/Snow-Retrievals-2014-2015(last access: 1 October 2018). The sounding data can beobtained from the University of Wyoming collection athttp://weather.uwyo.edu/upperair/sounding.html (last access:1 October 2018). The retrieval results, used to generate the plots,are available in numerical form from Jussi Leinonen (email:[email protected]).


https://doi.org/10.5067/GPMGV/OLYMPEX/APR3/DATA201https://doi.org/10.5067/GPMGV/OLYMPEX/NPOL/DATA301https://github.com/dopplerchase/Chase_et_al_2018https://github.com/dmoisseev/Snow-Retrievals-2014-2015http://weather.uwyo.edu/upperair/sounding.html


Appendix A: Fast derivation of error estimates forretrieved quantities

Consider a scalar Q(x) that is a function (not necessarily alinear function) of the vector x of normally distributed ran-dom variables, whose probability distribution p(x) is givenby the mean 〈x〉 and the covariance S. For example, Q canbe ln Wice or the logarithm of any variable introduced inSect. 2.4. Then, a probabilistic error estimate is given by thestandard deviation

1Q= Std[Q] =√〈Q2〉− 〈Q〉2, (A1)

where the expectation, denoted by 〈·〉, is taken over the PDFof x. The expectation can be estimated efficiently using aGauss–Hermite quadrature. For a three-variable x (general-ization to other numbers of variables is straightforward), theexpectation 〈Q〉 is obtained as follows:

〈Q〉 =

∫x

Q(x)p(x)dx ≈∑i,j,k

wiwjwkQ(xijk), (A2)

wi =1√πwGH,i, (A3)

xijk = 〈x〉+√

2V31/2[xGH,i xGH,j xGH,k

]T, (A4)

where

– V is a matrix whose columns contain the normalizedeigenvectors of S,

– 3 is a diagonal matrix containing the correspondingeigenvalues of S,

– xGH and wGH are the points and weights of a Gauss–Hermite quadrature that gives the approximation

∞∫−∞

exp(−x2)f (x)dx ≈N∑i=1

wGH,i f (xGH,i), (A5)

where the approximation is exact if f is a polynomialof at most degree 2N − 1; xGH and wGH can be foundin many tables (e.g., Beyer, 1987) and in scientific soft-ware packages (e.g., SciPy; Oliphant, 2007).

〈Q2〉 can also be estimated using the above method, thus giv-ing the error estimate when substituted into Eq. (A1). Thisis derived by computing the Gauss–Hermite quadrature forthe standard multivariate normal distribution with zero meanand identity covariance, then mapping the quadrature pointsto the corresponding points in the distribution of x.



Supplement. The supplement related to this article is availableonline at: https://doi.org/10.5194/amt-11-5471-2018-supplement.

Author contributions. JL planned the study, formulated and imple-mented the retrieval algorithm, and performed the analysis pre-sented in this paper. He also led the preparation of this article, withcontributions from all authors. MDL advised on the algorithm for-mulation and data analysis. ST and OOS calibrated and quality con-trolled the APR-3 data and provided support for using them; ST alsoparticipated in the collection of the APR-3 data during OLYMPEX–RADEX. BD processed the NPOL data, provided advice on theiruse and participated in the NPOL operations during OLYMPEX.RJC and JAF analyzed and processed the Citation data, collocatedthem with APR-3, and advised on the comparisons to the retrievals.AvL and DM coordinated the collection of the BAECC in situ dataand processed them for use in this study.

Competing interests. The authors declare that they have no con-flicts of interest.

Acknowledgements. We thank the two anonymous reviewers fortheir constructive comments. The research of Jussi Leinonen,Matthew D. Lebsock, Simone Tanelli and Ousmane O. Sy wascarried out at the Jet Propulsion Laboratory (JPL), CaliforniaInstitute of Technology, under contract with NASA. The workof Jussi Leinonen and Matthew D. Lebsock was supported bythe NASA Aerosol-Cloud-Ecosystem and CloudSat missionsunder RTOP WBS 103930/6.1 and 103428/8.A.1.6, respectively.Jussi Leinonen was partly funded under subcontract 1559252from JPL to UCLA. Simone Tanelli and Ousmane O. Sy ac-knowledge support from the GPM GV program and the ACEScience Working Group funding for the acquisition and initialprocessing of APR-3 data, and support from the Earth ScienceU.S. Participating Investigator program for the detailed analysis ofW-band Doppler data. Funding for the research of Brenda Dolan,Randy J. Chase and Joseph A. Finlon was provided by NASAPrecipitation Measurement Missions grants NNX16AI11G (BD)and NNX16AD80G (Randy J. Chase and Joseph A. Finlon)under Ramesh Kakar. Dmitri Moisseev acknowledges the fundingreceived through ERA-PLANET, trans-national project iCUPE(grant agreement 689443), funded under the EU Horizon 2020Framework Programme, and the Academy of Finland (grant nos.307331 and 305175). The research work of Annakaisa von Lerberwas funded by EU’s Horizon 2020 research and innovation program(EC-HORIZON2020-PR700099-ANYWHERE).

Edited by: Mark KulieReviewed by: two anonymous referees

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Retrieval of snowﬂake microphysical properties from ...olympex.atmos.washington.edu/publications/2018/AMT...In this paper, we introduce a method for retrieving certain microphysical