• Atlas, D., and C. Ulbrich, 2006: Drop size spectra and integral remote sensing parameters in the transition from convective to stratiform rain. Geophys. Res. Lett., 33, L16803, https://doi.org/10.1029/2006GL026824.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Barnes, S. L., 1964: A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3, 396409, https://doi.org/10.1175/1520-0450(1964)003<0396:ATFMDI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Baumgardner, D., and Coauthors, 2017: Cloud ice properties: In situ measurement challenges. Ice Formation and Evolution in Clouds and Precipitation: Measurement and Modeling Challenges, Meteor. Monogr., No. 58, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-16-0011.1.

    • Crossref
    • Export Citation
  • Bringi, V., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 636 pp.

    • Crossref
    • Export Citation
  • Brown, E. N., 1982: Ice detector evaluation for aircraft hazard warning and undercooled water content measurements. J. Aircr., 19, 980983, https://doi.org/10.2514/3.44800.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Brown, P., and P. Francis, 1995: Improved measurements of the ice water content in cirrus using a total-water probe. J. Atmos. Oceanic Technol., 12, 410414, https://doi.org/10.1175/1520-0426(1995)012<0410:IMOTIW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cao, Q., and G. Zhang, 2009: Errors in estimating raindrop size distribution parameters employing disdrometer and simulated raindrop spectra. J. Appl. Meteor. Climatol., 48, 406425, https://doi.org/10.1175/2008JAMC2026.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Casella, D., G. Panegrossi, P. Sanò, A. C. Marra, S. Dietrich, B. T. Johnson, and M. S. Kulie, 2017: Evaluation of the GPM-DPR snowfall detection capability: Comparison with CloudSat-CPR. Atmos. Res., 197, 6475, https://doi.org/10.1016/j.atmosres.2017.06.018.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., and V. N. Bringi, 1987: Simulation of radar reflectivity and surface measurements of rainfall. J. Atmos. Oceanic Technol., 4, 464478, https://doi.org/10.1175/1520-0426(1987)004<0464:SORRAS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., W. Li, and B. Zafar, 2005: Estimation of raindrop size distribution from spaceborne radar observations. IEEE Trans. Geosci. Remote Sens., 43, 10781086, https://doi.org/10.1109/TGRS.2005.846130.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chase, R. J., and Coauthors, 2018: Evaluation of triple-frequency radar retrieval of snowfall properties using coincident airborne in situ observations during OLYMPEX. Geophys. Res. Lett., 45, 57525760, https://doi.org/10.1029/2018GL077997.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cober, S. G., G. A. Isaac, A. V. Korolev, and J. W. Strapp, 2001: Assessing cloud-phase conditions. J. Appl. Meteor., 40, 19671983, https://doi.org/10.1175/1520-0450(2001)040<1967:ACPC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cohard, J., and J. Pinty, 2000: A comprehensive two-moment warm microphysical bulk scheme. I: Description and tests. Quart. J. Roy. Meteor. Soc., 126, 18151842, https://doi.org/10.1256/smsqj.56613.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delanoë, J. M. E., A. J. Heymsfield, A. Protat, A. Bansemer, and R. J. Hogan, 2014: Normalized particle size distribution for remote sensing application. J. Geophys. Res. Atmos., 119, 42044227, https://doi.org/10.1002/2013JD020700.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delene, D. and M. R Poellot, 2012: GPM Ground Validation UND Citation Cloud Microphysics GCPEX Version 1. NASA Global Hydrology Resource Center DAAC, accessed 1 October 2016, https://doi.org/10.5067/GPMGV/GCPEX/MULTIPLE/DATA202.

    • Crossref
    • Export Citation
  • Ferrier, B. S., 1994: A double-moment multiple-phase four-class bulk ice scheme. Part I: Description. J. Atmos. Sci., 51, 249280, https://doi.org/10.1175/1520-0469(1994)051<0249:ADMMPF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Field, P. R., A. J. Heymsfield, and A. Bansemer, 2006: Shattering and particle interarrival times measured by optical array probes in ice clouds. J. Atmos. Oceanic Technol., 23, 13571371, https://doi.org/10.1175/JTECH1922.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Field, P. R., A. J. Heymsfield, and A. Bansemer, 2007: Snow size distribution parameterization for midlatitude and tropical ice clouds. J. Atmos. Sci., 64, 43464365, https://doi.org/10.1175/2007JAS2344.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Finlon, J. A., G. M. McFarquhar, S. W. Nesbitt, R. M. Rauber, H. Morrison, W. Wu, and P. Zhang, 2019: A novel approach to characterize the variability in mass–dimension relationships: Results from MC3E. Atmos. Chem. Phys., 19, 36213643, https://doi.org/10.5194/acp-19-3621-2019.

    • Search Google Scholar
    • Export Citation
  • Grecu, M., L. Tian, W. Olson, and S. Tanelli, 2011: A robust dual-frequency radar profiling algorithm. J. Appl. Meteor. Climatol., 50, 15431557, https://doi.org/10.1175/2011JAMC2655.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haddad, Z. S., S. L. Durden, and E. Im, 1996: Parameterizing the raindrop size distribution. J. Appl. Meteor., 35, 313, https://doi.org/10.1175/1520-0450(1996)035<0003:PTRSD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., 2003: Properties of tropical and midlatitude ice cloud particle ensembles. Part I: Median mass diameters and terminal velocities. J. Atmos. Sci., 60, 25732591, https://doi.org/10.1175/1520-0469(2003)060<2573:POTAMI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., and J. L. Parrish, 1978: A computational technique for increasing the effective sampling volume of the PMS two-dimensional particle size spectrometer. J. Appl. Meteor., 17, 15661572, https://doi.org/10.1175/1520-0450(1978)017<1566:ACTFIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., P. Field, and A. Bansemer, 2008: Exponential size distributions for snow. J. Atmos. Sci., 65, 40174031, https://doi.org/10.1175/2008JAS2583.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., C. Schmitt, A. Bansemer, and C. H. Twohy, 2010: Improved representation of ice particle masses based on observations in natural clouds. J. Atmos. Sci., 67, 33033318, https://doi.org/10.1175/2010JAS3507.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., and C. D. Westbrook, 2014: Equation for the microwave backscatter cross section of aggregate snowflakes using the self-similar Rayleigh–Gans approximation. J. Atmos. Sci., 71, 32923301, https://doi.org/10.1175/JAS-D-13-0347.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., M. P. Mittermaier, and A. J. Illingworth, 2006: The retrieval of ice water content from radar reflectivity factor and temperature and its use in evaluating a mesoscale model. J. Appl. Meteor. Climatol., 45, 301317, https://doi.org/10.1175/JAM2340.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., L. Tian, P. R. A. Brown, C. D. Westbrook, A. J. Heymsfield, and J. D. Eastment, 2012: Radar scattering from ice aggregates using the horizontally aligned oblate spheroid approximation. J. Appl. Meteor. Climatol., 51, 655671, https://doi.org/10.1175/JAMC-D-11-074.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hong, S., J. Dudhia, and S. Chen, 2004: A revised approach to ice microphysical processes for the bulk parameterization of clouds and precipitation. Mon. Wea. Rev., 132, 103120, https://doi.org/10.1175/1520-0493(2004)132<0103:ARATIM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hou, A., and Coauthors, 2014: The Global Precipitation Measurement Mission. Bull. Amer. Meteor. Soc., 95, 701722, https://doi.org/10.1175/BAMS-D-13-00164.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hudak, D., 2013: GPM ground validation dual polarized C-band Doppler radar King City GCPEx. NASA EOSDIS Global Hydrology Resource Center DAAC, accessed 1 November 2016, https://doi.org/10.5067/GPMGV/GCPEX/MUTIPLE/DATA201.

    • Crossref
    • Export Citation
  • Jackson, R. C., and Coauthors, 2012: The dependence of ice microphysics on aerosol concentration in arctic mixed-phase stratus clouds during ISDAC and M-PACE. J. Geophys. Res., 117, D15207, https://doi.org/10.1029/2012JD017668.

    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., and P. R. Field, 2015: Assessment of the performance of the inter-arrival time algorithm to identify ice shattering artifacts in cloud particle probe measurements. Atmos. Meas. Tech., 8, 761777, https://doi.org/10.5194/amt-8-761-2015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., J. W. Strapp, G. A. Isaac, and A. N. Nevzorov, 1998: The Nevzorov airborne hot-wire LWC–TWC probe: Principle of operation and performance characteristics. J. Atmos. Oceanic Technol., 15, 14951510, https://doi.org/10.1175/1520-0426(1998)015<1495:TNAHWL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., E. F. Emery, J. W. Strapp, S. G. Cober, and G. A. Isaac, 2013: Quantification of the effects of shattering on airborne ice particle measurements. J. Atmos. Oceanic Technol., 30, 25272553, https://doi.org/10.1175/JTECH-D-13-00115.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kumar, L. S., Y. H. Lee, and J. T. Ong, 2011: Two-parameter gamma drop size distribution models for Singapore. IEEE Trans. Geosci. Remote Sens., 49, 33713380, https://doi.org/10.1109/TGRS.2011.2124464.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kummerow, C., W. Barnes, T. Kozu, J. Shiue, and J. Simpson, 1998: The Tropical Rainfall Measuring Mission (TRMM). J. Atmos. Oceanic Technol., 15, 809817, https://doi.org/10.1175/1520-0426(1998)015<0809:TTRMMT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lang, S. E., W. Tao, J. Chern, D. Wu, and X. Li, 2014: Benefits of a fourth ice class in the simulated radar reflectivities of convective systems using a bulk microphysics scheme. J. Atmos. Sci., 71, 35833612, https://doi.org/10.1175/JAS-D-13-0330.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • L’Ecuyer, T., W. Petersen, and D. Moiseev, 2010: Light Precipitation Validation Experiment (LPVEx). National Aeronautics and Space Administration Doc., 28 pp., https://pmm.nasa.gov/sites/default/files/document_files/lpvex_science_plan_Jan29_2010.pdf.

  • Liao, L., and R. Meneghini, 2005: A study of air/space-borne dual-wavelength radar for estimation of rain profiles. Adv. Atmos. Sci., 22, 841851, https://doi.org/10.1007/BF02918684.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liao, L., R. Meneghini, and A. Tokay, 2014: Uncertainties of GPM DPR rain estimates caused by DSD parameterizations. J. Appl. Meteor., 53, 25242537, https://doi.org/10.1175/JAMC-D-14-0003.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lin, Y., R. D. Farley, and H. D. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Climate Appl. Meteor., 22, 10651092, https://doi.org/10.1175/1520-0450(1983)022<1065:BPOTSF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mardiana, R., T. Iguchi, and N. Takahashi, 2004: A dual-frequency rain profiling method without the use of a surface reference technique. IEEE Trans. Geosci. Remote Sens., 42, 22142225, https://doi.org/10.1109/TGRS.2004.834647.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marks, D. A., D. B. Wolff, L. D. Carey, and A. Tokay, 2011: Quality control and calibration of the dual-polarization radar at Kwajalein, RMI. J. Atmos. Oceanic Technol., 28, 181196, https://doi.org/10.1175/2010JTECHA1462.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., M. S. Timlin, R. M. Rauber, B. F. Jewett, J. A. Grim, and D. P. Jorgensen, 2007: Vertical variability of cloud hydrometeors in the stratiform region of mesoscale convective systems and bow echoes. Mon. Wea. Rev., 135, 34053428, https://doi.org/10.1175/MWR3444.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., J. Um, and R. Jackson, 2013: Small cloud particle shapes in mixed-phase clouds. J. Appl. Meteor. Climatol., 52, 12771293, https://doi.org/10.1175/JAMC-D-12-0114.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., T. Hsieh, M. Freer, J. Mascio, and B. F. Jewett, 2015: The characterization of ice hydrometeor gamma size distributions as volumes in N0λ-μ phase space: Implications for microphysical process modeling. J. Atmos. Sci., 72, 892909, https://doi.org/10.1175/JAS-D-14-0011.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., and Coauthors, 2017: Processing of ice cloud in situ data collected by bulk water, scattering, and imaging probes: Fundamentals, uncertainties, and efforts toward consistency. Ice Formation and Evolution in Clouds and Precipitation: Measurement and Modeling Challenges, Meteor. Monogr., No. 58, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-16-0007.1.

    • Crossref
    • Export Citation
  • Meneghini, R., T. Kozu, H. Kumagai, and W. C. Boncyk, 1992: A study of rain estimation methods from space using dual-wavelength radar measurements at near-nadir incidence over ocean. J. Atmos. Oceanic Technol., 9, 364382, https://doi.org/10.1175/1520-0426(1992)009<0364:ASOREM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meyers, M. P., R. L. Walko, J. Y. Harrington, and W. R. Cotton, 1997: New RAMS cloud microphysics. Part II: The two-moment scheme. Atmos. Res., 45, 339, https://doi.org/10.1016/S0169-8095(97)00018-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Milbrandt, J. A., and M. K. Yau, 2005: A multimoment bulk microphysics parameterization. Part I: Analysis of the role of the spectral shape parameter. J. Atmos. Sci., 62, 30513064, https://doi.org/10.1175/JAS3534.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., 1996: Use of mass- and area-dimensional power laws for determining precipitation particle terminal velocities. J. Atmos. Sci., 53, 17101723, https://doi.org/10.1175/1520-0469(1996)053<1710:UOMAAD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., R. Zhang, and R. L. Pitter, 1990: Mass-dimensional relationships for ice particles and the influence of riming on snowfall rates. J. Appl. Meteor., 29, 153163, https://doi.org/10.1175/1520-0450(1990)029<0153:MDRFIP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moisseev, D. N., and V. Chandrasekar, 2007: Examination of the μλ relation suggested for drop size distribution parameter. J. Atmos. Oceanic Technol., 24, 847855, https://doi.org/10.1175/JTECH2010.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., and J. A. Milbrandt, 2015: Parameterization of cloud microphysics based on the prediction of bulk ice particle properties. Part I: Scheme description and idealized tests. J. Atmos. Sci., 72, 287311, https://doi.org/10.1175/JAS-D-14-0065.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., G. Thompson, and V. Tatarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Mon. Wea. Rev., 137, 9911007, https://doi.org/10.1175/2008MWR2556.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Munchak, S. J., and A. Tokay, 2008: Retrieval of raindrop size distribution from simulated dual-frequency radar measurements. J. Appl. Meteor. Climatol., 47, 223239, https://doi.org/10.1175/2007JAMC1524.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Naeger, A. R., B. A. Colle, and A. Molthan, 2017: Evaluation of cloud microphysical schemes for a warm frontal snowband during the GPM Cold Season Precipitation Experiment (GCPEx). Mon. Wea. Rev., 145, 46274650, https://doi.org/10.1175/MWR-D-17-0081.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Petersen, W., and M. Schwaller, 2008: Global precipitation mission (GPM) ground validation science implementation plan. National Aeronautics and Space Administration Doc., 41 pp., https://pmm.nasa.gov/sites/default/files/document_files/GPM_GVS_imp_plan_Jul08.pdf.

  • Seto, S., T. Iguchi, and T. Oki, 2013: The basic performance of a precipitation retrieval algorithm for the global precipitation measurement mission’s single/dual-frequency radar measurements. IEEE Trans. Geosci. Remote Sens., 51, 52395251, https://doi.org/10.1109/TGRS.2012.2231686.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Skofronick-Jackson, G., and Coauthors, 2015: Global Precipitation Measurement Cold Season Precipitation Experiment (GCPEX): For measurement’s sake, let it snow. Bull. Amer. Meteor. Soc., 96, 17191741, https://doi.org/10.1175/BAMS-D-13-00262.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Skofronick-Jackson, G., and Coauthors, 2017: The Global Precipitation Measurement (GPM) Mission for science and society. Bull. Amer. Meteor. Soc., 98, 16791695, https://doi.org/10.1175/BAMS-D-15-00306.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tang, G., Y. Wen, J. Gao, D. Long, Y. Ma, W. Wan, and Y. Hong, 2017: Similarities and differences between three coexisting spaceborne radars in global rainfall and snowfall estimation. Water Resour. Res., 53, 38353853, https://doi.org/10.1002/2016WR019961.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Testud, J., S. Oury, R. A. Black, P. Amayenc, and X. Dou, 2001: The concept of “normalized” distribution to describe the raindrop spectra: A tool for cloud physics and remote sensing. J. Appl. Meteor., 40, 11181140, https://doi.org/10.1175/1520-0450(2001)040<1118:TCONDT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Wea. Rev., 136, 50955115, https://doi.org/10.1175/2008MWR2387.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Toyoshima, K., H. Masunaga, and F. A. Furuzawa, 2015: Early evaluation of Ku- and Ka-band sensitivities for the Global Precipitation Measurement (GPM) Dual-Frequency Precipitation Radar (DPR). SOLA, 11, 1417, https://doi.org/10.2151/sola.2015-004.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ulbrich, C., 1983: Natural variations in the analytical form of the raindrop size distribution. J. Climate Appl. Meteor., 22, 17641775, https://doi.org/10.1175/1520-0450(1983)022<1764:NVITAF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Williams, C. R., and Coauthors, 2014: Describing the shape of raindrop size distributions using uncorrelated raindrop mass spectrum parameters. J. Appl. Meteor. Climatol., 53, 12821296, https://doi.org/10.1175/JAMC-D-13-076.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wu, W., and G. M. McFarquhar, 2018: Statistical theory on the functional form of cloud particle size distributions. J. Atmos. Sci., 75, 28012814, https://doi.org/10.1175/JAS-D-17-0164.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, G., J. Vivekanandan, and E. A. Brandes, 2001: A method for estimating rain rate and drop size distribution from polarimetric radar measurements. IEEE Trans. Geosci. Remote Sens., 39, 830841, https://doi.org/10.1109/36.917906.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, G., J. Vivekanandan, E. A. Brandes, R. Meneghini, and R. Kozu, 2003: The shape-slope relation in observed gamma raindrop size distributions: Statistical error or useful information? J. Atmos. Oceanic Technol., 20, 11061119, https://doi.org/10.1175/1520-0426(2003)020<1106:TSRIOG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, G., J. Sun, and E. A. Brandes, 2006: Improving parameterization of rain microphysics with disdrometer and radar observations. J. Atmos. Sci., 63, 12731290, https://doi.org/10.1175/JAS3680.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • View in gallery

    PDF of (left) number concentration, (center) height, and (right) temperature for different temperatures ranging from below −30°C to −2°C, increasing every 10°C, for all of the observations collected by the UND Citation (red) and for the ice-phase-only cases analyzed in this work (blue). Distributions for temperature for which the absence of liquid water could not be verified (i.e.: T > −2°C) are not shown.

  • View in gallery

    As in Fig. 1, but for (left) Dm, (center) σm, and (right) μ.

  • View in gallery

    Joint frequency of occurrence of (a) σm and (b) μ as a function Dm for the entire ice-phase dataset. (c),(d) As in (a) and (b), but using the melted equivalent diameter. The statistically independent relationship between the different variables for ice and snow cases from GCPEx in situ observations (red line) and for rainfall from Williams et al. (2014) (black line) and their equations are also shown.

  • View in gallery

    As in Fig. 3, but for GPM-detectable particles (Ze ≥ 12 dBZ).

  • View in gallery

    (a) Joint frequency of occurrence of Ze as a function of Dm, and (b) median μ as a function of Ze and Dm. The minimum detectable signal for the DPR on board the GPM (12 dBZ) is shown with a dashed red line.

  • View in gallery

    Scatterplot of (a) σm and (b) μ as a function of Dm, with the colors representing the temperature at which the observations were made.

  • View in gallery

    Normalized joint frequency of occurrence of the ratio of IWC between observations from the Nevzorov probe on board the UND Citation and that estimated from different techniques, as a function of Dm, showing the difference with respect to estimations from (a) the observed PSD, (b) the Field et al. (2007) method, (c) the Williams et al. (2014) method, and (d) IGF estimates.

  • View in gallery

    As in Fig. 7, but for particles in the GPM detectability range (i.e., Ze ≥ 12 dBZ).

  • View in gallery

    As in Fig. 7, but for Ze. The observed Ze corresponds to the matched effective reflectivity from the King City C-band radar along the path of the UND Citation.

  • View in gallery

    As in Fig. 9, but for particles in the GPM detectability range (i.e., Ze ≥ 12 dBZ).

  • View in gallery

    Influence of different a and b parameters, via the mass–diameter relation, in (a) Dm, (b) σm, (c) μ, (d) IWC, and (e) Ze. This effect is quantified by estimating the mass for different a and b and calculating the associated moments and variables normalized by the ones using ao and bo. Therefore, a value of 2 or 0.5 indicates that, for the corresponding a and b value, there is a 100% positive or 50% negative bias, respectively, in the estimated variable. The a and b values used in this work (Heymsfield et al. 2010) are indicated with the black lines.

  • View in gallery

    Normalized joint frequency of occurrence of the ratio between the estimated Ze with varying μ and Ze estimated with (a) μ = 0, (b) μ =3, (c) μ =10, and (d) μ(Dm)=5.098×Dm0.414 as a function of Dm. Frequencies larger than 50% are enclosed by the solid black line, and a zero difference between the estimated reflectivities is marked with a dotted red line.

  • View in gallery

    As in Fig. 12, but for particles in the GPM detectability range (i.e., Ze ≥ 12 dBZ) so that the relationship used in (d) is μ(Dm)=4.49×Dm0.254 [Eq. (11b)].

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 143 132 8
PDF Downloads 160 156 16

Improved Parameterization of Ice Particle Size Distributions Using Uncorrelated Mass Spectrum Parameters: Results from GCPEx

View More View Less
  • 1 Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois
  • | 2 NOAA/Climate Prediction Center, College Park, Maryland
  • | 3 Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois
  • | 4 Cooperative Institute of Mesoscale Meteorological Studies and School of Meteorology, University of Oklahoma, Norman, Oklahoma
© Get Permissions
Full access

Abstract

Satellite retrieval algorithms and model microphysical parameterizations require guidance from observations to improve the representation of ice-phase microphysical quantities and processes. Here, a parameterization for ice-phase particle size distributions (PSDs) is developed using in situ measurements of cloud microphysical properties collected during the Global Precipitation Measurement (GPM) Cold-Season Precipitation Experiment (GCPEx). This parameterization takes advantage of the relation between the gamma-shape parameter μ and the mass-weighted mean diameter Dm of the ice-phase PSD sampled during GCPEx. The retrieval of effective reflectivity Ze and ice water content (IWC) from the reconstructed PSD using the μDm relationship was tested with independent measurements of Ze and IWC and overall leads to a mean error of 8% in both variables. This represents an improvement when compared with errors using the Field et al. parameterization of 10% in IWC and 37% in Ze. Current radar precipitation retrieval algorithms from GPM assume that the PSD follows a gamma distribution with μ = 3. This assumption leads to a mean overestimation of 5% in the retrieved Ze, whereas applying the μDm relationship found here reduces this bias to an overestimation of less than 1%. Proper selection of the a and b coefficients in the mass–dimension relationship is also of crucial importance for retrievals. An inappropriate selection of a and b, even from values observed in previous studies in similar environments and cloud types, can lead to more than 100% bias in IWC and Ze for the ice-phase particles analyzed here.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Paloma Borque, paloma@illinois.edu

This article is included in the Global Precipitation Measurement (GPM) special collection.

Abstract

Satellite retrieval algorithms and model microphysical parameterizations require guidance from observations to improve the representation of ice-phase microphysical quantities and processes. Here, a parameterization for ice-phase particle size distributions (PSDs) is developed using in situ measurements of cloud microphysical properties collected during the Global Precipitation Measurement (GPM) Cold-Season Precipitation Experiment (GCPEx). This parameterization takes advantage of the relation between the gamma-shape parameter μ and the mass-weighted mean diameter Dm of the ice-phase PSD sampled during GCPEx. The retrieval of effective reflectivity Ze and ice water content (IWC) from the reconstructed PSD using the μDm relationship was tested with independent measurements of Ze and IWC and overall leads to a mean error of 8% in both variables. This represents an improvement when compared with errors using the Field et al. parameterization of 10% in IWC and 37% in Ze. Current radar precipitation retrieval algorithms from GPM assume that the PSD follows a gamma distribution with μ = 3. This assumption leads to a mean overestimation of 5% in the retrieved Ze, whereas applying the μDm relationship found here reduces this bias to an overestimation of less than 1%. Proper selection of the a and b coefficients in the mass–dimension relationship is also of crucial importance for retrievals. An inappropriate selection of a and b, even from values observed in previous studies in similar environments and cloud types, can lead to more than 100% bias in IWC and Ze for the ice-phase particles analyzed here.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Paloma Borque, paloma@illinois.edu

This article is included in the Global Precipitation Measurement (GPM) special collection.

1. Introduction

Precipitation is a key component of Earth’s water and energy cycles with direct consequences for life on Earth. The understanding of precipitation processes has a variety of implications on local and global scales including the accurate quantification of freshwater supply and the monitoring of extreme precipitation events. Models and observations of ice-phase precipitation, used to address these challenges, require assumptions of the characteristics of the particle size distribution (PSD) to perform accurate simulations and retrievals. In this work, a new parameterization for the ice-phase PSD based on aircraft measurements collected during the Global Precipitation Measurement (GPM) Cold-Season Precipitation Experiment (GCPEx) is developed. This new parameterization leverages the uncorrelated mass parameter PSD estimation technique developed by Williams et al. (2014), as described below to eliminate a free parameter in assumed PSD characteristics, which can provide reduced uncertainties in ice-phase microphysical models and retrievals compared with currently used schema.

One key application of an ice-phase PSD parameterization is for spaceborne active and passive ice cloud retrievals. The GPM (Hou et al. 2014; Skofronick-Jackson et al. 2017) Core Observatory satellite was launched on 28 February 2014 by the National Aeronautics and Space Administration (NASA) and Japanese Aerospace Exploration Agency as a follow-up to the successful Tropical Rainfall Measuring Mission (TRMM; Kummerow et al. 1998). The dual-frequency precipitation radar (DPR) aboard the GPM Core Observatory satellite consists of a Ku-band radar operating at 13.6 GHz with a minimum detectable equivalent reflectivity factor of 12–14 dBZ (or 0.5 mm h−1) and a Ka-band radar operating at 35.5 GHz with a minimum detectable reflectivity of 12 dBZ (or 0.2 mm h−1; Toyoshima et al. 2015). The increased sensitivity and the addition of the Ka-band frequency represents a new capability of GPM over TRMM to measure snowfall and light precipitation from a space-based perspective. Further, the sample of precipitation with both DPR frequencies in an overlapping swath can lead to quantitative estimates of PSD characteristics that can be used to improve the associated GPM retrieval algorithms (Hou et al. 2014). This can be achieved by a better determination of the PSD-derived physical quantities such as ice water content (IWC) and effective reflectivity Ze.

In current GPM algorithms, PSDs are assumed to follow gamma distributions with three unknown mathematical parameters: the intercept N0, the shape μ, and the slope λ (e.g., Ulbrich 1983; Seto et al. 2013; Williams et al. 2014; Wu and McFarquhar 2018). Given two independent measurements and an assumption or a constraint on the third parameter, the PSD can be retrieved. There has been extensive work on the constraint of one of the three gamma parameters both from a ground-based radar perspective (Zhang et al. 2001, 2003, 2006; Moisseev and Chandrasekar 2007; Bringi and Chandrasekar 2001; Delanoë et al. 2014) and as part of satellite retrieval algorithms (Seto et al. 2013; Williams et al. 2014; Grecu et al. 2011; Liao et al. 2014). Currently, the DPR retrieval algorithm provides two measurements [Ze and dual-frequency ratio (DFR)] and assumes a constant value of μ = 3 (Seto et al. 2013). Recent studies show success in reducing biases in the algorithms by estimating uncorrelated mass parameters in rainfall events (Williams et al. 2014; Liao et al. 2014). In particular, Williams et al. (2014) avoided mathematical artifacts by developing joint probability distribution functions of statistically independent PSD attributes for the raindrop mass spectrum.

Beyond satellite retrieval needs of parameterizing the PSD in ice-phase clouds, multiscale numerical models need a priori assumptions of ice and snow hydrometeor species within bulk and bin microphysical parameterizations. At this time, most single-moment PSD schemes (e.g., Lin et al. 1983; Hong et al. 2004; Thompson et al. 2008; Lang et al. 2014) prescribe PSD characteristics with a fixed intercept and μ parameter (the latter value typically prescribed to be μ = 0, yielding an exponential size distribution), with a slope parameter that is a function of predicted hydrometeor mixing ratio. In most two-moment schemes, μ is usually assumed to be constant (e.g., Ferrier 1994; Meyers et al. 1997; Cohard and Pinty 2000; Morrison et al. 2009). However, others, such as the predicted particle properties (P3) scheme, use variable μ for ice particles but follow in situ observations of Heymsfield (2003) for tropical and midlatitude ice cloud particles, with μ limited to positive values below 6 (Morrison and Milbrandt 2015). Bin microphysics schemes have been developed more recently in an attempt to more accurately represent PSD characteristics in model simulations. However, the behavior of these schemes in ice-phase precipitation has only rarely been quantitatively evaluated (e.g., Naeger et al. 2017; Milbrandt and Yau 2005).

It is clear that satellite retrievals of ice-phase particles and model microphysical parameterizations are in need of guidance from in situ observations as a potential way forward to improve the representation of ice-phase microphysical processes. In this study, a catalog of in situ measurements collected during an intensive cold-season precipitation field campaign is used to evaluate the uncorrelated mass parameters approach to estimate ice-phase PSDs. Furthermore, a new PSD parameterization is introduced aiming to improve the consistency of ice-phase precipitation retrievals.

2. Data

GPM–Ground Validation (GV) campaigns collected precipitation measurements over the course of multiple field campaigns in an effort to mitigate the uncertainty in GPM retrieval algorithm assumptions, especially in precipitation regimes that were previously challenging to retrieval algorithms (Petersen and Schwaller 2008). Comparisons of precipitation products from CloudSat, Aqua, TRMM, and GPM show a clear disagreement in precipitation amounts at latitudes poleward of 30°. These discrepancies are most likely a result of different sensitivities of the sensors [CloudSat Cloud-Profiling Radar: −27 dBZ, TRMM Precipitation Radar (PR): 18 dBZ, and GPM DPR: 12 dBZ) in conjunction with the majority of precipitation occurring in the form of snowfall and of light rainfall with shallow freezing levels and the inability of those satellite precipitation sensors to accurately detect and estimate precipitation in the regimes found there (L’Ecuyer et al. 2010; Tang et al. 2017; Casella et al. 2017). Differences among the various retrieval algorithms that determine the phase of precipitation could also play a role in these differences.

a. Aircraft in situ data

One of the overarching goals of GPM-GV is to characterize the ability of multifrequency active and passive microwave sensors to detect and estimate ice-phase precipitation rate. This effort entails numerous GPM-specific and joint-agency field campaigns with state of the art cloud and precipitation observational capabilities (e.g., polarimetric radars, wind profiler radars, rain gauges, disdrometers, and aircraft in situ observations). One such field campaign was GCPEx conducted in cooperation with Environment and Climate Change Canada (ECCC) and NASA in January–February of 2012 near Barrie, Ontario, Canada (Skofronick-Jackson et al. 2015). GCPEx collected in situ cloud microphysics and thermodynamic measurements using the University of North Dakota (UND) Cessna Citation II research aircraft (Delene and Poellot 2012). Aboard the Citation, the King probe was used to quantify the liquid water content (LWC), the Nevzorov probe (Korolev et al. 1998) was used to measure the LWC and total water content (TWC), and the Rosemount Icing Detector (RICE) was used to detect the presence of supercooled liquid water (Brown 1982). Baumgardner et al. (2017) and references therein describe the different in situ probes and measurements that are used to characterize the microphysical properties of ice clouds.

Two-dimensional cloud probe data from the UND Citation were processed by the System for Optical Array Probe Data Analysis (SODA) software developed by the National Center for Atmospheric Research. PSDs were obtained from a combination of observations from the Cloud Imaging Probe (CIP) and the High Volume Precipitation Spectrometer, version 3 (HVPS3). The CIP is an airborne particle spectrometer that nominally measures the size and shape of cloud particles from 25 to 1550 μm, and the HVPS3 measures the size and shape of cloud particles from about 150 μm to 19.2 mm. Following Heymsfield et al. (2008), merged PSDs were constructed with the CIP data for particles with maximum dimension D smaller than 1 mm and with the HVPS3 data for particles with D > 1 mm. Particles with D < 100 μm were ignored because of uncertainties in the probe’s sample area (e.g., Korolev and Field 2015; Field et al. 2006). Particle reconstruction for particles whose geometrical center was in the photodiode array was used to expand the sampling range of both the CIP and HVPS3 following the method described in Heymsfield and Parrish (1978). Antishattering tips and the algorithms described in McFarquhar et al. (2017) were used to remove shattered artifacts from the sampled PSDs. The use of the HVPS3, capable of measuring large aggregates, and the handling of shattered artifacts through improved probe design and software quality control diminished the source of error in the sampled particles and allows for an improved characterization of the PSD in comparison with previous studies (e.g., Field et al. 2007).

During 12 intensive observation periods (IOPs) in GCPEx, the UND Citation sampled a variety of winter precipitation systems, with the majority representing synoptically forced snowfall over southern Ontario in Canada. This study seeks to characterize the PSDs in exclusively ice-phase conditions. To achieve this, a technique has been implemented to only include data when the research aircraft sampled regions with purely ice particles, namely excluding times when liquid or mixed-phase conditions were present. Thus, in situ observations were only considered if they met the following three criteria: temperature T < −2°C, total particle total number concentration > 103 m−3, and no changes in the voltage measured by the RICE above or below one standard deviation of the median voltage of the flight leg. Although other studies apply more sophisticated phase detection schemes to delineate ice, mixed phase, and liquid clouds (e.g., Cober et al. 2001; McFarquhar et al. 2007; Jackson et al. 2012), the approach used here proved to be sufficient to eliminate cases where liquid was present in the sampled volume. Visual inspection of the high-resolution Cloud Particle Imager (CPI) imagery confirmed that periods in which liquid drops large enough to be observed by the CPI were present are well correlated with period of variations in the observed RICE output, thus confirming that the aforementioned exclusion criteria is capable of detecting the presence of supercooled liquid water particles larger than approximately 35 μm (McFarquhar et al. 2013). Furthermore, the RICE criterion is a very conservative condition and it could filter out time periods in which the presence of super cooled liquid drops is not visible in CPI imagery, or when the RICE probe is going in the reheating cycle, provided the temperature was less than about −2°C. However, it will achieve its purpose of retaining mostly periods of only ice-phase particles in the sample volume.

To describe the environmental and PSD characteristics sampled in GCPEx as well as those that were subselected in only ice-phase conditions, Fig. 1 shows probability distribution functions (PDFs) of observed quantities for all flights. Observations were collected from about 500 m AGL to heights below 8 km for temperatures ranging from slightly above 0°C to below −40°C. Height distributions follow the expected inverse relation with temperature, and none of these variables show a clear preferred value as a function of specific temperature ranges (Fig. 1). It is important to note that restricting the analysis to ice-phase conditions implies that the TWC measured by the Nevzorov probe should be identical to the IWC.

Fig. 1.
Fig. 1.

PDF of (left) number concentration, (center) height, and (right) temperature for different temperatures ranging from below −30°C to −2°C, increasing every 10°C, for all of the observations collected by the UND Citation (red) and for the ice-phase-only cases analyzed in this work (blue). Distributions for temperature for which the absence of liquid water could not be verified (i.e.: T > −2°C) are not shown.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

b. Radar measurements

Ground-based radar observations were also an integral part of GCPEx; such observations are of fundamental importance to link the in situ measured particle properties to remotely sensed bulk characteristics. As part of GCPEx, ECCC’s dual-polarization scanning C-band radar located in King City, Ontario, Canada, performed (in addition to their operational scanning) range–height indicator sector scans and plan position indicator scans (Hudak 2013). For some IOPs, the UND Citation flew in the coverage region of the King City radar, allowing for more than 3700 coincident (as defined in section 2c) 10-s averaged measurements between ground-based and in situ instrumentation. Following Marks et al. (2011), radar reflectivity data were quality controlled by removing points with copolar correlation coefficient ρHV < 0.75 and range standard deviation of differential phase ΦDP over a gate-centered window of 1.5 km > 12°.

c. In situ and ground-based data matching

The volumes sampled by probes onboard the UND Citation and by the King City radar are not coincident. Therefore, it is necessary to determine what temporal and spatial separation can be tolerated to assume the in situ measurements as linked to a corresponding radar volume. In the present study, the volumes sampled by the King City Radar and the UND Citation were defined as collocated in time when the Citation was sampling within the start and end time of each 10-min volume coverage pattern. Once a collocated volume was defined a spatial matching algorithm between the radar volume and Citation observations was used. Points were weighted by constructing a collocated reflectivity along the aircraft track averaged in linear units (mm6 m−3) using an eight-closest point Barnes (1964) interpolation technique as in Finlon et al. (2019). Reflectivities are weighted according to distance r from the aircraft using wi = exp(−r2/κ2), with weighting parameter κ given a value of 500 m. A sensitivity analysis of the radius of influence in the matching technique showed that varying κ over selected values from 100 m to 1.5 km for all GCPEx flights only slightly changed the bias and root-mean-square differences of the matched radar reflectivities relative to the value of κ used. Matched reflectivity values were biased on average by less than 0.3 dBZ with a mean root mean squared difference of 1.2 dBZ for all radii tested except for radii less than 250 m (which was the radial gate spacing of the raw radar data).

3. PSD reconstruction

a. Definitions of PSD parameters

Particle size distributions are often assumed to follow a gamma-shaped distribution given by
N(D)=N0DμeλD,
where N(D) is the number distribution function and D, N0, μ, and λ are as defined above (e.g., Ulbrich 1983; Seto et al. 2013; Williams et al. 2014). Therefore, three parameters (N0, μ, and λ) are needed to characterize a PSD when represented as a gamma function. A well-documented caveat with this PSD formulation is the statistical dependence among these parameters (e.g., Ulbrich 1983; Chandrasekar and Bringi 1987; Moisseev and Chandrasekar 2007; Williams et al. 2014; McFarquhar et al. 2015). Constraining one of the parameters, such as by introducing a μλ relationship, reduces the number of free parameters within the gamma PSD. Previous studies (e.g., Zhang et al. 2001, 2003) have shown that the application of such a relation between μ and λ can improve rain-rate estimations. However, others (e.g., Atlas and Ulbrich 2006; Moisseev and Chandrasekar 2007; Zhang et al. 2003; Cao and Zhang 2009) have debated whether this statistical dependence can be used to discern physical properties or if it is a statistical artifact due to the initial assumption of the gamma size distribution. For liquid particles, the DPR algorithm avoids such assumptions by calculating the PSD with a normalized gamma function (Testud et al. 2001) in terms of μ and two physical quantities: the normalized intercept Nw (which is a function of LWC) and the mass-weighted mean diameter Dm (Chandrasekar et al. 2005; Seto et al. 2013), where
N(D)=Nw6(μ+b+1)(μ+b+1)4(b+1)Γ(μ+b+1)(DDm)μe(μ+b+1)(D/Dm),
with
Dm=0mN(D)DdD0mN(D)dDand
Nw=4b+1πρw(LWCDmb+1).
Herein, a and b are the parameters of the power-law particle mass–dimension relation of the form m = aDb described in more detail in section 4. Delanoë et al. (2014) showed that by combining N0, Dm, and a modified gamma function the IWC could be estimated with an absolute mean relative error smaller than 20%. Current TRMM PR and DPR algorithms retrieve Nw and Dm given an assumption on μ (e.g., Meneghini et al. 1992; Zhang et al. 2001; Mardiana et al. 2004; Liao and Meneghini 2005; Munchak and Tokay 2008; Kumar et al. 2011; Grecu et al. 2011; Seto et al. 2013; Williams et al. 2014). In particular, current GPM version-5 algorithms assume μ to be constant and equal to 3 (Seto et al. 2013). However, the rain rates estimated from these retrieval algorithms are sensitive to μ, and small changes in its value lead to large changes in precipitation estimates. For example, Grecu et al. (2011) showed that changing μ from 0 to 1 led to 15% differences in the estimations of precipitation water content, whereas Liao et al. (2014) showed that larger μ values (3, 6, and 10) lead to a relative bias of rain-rate estimates of larger than 10%. Williams et al. (2014) proposed a method to parameterize gamma PSD characteristics by calculating a statistically independent μDm constraint for the μ parameter. Their results showed that the rain-rate bias never exceeded 3.5% when the statistically independent μDm constraint was applied. Our study expands the Williams et al. (2014) technique to ice particles, and thus, given the variable effective density of ice-phase particles, the coefficients of a and b in the mass–dimension relation must be considered. Therefore, expanding Eq. (3) leads to
Dm=0aDbN(D)DdD0aDbN(D)dD=0Db+1N(D)dD0DbN(D)dDDminDmaxDb+1N(D)ΔDDminDmaxDbN(D)ΔD,
and the mass spectrum standard deviation σm, which physically represents the width of the PSD mass spectrum, is given by
σm2=0mN(D)(DDm)2dD0mN(D)dD=0aDbN(D)(DDm)2dD0aDbN(D)dDDminDmaxDbN(D)(DDm)2ΔDDminDmaxDbN(D)ΔD,
where Dmin and Dmax are the minimum and maximum dimensions of measured ice particles, respectively, and ∆D is the width of the size bin.
Combining Eqs. (5) and (6) leads to
μ=(Dm2/σm2)(b+1).
Following Williams et al. (2014), a statistically independent parameter σm can be defined as
σm=σm/Dmbm,
where bm is adjusted until the Pearson correlation coefficient between σm and Dm is 0 (Haddad et al. 1996; Williams et al. 2014). This leads to the following statistically independent relationship for μ:
μ=[Dm2(1bm)/σm2](b+1).
The parameters of the mD relationship used in this work (a = 3.59 × 10−3 g cm−2.1 and b = 2.1) correspond to the values found by Heymsfield et al. (2010) for observations analyzed in a similar region to where GCPEx took place [Alliance Icing Research Study (AIRS) in Toronto, Ontario, Canada, and the Canadian CloudSat/Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) Validation Programme (C3VP) near Barrie].

Figure 2 shows PDFs of Dm, σm, and μ derived from the in situ observations for all of the flights that took place during GCPEx. The properties have different distributions depending on temperature (Fig. 2). There is a noticeable increase in Dm as temperature increases, from a median of around 0.25 mm for temperatures lower than −30°C to 1.5 mm in the interval from −2° to −10°C. The PSD becomes narrower as temperature decreases (with a normal distribution of σm centered at 0.8 mm for −2° < T < −10°C to σm close to zero for T < 30°C). The shape parameter μ is centered around zero for temperatures between −2° and −20°C but shifts toward larger values as temperature decreases such that for temperatures between −20° and −30°C it is centered at μ = 2. The width of the μ PDF also increases with decreasing temperature, from a narrow distribution at the highest temperature to a very broad distribution for temperatures below −30°C (Fig. 2). It is important to note that restricting the dataset to samples in which no liquid was present did not alter the main characteristics of the PDFs presented in Figs. 1 and 2.

Fig. 2.
Fig. 2.

As in Fig. 1, but for (left) Dm, (center) σm, and (right) μ.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

b. Characterization of Dm, σm, and μ distributions observed during GCPEx

The statistical relationships found for ice-phase particles between σm and Dm and between μ and Dm have a similar shape as for rainfall observations (Williams et al. 2014) but with different absolute values (Fig. 3). In particular, for small Dm, μ has a large spread of values ranging from −3 to greater than 10 (Fig. 3b).

Fig. 3.
Fig. 3.

Joint frequency of occurrence of (a) σm and (b) μ as a function Dm for the entire ice-phase dataset. (c),(d) As in (a) and (b), but using the melted equivalent diameter. The statistically independent relationship between the different variables for ice and snow cases from GCPEx in situ observations (red line) and for rainfall from Williams et al. (2014) (black line) and their equations are also shown.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

To have a consistent comparison between the results presented here and those for rainfall presented by Williams et al. (2014), the frequency of occurrence plots are redone for the PSD parameters determined using the melted equivalent diameter
Dmelted=[6m/(πρ)]1/3,
where ρ is the density of water. Figures 3c and 3d show the effect of replacing D by Dmelted in the calculations of Dm, σm, and μ (herein, Dmmelted, σmmelted, and μmelted). Similar to what was described for these parameters computed with the maximum dimension, the statistical relationships found for ice-phase particles between σmmelted and Dmmelted and between μmelted and Dmmelted have a similar shape as for rainfall observations (Williams et al. 2014) but with different absolute values (Figs. 3c,d). There is a high concentration of samples (more than 40%) for Dmmelted between 0.2 and 0.4 mm with μmelted close to 3. This is represented by a median μmelted of 3.3. However, it is important to note that there is a clear variability of μmelted with Dmmelted. In particular, μmelted ranges from negative values to larger than 25 for Dm < 0.2 mm, and it is likely to tend to 0 for Dm > 0.4 (Fig. 3d). This variability is also represented by the mean and standard deviation of μmelted being 5.3 and 13.3 respectively. Thus, it is important to consider this variability in the representation of μ, computed with either the maximum dimension or the melted equivalent diameter.

Figures 3 and 4 show that the previously found relationship between μ and Dm for rainfall cases (Williams et al. 2014) is not representative of ice-phase particles and that there is a clear necessity to use different relationships for retrieving particles distributions of different precipitation types. Assuming a constant μ represents a large uncertainty for any retrieval algorithm for ice-phase particles. However, when considering particles within the detectability range of the GPM DPR of Ze > 12 dBZ, this broad range of μ values decreases (Fig. 4b). The ice-phase PSDs observed during GCPEx that are within the detectability range of the GPM DPR (above the red line in Fig. 5) were not likely to have μ > 1. In particular, only 3% of the ice-phase PSDs observed during GCPEx within the detectability range of the GPM DPR have μ ≥ 3. Thus, the current GPM algorithm assumption of μ = 3 overestimates the vast majority of μ values observed during GCPEx. Given that a large value of μ indicates a large tail in the PSD, the current GPM algorithm assumes the presence of larger particles in the retrieved PSD than are present (Fig. 4b).

Fig. 4.
Fig. 4.

As in Fig. 3, but for GPM-detectable particles (Ze ≥ 12 dBZ).

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

Fig. 5.
Fig. 5.

(a) Joint frequency of occurrence of Ze as a function of Dm, and (b) median μ as a function of Ze and Dm. The minimum detectable signal for the DPR on board the GPM (12 dBZ) is shown with a dashed red line.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

The data collected during GCPEx show that the best statistical relationship, as determined by a polynomial curve fitting that is the best fit (in a least squares sense) between μ and Dm for ice-phase particles is given by
μ(Dm)=5.10mm0.41Dm0.414.
The best statistical relationship between μ and Dm for ice-phase particles in the detectability range of the GPM DPR algorithm is given by
μ(Dm)=4.49mm0.25Dm0.254.
The mean-mass diameter shows a positive relationship with temperature for both the μDm and σmDm distributions, implying that the temperature dependency of the shape and width of the distribution is likely to be implicitly included when considering the PSD dependency on Dm (Fig. 6).
Fig. 6.
Fig. 6.

Scatterplot of (a) σm and (b) μ as a function of Dm, with the colors representing the temperature at which the observations were made.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

c. Comparison of different PSD estimations

The validity of the framework described in the previous section for the representation of PSDs is tested with independent measurements of IWC and Ze. Reflectivity can be estimated from the PSD as
Ze=1018λ40.93π5DminDmaxn(D)σbΔD,
where λ is the wavelength; in this case, λ = 5.33 cm to compare the estimations with the King City C-Band radar, and 〈σb〉 is the ensemble-average backscatter cross section.
Following the self-similar Rayleigh–Gans approximation (Hogan and Westbrook 2014),
σb=9πk4|K|2V216{cos2(x)[(1+κ3)(12x+π12xπ)κ(12x+3π12x3π)]2+βj=1n(2j)γsin2(x)[1(2x+2πj)2+1(2x2πj)2]}
where k is the wavenumber, x = kD, and K = (ε − 1)/(ε + 2) with ε being the dielectric constant of solid ice; κ, β, and γ are the coefficients 0.19, 0.23, and 5/3 respectably, and V is the volume of the precipitating particle calculated assuming the particle shape can be represented by an ellipsoid of axis ration of 0.6 (Hogan et al. 2012) so that
V=aDbρi=a(0.6D)bρi.
Ice water content can be calculated as
IWC=0mass×N(D)dD=0aDbN0DμeλDdD=aN0Γb+μ+1λb+μ+1.
IWC comparisons are made between the observed values of TWC from the Nevzorov probe mounted on the UND Citation and estimates [following Eq. (15)] from in situ PSDs as sampled by the Citation, and from reconstructed PSDs using the moment-based method (following the method described in section 3), the Field et al. (2007) parameterization, and the lowest most likely estimate (i.e., lowest χ2 goodness of fit) using the incomplete gamma-fitting technique (IGF; McFarquhar et al. 2015). The IGF technique estimates a volume of equally realizable solutions in (N0, λ, μ) phase space that characterized every single PSD or a family of PSDs obtained in similar conditions. This is done by minimizing the χ2 difference between observed moments of the PSD (the first, third, and sixth moments of the PSD are used here) and those estimated from the fit parameters (McFarquhar et al. 2015). From the generated volume the pair of N0, λ that give the minimum χ2 for a specific fixed value of μ can be determined.

The estimated TWC from the IGF method is restricted to the minimum and maximum dimension of the sampled particles (Dmin = 75 μm and Dmax = 27.5 mm). On the other hand, PSD-based estimates are computed following the complete gamma function definition, and thus are integrating over all sizes from zero to infinity [Eq. (15)]. When comparing the difference between the incomplete (between Dmin and Dmax) and the complete gamma functions for the observed GCPEx PSDs, it can be shown that the contribution of the particles with a diameter smaller than Dmin and larger than Dmax to IWC is smaller than 0.2%. Therefore, the comparison shown here between the observed TWC and the different estimates from reconstructed PSDs following Eq. (15) is appropriate.

The retrieval of IWC from the sampled PSD and the different PSD reconstructions lead to similar estimates, with a mean underestimation of IWC < 10% with respect to the value sampled by the Nevzorov probe (Fig. 7 and Table 1). Considering observations within the detectability range of the GPM satellite (Ze ≥ 12 dBZ) the inaccuracy increases to 30% positive bias for IWC estimations from reconstructed PSDs (Table 1). Overall, the estimated IWCs following the IGF and moment-based methods are slightly closer to the observed value, by 2% and 4% for all ice-phase particle and GPM-detectable PSDs, respectively, than applying the Field et al. (2007) parameterization (Table 1). In particular, for Dm > 3 mm, all IWC estimations analyzed in this work are likely to overestimate the observed IWC when compared with the Nevzorov probe (Figs. 7 and 8). One possible reason for this is that the Nevzorov probe could be underestimating IWC during conditions containing PSDs with large particles (and thus large Dm) because of particle shattering and bounce out (Korolev et al. 2013). Another possible reason is the dependence of the mD relationship on particle habit, which implies that an incorrect selection of the a and b parameters could lead to an incorrect estimation of the particles’ mass and thus to an incorrect estimation of IWC that could be particularly important for large particles where large aggregates could be misrepresented.

Fig. 7.
Fig. 7.

Normalized joint frequency of occurrence of the ratio of IWC between observations from the Nevzorov probe on board the UND Citation and that estimated from different techniques, as a function of Dm, showing the difference with respect to estimations from (a) the observed PSD, (b) the Field et al. (2007) method, (c) the Williams et al. (2014) method, and (d) IGF estimates.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

Table 1.

Mean and median values of the ratio between TWC from the Nevzorov probe on board the UND Citation and the IWC estimated from the observed PSD (IWCobs), the Field et al. (2007) parameterization (IWCField), the Williams et al. (2014) method, (IWCmoment), and the IGF estimates (IWCIGF).

Table 1.
Fig. 8.
Fig. 8.

As in Fig. 7, but for particles in the GPM detectability range (i.e., Ze ≥ 12 dBZ).

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

Reflectivity estimated from the different PSDs [following Eq. (12)] are compared with those from the ground-based King City radar extracted along the path of the Citation following the matching criteria described in section 2c. For all sampled ice-phase particles, Fig. 9 and Table 2 show that the estimates of Ze based on the observed PSD, moment-based, and IGF methods have a mean and median positive bias around 8% with respect to the remotely sensed value. On the other hand, estimates using the Field et al. (2007) parameterization lead to a 37% mean and a 17% median overestimation of the remotely sensed reflectivity. All Ze estimates have a large spread for smaller values of Dm (Fig. 9) with some outliers representing differences larger than 10 dBZ. For Dm > 1 mm the estimation from the moment-based and IGF methodologies tend to the observed Ze (Fig. 9). Thus, considering only the observations within the detectability range of the GPM satellite, the median overestimation decreases to a 2% positive bias for estimations from the observed PSD and the reconstructed PSD following the moment-based and IGF methods (Fig. 10 and Table 2). However, applying the Field et al. (2007) parameterization leads to a mean (median) underestimation of the observed Ze of 78% (45%) for the observations with Ze ≥ 12 dBZ (Fig. 10b and Table 2).

Fig. 9.
Fig. 9.

As in Fig. 7, but for Ze. The observed Ze corresponds to the matched effective reflectivity from the King City C-band radar along the path of the UND Citation.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

Table 2.

As in Table 1,but for effective reflectivity. The observed Ze corresponds to the matched effective reflectivity from the King City C-band radar along the path of the UND Citation.

Table 2.
Fig. 10.
Fig. 10.

As in Fig. 9, but for particles in the GPM detectability range (i.e., Ze ≥ 12 dBZ).

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

Discrepancies when comparing reflectivity from a ground-based radar with estimates from in situ PSDs are expected. Similar to the errors found in the IWC estimates, the uncertainty in the mass–dimension relationship can contribute (this factor is analyzed in more detail in section 4); in the case of Ze, the different sample volumes and the poor sensitivity of a C-band radar to small particles can also have an impact.

4. Analysis of the mass–dimension relationship

Previous studies have parameterized the mass m of ice particles as a function of their maximum dimension D by m(D) = aDb, with a and b being empirically derived (e.g., Mitchell et al. 1990; Mitchell 1996; McFarquhar et al. 2007; Heymsfield et al. 2008, 2010; Finlon et al. 2019). There is considerable variation in published a and b values with the Brown and Francis (1995) relation, or its modified version for maximum dimension (Hogan et al. 2012), frequently used in algorithm development (e.g., Hogan et al. 2006; Heymsfield et al. 2008; Chase et al. 2018). However, fixed a and b values only characterize the average representation of a particle’s mass but fail to capture the dependence of these parameters on habit, temperature, and cloud type (Heymsfield et al. 2010). Thus, Heymsfield et al. (2010) defined four different mD relationships with a constant b of 2.1 but a depending on the environment. Specifically, one subset of the dataset they analyzed corresponds to observations from field campaigns in a similar region to where GCPEx took place and therefore, the mD relationship found for that dataset [m(D) = 3.59 × 10−3 cm−2.1 D2.1] is used in this work and as the reference values to the estimations computed in this section.

Any errors in the mD relationship propagate through the σm and Dm calculations and affect IWC and Ze estimates. Thus, it is valuable to look at the level of uncertainty that fixed a and b coefficients introduced by the mD relationship add to the analysis. Here, the uncertainty is given by the range of possible σm, Dm, μ, IWC, and Ze values estimated using the GCPEx PSDs while adjusting for possible a and b values following the observations shown in Heymsfield et al. (2010) for midlatitude stratiform-type clouds in the region similar to the observations analyzed here (a ranges from 3 × 10−3 g cm−2.1 to 3 × 10−2 g cm−2.1, and b ranges from 1.5 to 3.5). The a and b proposed in Brown and Francis (1995) of a = 2.94 × 10−3 g cm−2.1 and b = 1.9 are also analyzed. From Eqs. (5), (6), and (9), it is seen that Dm, σm, and μ do not depend on a but that changes in the exponent b affect the resulting Dm, σm, and μ. Therefore, assuming that the finding from Heymsfield et al. (2010) of b in the mD relationship is correct, there would be no additional error propagated through the Dm, σm, and μ calculations as a result of using different a assumptions (Figs. 11a–c). On the other hand, an unfitting selection of coefficient b leads to minor biases in μ, Dm and σm. Considering the variety of b coefficient present in the literature, applying a small value of b (b = 1.5) is likely to cause an underestimation of 12% for Dm and 9% for σm and an overestimation of 20% for μ (Figs. 11a–c). Likewise, applying a large value of b (b = 3) is likely to be associated with an overestimation of 19% for Dm and 20% for σm and an underestimation of 4% for μ (Figs. 11a–c).

Fig. 11.
Fig. 11.

Influence of different a and b parameters, via the mass–diameter relation, in (a) Dm, (b) σm, (c) μ, (d) IWC, and (e) Ze. This effect is quantified by estimating the mass for different a and b and calculating the associated moments and variables normalized by the ones using ao and bo. Therefore, a value of 2 or 0.5 indicates that, for the corresponding a and b value, there is a 100% positive or 50% negative bias, respectively, in the estimated variable. The a and b values used in this work (Heymsfield et al. 2010) are indicated with the black lines.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

The impact of a and b becomes significant when calculating different moments of the distribution, such as IWC and Ze, for which applying a large value of b (b = 3) is associated with a more than 50% overestimation of Ze and an underestimation of 3% for IWC (Figs. 11d,e) and applying a small b leads to an underestimation of 44% for Ze and an overestimation of almost 4 times the reference value of IWC (Figs. 11d,e). As previously mentioned, applying different a values in the mD relationship has no impact on Dm, σm, and μ but does affect the estimates of Ze and IWC. Applying a small value of a (a = 3 × 10−4 g cm−2.1) is likely to cause a slight underestimation of 9% for IWC but considerably decreases Ze, going from the estimated referenced value of 14 dBZ to a value of −7.6 dBZ (Figs. 11d,e). On the other hand, applying a large value of a (a = 3 × 10−2 g cm−2.1) leads to an overestimation of more than 8 times the referenced value of IWC and 2 times the referenced Ze values (Figs. 11d,e). In particular, when comparing the results of using the a and b coefficients from Heymsfield et al. (2010) versus the ones introduced by Brown and Francis (1995) there is less than a 5% discrepancy in Dm, σm, and μ, but as it propagates into the calculations for the different variables the error amplifies to 40% overestimation in IWC and 25% underestimation in Ze. It is important to note, however, that the covariability of a and b was not considered here (e.g., Finlon et al. 2019).

The large spread in the estimations of Ze and IWC shows the severe impact that the selection of a and b in the mass–dimension relationship, even from realistic values previously used in the literature, has in the calculations of these variables. This could represent a reason for the differences found when comparing estimates of Ze and IWC from the sampled and reconstructed PSDs with the independently observed values (Figs. 710).

5. PSD parameterization for ice-phase particles

Herein a parameterization of the PSD that improves the consistency of the retrievals for ice-phase precipitation is presented using GCPEx PSD observations. This is of fundamental use for the atmospheric modeling community and should help to guide the assumptions for retrievals development in double-moment microphysical schemes. Expanding Eq. (3) leads to
Dm=0mN(D)DdD0mN(D)dD=0aN0Db+μ+1eλDdD0aN0Db+μeλDdD=Γ(b+1+μ+1)λb+1+μ+1λb+μ+1Γ(b+μ+1)=b+μ+1λ,
which can be rearranged to obtain
λ=(b+μ+1)/Dm.
From Eq. (15) it follows that
N0=IWCλb+μ+1aΓb+μ+1.

Therefore, from the best fit between μ and Dm for ice-phase particles found in this work [Eq. (11)] and Eqs. (17) and (18), the PSD for ice-phase particles [N(D) = N0DμeλD] can be described with only two input variables, Dm and IWC. Note that in the derivation of the best fit between μ and Dm the exponent in the mass–dimension relationship was assumed to be 2.1. However, any value of a and b in a mass–dimension relation could be used within the relations above.

6. Estimated Ze using different μ constraints

The difference between Ze estimated from the moment-based method with varying μ and those estimated for different constant values of μ and for μ as a function of Dm [Eqs. (10a) and (10b)] is shown in Figs. 12 and 13. Figure 12 shows the impact of varying μ for the entire ice-phase dataset collected during GCPEx. As expected from Fig. 3, using a constant μ or μ as a function of Dm for the entire dataset gives a large spread in the estimated Ze, especially for Dm < 2 mm. For example, for Dm = 1 mm all estimates of Ze can present more than a 50% error (Fig. 12). The moment-based method with μ = 0 slightly overestimates Ze with a 3% mean bias (Fig. 12a). This overestimation increases as the selected μ value increases. For example, using μ = 3, as currently used in the GPM algorithm, leads to a 1% bias in the estimated Ze, and using μ = 10 represents a 3% mean overestimation (Fig. 12 and Table 3). Applying the best fit between μ and Dm for ice-phase particles [Eq. (11a)] leads to a better representation of the small particles (Dm < 2.5mm) and a median bias smaller than 1% (Fig. 12d and Table 3). As Dm increases, the spread between Ze estimates with varying μ and the estimates under different μ assumptions decreases (Fig. 12). Therefore, restricting the dataset to observations in the detectability range of the GPM DPR leads to overall a smaller spread and a better fit given an appropriate selection of μ (Fig. 13 and Table 3). Increasing the μ value used in the calculations of Ze leads to lower estimates of Ze, using a constant value of μ = 0 leads to a 2% mean bias, using μ = 3 leads to a bias of 5%, and using μ = 10 leads to a 7% bias (Figs. 13a–c and Table 3). These findings are consistent with the median of the distribution for the different estimates (Table 3). Moreover, applying the best fit between μ and Dm for ice-phase particles in the detectability range of the GPM DPR [Eq. (11b)] leads to the lowest discrepancies, with a mean and median bias smaller than 1% (Fig. 13d and Table 3). It is important to note that these statistical values show that, on average, choosing an exponential distribution (i.e.: μ = 0) versus a gamma distribution seem comparable. However, the variability of the μ parameter is a function of Dm and the benefits of using the gamma distribution is present for small mass-weighted mean diameter. For example, assuming that the PSD follows an exponential distribution for Dm < 1 mm leads to an error 3 times as large as that obtained when assuming that the PSD follows a gamma distribution with μ as a function of Dm.

Fig. 12.
Fig. 12.

Normalized joint frequency of occurrence of the ratio between the estimated Ze with varying μ and Ze estimated with (a) μ = 0, (b) μ =3, (c) μ =10, and (d) μ(Dm)=5.098×Dm0.414 as a function of Dm. Frequencies larger than 50% are enclosed by the solid black line, and a zero difference between the estimated reflectivities is marked with a dotted red line.

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

Fig. 13.
Fig. 13.

As in Fig. 12, but for particles in the GPM detectability range (i.e., Ze ≥ 12 dBZ) so that the relationship used in (d) is μ(Dm)=4.49×Dm0.254 [Eq. (11b)].

Citation: Journal of Applied Meteorology and Climatology 58, 8; 10.1175/JAMC-D-18-0203.1

Table 3.

Mean and median values of the ratio between estimated Ze with varying μ and that estimated with μ = 0 [Ze/Ze(μ = 0)], μ = 3 [Ze/Ze(μ = 3)], μ =10 [Ze/Ze(μ = 10)], and μ(Dm)=5.098×Dm0.414 [Eq. (11a)] for all ice-phase data and μ(Dm)=4.49×Dm0.254 [Eq. (11b)] for Ze > 12 dBZ [Ze/Ze(μ = best fit)].

Table 3.

7. Summary and conclusions

Both satellite retrieval algorithms and model microphysical parameterizations are in need of guidance from observations to improve the representation of ice-phase quantities and processes. Here, in situ observations collected during GCPEx were analyzed to expand the method originally proposed by Williams et al. (2014) for raindrop mass distributions to ice-phase particles. This technique parameterizes gamma PSD characteristics by calculating an independent μDm constraint for the μ parameter. The statistical relationships between μ and Dm found for ice-phase particles [either by using the maximum dimension or the equivalent melted diameter in Eqs. (3)(10)] differ from the one found for rainfall observations (Figs. 3 and 4). In particular, using the maximum dimension of the sampled particle, for GCPEx there is a large spread of μ, ranging from −3 to greater than 10 for Dm < 1 mm (Fig. 3). Thus, any algorithm designed for ice-phase particles that assumes a constant μ will have a large uncertainty in any retrieved variable, especially for small Dm for conditions similar to those sampled during GCPEx. Furthermore, given that the relationship previously found between μ and Dm for rainfall is not representative of ice-phase particles, the necessity to use different relationships for retrievals of different precipitation types is evident. Thus, more realistic microphysical parameterizations across the full range of observable ice-phase clouds require an adaptable constraint that considers the observed variability in μ. Here, the best statistical relationship between μ and Dm for all ice-phase particles collected during GCPEx is given in Eq. (11a). When considering only observations within the detectability range of the GPM DPR of Ze ≥ 12 dBZ, μ does not exhibit a broad range of values and is likely to be around 0 (Fig. 4). This result indicates that it is critical to allow μ close to 0 for a better representation of GPM-detectable ice-phase PSDs and that a different relationship between μ and Dm is needed [Eq. (11b)].

The effect of introducing the relationship between μ and Dm in the estimation of effective reflectivity and ice water content for ice-phase particles was analyzed in section 3c. When compared with independently observed values of Ze and IWC, the retrieved variables from the reconstructed PSD including the μ and Dm constraint had an 8% mean expected error in both variables (Tables 1 and 2). Furthermore, this error is smaller than or equivalent to the expected error when estimating these variables following other techniques or parameterizations (Tables 1 and 2). In particular, applying the Field et al. (2007) parameterization to estimate Ze leads to a mean 37% overall overestimation and a 78% mean overestimation for particles with Ze ≥ 12 dBZ (Fig. 10b and Table 2). This is in contraposition to the 8% and 22% when applying the μ and Dm constraint to all ice-phase particle and GPM-detectable PSDs, respectably (Table 2). This shows that, especially for Ze estimates, parameterizing the gamma PSD by calculating a statistically independent μDm constraint for the μ parameter represents a clear improvement to currently used algorithms in ice-phase precipitation. It is important to add that the Field et al. (2007) parameterization requires only one moment to generate a PSD estimate, whereas the parameterization presented in this work requires two inputs, such as Ze and DFR. In real data, these two inputs could contain measurement error that may impact the retrieved PSD parameters; the impacts of measurement errors on PSD retrievals will be a topic for future study.

This work further illustrates the variability of μ in observed ice-phase clouds and further demonstrates the impact of a constant μ on Ze estimates (Figs. 12 and 13). The simple assumption used in the current GPM algorithms of μ = 3 leads to an erroneous retrieval of Ze, with a mean underestimation of Ze of 5%, whereas applying the best fit found in this work between μ and Dm reduces this to a smaller-than-1% overestimation (Figs. 13 and Table 3). Therefore, a straightforward improvement that can be done in the current GPM algorithm is to use values of μ that are more representative of snowfall conditions for high-latitude retrievals. However, further study is needed before applying these results on a global scale because the μ parameter relationship may be regime dependent, including in mixed-phase conditions that were excluded in this study.

The impact of the selection of the a and b coefficients in the mass–dimension relation m(D) = aDb on the retrieved IWC and Ze was also presented. This study compared the impact of the selected a and b coefficients used in this work with estimates of IWC and Ze using a range of a and b values previously used in the literature. It was found that applying a large value of b (b = 3) is likely to cause more than 50% overestimation of Ze and an underestimation of 3% of IWC and that use of a small b (b = 1.5) leads to an underestimation of 44% for Ze and an overestimation of almost a factor of 4 in IWC (Fig. 11). The impact of the a coefficient in the calculation of these variables is also significant; applying a small value of a (a = 3 × 10−4 g cm−2.1) is likely to cause an underestimation of 9% for IWC and a considerable underestimation of Ze (from an expected referenced value of 14 dBZ to −7.6 dBZ), and applying a large value of a (a = 3 × 10−2 g cm−2.1) leads to a major overestimation in both variables: more than a factor of 8 for IWC and a factor of 2 for Ze (Fig. 11). This spread in the estimations of Ze and IWC demonstrates the need to use a and b parameters that are most representative of the sampled particles so as to minimize any errors in the mass calculation and thus avoid further error propagation and amplification into the calculations of different variables, including a wider variety of precipitation conditions sampled in other regions.

Acknowledgments

The authors thank two anonymous reviewers and Dmitri Moisseev for their helpful comments and suggestions that helped to improve the paper. Funding for this research was provided by NASA Precipitation Measurement Missions Grant NNX16AD80G under Ramesh Kakar. We thank all of the participants of GCPEx for collecting the data used in this study and Andrew Heymsfield and Aaron Bansemer for providing the microphysical data. Data used in this investigation are available on the NASA GHRC GCPEx data archive.

REFERENCES

  • Atlas, D., and C. Ulbrich, 2006: Drop size spectra and integral remote sensing parameters in the transition from convective to stratiform rain. Geophys. Res. Lett., 33, L16803, https://doi.org/10.1029/2006GL026824.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Barnes, S. L., 1964: A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3, 396409, https://doi.org/10.1175/1520-0450(1964)003<0396:ATFMDI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Baumgardner, D., and Coauthors, 2017: Cloud ice properties: In situ measurement challenges. Ice Formation and Evolution in Clouds and Precipitation: Measurement and Modeling Challenges, Meteor. Monogr., No. 58, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-16-0011.1.

    • Crossref
    • Export Citation
  • Bringi, V., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 636 pp.

    • Crossref
    • Export Citation
  • Brown, E. N., 1982: Ice detector evaluation for aircraft hazard warning and undercooled water content measurements. J. Aircr., 19, 980983, https://doi.org/10.2514/3.44800.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Brown, P., and P. Francis, 1995: Improved measurements of the ice water content in cirrus using a total-water probe. J. Atmos. Oceanic Technol., 12, 410414, https://doi.org/10.1175/1520-0426(1995)012<0410:IMOTIW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cao, Q., and G. Zhang, 2009: Errors in estimating raindrop size distribution parameters employing disdrometer and simulated raindrop spectra. J. Appl. Meteor. Climatol., 48, 406425, https://doi.org/10.1175/2008JAMC2026.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Casella, D., G. Panegrossi, P. Sanò, A. C. Marra, S. Dietrich, B. T. Johnson, and M. S. Kulie, 2017: Evaluation of the GPM-DPR snowfall detection capability: Comparison with CloudSat-CPR. Atmos. Res., 197, 6475, https://doi.org/10.1016/j.atmosres.2017.06.018.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., and V. N. Bringi, 1987: Simulation of radar reflectivity and surface measurements of rainfall. J. Atmos. Oceanic Technol., 4, 464478, https://doi.org/10.1175/1520-0426(1987)004<0464:SORRAS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., W. Li, and B. Zafar, 2005: Estimation of raindrop size distribution from spaceborne radar observations. IEEE Trans. Geosci. Remote Sens., 43, 10781086, https://doi.org/10.1109/TGRS.2005.846130.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chase, R. J., and Coauthors, 2018: Evaluation of triple-frequency radar retrieval of snowfall properties using coincident airborne in situ observations during OLYMPEX. Geophys. Res. Lett., 45, 57525760, https://doi.org/10.1029/2018GL077997.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cober, S. G., G. A. Isaac, A. V. Korolev, and J. W. Strapp, 2001: Assessing cloud-phase conditions. J. Appl. Meteor., 40, 19671983, https://doi.org/10.1175/1520-0450(2001)040<1967:ACPC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cohard, J., and J. Pinty, 2000: A comprehensive two-moment warm microphysical bulk scheme. I: Description and tests. Quart. J. Roy. Meteor. Soc., 126, 18151842, https://doi.org/10.1256/smsqj.56613.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delanoë, J. M. E., A. J. Heymsfield, A. Protat, A. Bansemer, and R. J. Hogan, 2014: Normalized particle size distribution for remote sensing application. J. Geophys. Res. Atmos., 119, 42044227, https://doi.org/10.1002/2013JD020700.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Delene, D. and M. R Poellot, 2012: GPM Ground Validation UND Citation Cloud Microphysics GCPEX Version 1. NASA Global Hydrology Resource Center DAAC, accessed 1 October 2016, https://doi.org/10.5067/GPMGV/GCPEX/MULTIPLE/DATA202.

    • Crossref
    • Export Citation
  • Ferrier, B. S., 1994: A double-moment multiple-phase four-class bulk ice scheme. Part I: Description. J. Atmos. Sci., 51, 249280, https://doi.org/10.1175/1520-0469(1994)051<0249:ADMMPF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Field, P. R., A. J. Heymsfield, and A. Bansemer, 2006: Shattering and particle interarrival times measured by optical array probes in ice clouds. J. Atmos. Oceanic Technol., 23, 13571371, https://doi.org/10.1175/JTECH1922.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Field, P. R., A. J. Heymsfield, and A. Bansemer, 2007: Snow size distribution parameterization for midlatitude and tropical ice clouds. J. Atmos. Sci., 64, 43464365, https://doi.org/10.1175/2007JAS2344.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Finlon, J. A., G. M. McFarquhar, S. W. Nesbitt, R. M. Rauber, H. Morrison, W. Wu, and P. Zhang, 2019: A novel approach to characterize the variability in mass–dimension relationships: Results from MC3E. Atmos. Chem. Phys., 19, 36213643, https://doi.org/10.5194/acp-19-3621-2019.

    • Search Google Scholar
    • Export Citation
  • Grecu, M., L. Tian, W. Olson, and S. Tanelli, 2011: A robust dual-frequency radar profiling algorithm. J. Appl. Meteor. Climatol., 50, 15431557, https://doi.org/10.1175/2011JAMC2655.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Haddad, Z. S., S. L. Durden, and E. Im, 1996: Parameterizing the raindrop size distribution. J. Appl. Meteor., 35, 313, https://doi.org/10.1175/1520-0450(1996)035<0003:PTRSD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., 2003: Properties of tropical and midlatitude ice cloud particle ensembles. Part I: Median mass diameters and terminal velocities. J. Atmos. Sci., 60, 25732591, https://doi.org/10.1175/1520-0469(2003)060<2573:POTAMI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., and J. L. Parrish, 1978: A computational technique for increasing the effective sampling volume of the PMS two-dimensional particle size spectrometer. J. Appl. Meteor., 17, 15661572, https://doi.org/10.1175/1520-0450(1978)017<1566:ACTFIT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., P. Field, and A. Bansemer, 2008: Exponential size distributions for snow. J. Atmos. Sci., 65, 40174031, https://doi.org/10.1175/2008JAS2583.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., C. Schmitt, A. Bansemer, and C. H. Twohy, 2010: Improved representation of ice particle masses based on observations in natural clouds. J. Atmos. Sci., 67, 33033318, https://doi.org/10.1175/2010JAS3507.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., and C. D. Westbrook, 2014: Equation for the microwave backscatter cross section of aggregate snowflakes using the self-similar Rayleigh–Gans approximation. J. Atmos. Sci., 71, 32923301, https://doi.org/10.1175/JAS-D-13-0347.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., M. P. Mittermaier, and A. J. Illingworth, 2006: The retrieval of ice water content from radar reflectivity factor and temperature and its use in evaluating a mesoscale model. J. Appl. Meteor. Climatol., 45, 301317, https://doi.org/10.1175/JAM2340.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., L. Tian, P. R. A. Brown, C. D. Westbrook, A. J. Heymsfield, and J. D. Eastment, 2012: Radar scattering from ice aggregates using the horizontally aligned oblate spheroid approximation. J. Appl. Meteor. Climatol., 51, 655671, https://doi.org/10.1175/JAMC-D-11-074.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hong, S., J. Dudhia, and S. Chen, 2004: A revised approach to ice microphysical processes for the bulk parameterization of clouds and precipitation. Mon. Wea. Rev., 132, 103120, https://doi.org/10.1175/1520-0493(2004)132<0103:ARATIM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hou, A., and Coauthors, 2014: The Global Precipitation Measurement Mission. Bull. Amer. Meteor. Soc., 95, 701722, https://doi.org/10.1175/BAMS-D-13-00164.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hudak, D., 2013: GPM ground validation dual polarized C-band Doppler radar King City GCPEx. NASA EOSDIS Global Hydrology Resource Center DAAC, accessed 1 November 2016, https://doi.org/10.5067/GPMGV/GCPEX/MUTIPLE/DATA201.

    • Crossref
    • Export Citation
  • Jackson, R. C., and Coauthors, 2012: The dependence of ice microphysics on aerosol concentration in arctic mixed-phase stratus clouds during ISDAC and M-PACE. J. Geophys. Res., 117, D15207, https://doi.org/10.1029/2012JD017668.

    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., and P. R. Field, 2015: Assessment of the performance of the inter-arrival time algorithm to identify ice shattering artifacts in cloud particle probe measurements. Atmos. Meas. Tech., 8, 761777, https://doi.org/10.5194/amt-8-761-2015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., J. W. Strapp, G. A. Isaac, and A. N. Nevzorov, 1998: The Nevzorov airborne hot-wire LWC–TWC probe: Principle of operation and performance characteristics. J. Atmos. Oceanic Technol., 15, 14951510, https://doi.org/10.1175/1520-0426(1998)015<1495:TNAHWL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Korolev, A. V., E. F. Emery, J. W. Strapp, S. G. Cober, and G. A. Isaac, 2013: Quantification of the effects of shattering on airborne ice particle measurements. J. Atmos. Oceanic Technol., 30, 25272553, https://doi.org/10.1175/JTECH-D-13-00115.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kumar, L. S., Y. H. Lee, and J. T. Ong, 2011: Two-parameter gamma drop size distribution models for Singapore. IEEE Trans. Geosci. Remote Sens., 49, 33713380, https://doi.org/10.1109/TGRS.2011.2124464.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kummerow, C., W. Barnes, T. Kozu, J. Shiue, and J. Simpson, 1998: The Tropical Rainfall Measuring Mission (TRMM). J. Atmos. Oceanic Technol., 15, 809817, https://doi.org/10.1175/1520-0426(1998)015<0809:TTRMMT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lang, S. E., W. Tao, J. Chern, D. Wu, and X. Li, 2014: Benefits of a fourth ice class in the simulated radar reflectivities of convective systems using a bulk microphysics scheme. J. Atmos. Sci., 71, 35833612, https://doi.org/10.1175/JAS-D-13-0330.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • L’Ecuyer, T., W. Petersen, and D. Moiseev, 2010: Light Precipitation Validation Experiment (LPVEx). National Aeronautics and Space Administration Doc., 28 pp., https://pmm.nasa.gov/sites/default/files/document_files/lpvex_science_plan_Jan29_2010.pdf.

  • Liao, L., and R. Meneghini, 2005: A study of air/space-borne dual-wavelength radar for estimation of rain profiles. Adv. Atmos. Sci., 22, 841851, https://doi.org/10.1007/BF02918684.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liao, L., R. Meneghini, and A. Tokay, 2014: Uncertainties of GPM DPR rain estimates caused by DSD parameterizations. J. Appl. Meteor., 53, 25242537, https://doi.org/10.1175/JAMC-D-14-0003.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lin, Y., R. D. Farley, and H. D. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Climate Appl. Meteor., 22, 10651092, https://doi.org/10.1175/1520-0450(1983)022<1065:BPOTSF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mardiana, R., T. Iguchi, and N. Takahashi, 2004: A dual-frequency rain profiling method without the use of a surface reference technique. IEEE Trans. Geosci. Remote Sens., 42, 22142225, https://doi.org/10.1109/TGRS.2004.834647.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Marks, D. A., D. B. Wolff, L. D. Carey, and A. Tokay, 2011: Quality control and calibration of the dual-polarization radar at Kwajalein, RMI. J. Atmos. Oceanic Technol., 28, 181196, https://doi.org/10.1175/2010JTECHA1462.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., M. S. Timlin, R. M. Rauber, B. F. Jewett, J. A. Grim, and D. P. Jorgensen, 2007: Vertical variability of cloud hydrometeors in the stratiform region of mesoscale convective systems and bow echoes. Mon. Wea. Rev., 135, 34053428, https://doi.org/10.1175/MWR3444.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., J. Um, and R. Jackson, 2013: Small cloud particle shapes in mixed-phase clouds. J. Appl. Meteor. Climatol., 52, 12771293, https://doi.org/10.1175/JAMC-D-12-0114.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., T. Hsieh, M. Freer, J. Mascio, and B. F. Jewett, 2015: The characterization of ice hydrometeor gamma size distributions as volumes in N0λ-μ phase space: Implications for microphysical process modeling. J. Atmos. Sci., 72, 892909, https://doi.org/10.1175/JAS-D-14-0011.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McFarquhar, G. M., and Coauthors, 2017: Processing of ice cloud in situ data collected by bulk water, scattering, and imaging probes: Fundamentals, uncertainties, and efforts toward consistency. Ice Formation and Evolution in Clouds and Precipitation: Measurement and Modeling Challenges, Meteor. Monogr., No. 58, Amer. Meteor. Soc., https://doi.org/10.1175/AMSMONOGRAPHS-D-16-0007.1.

    • Crossref
    • Export Citation
  • Meneghini, R., T. Kozu, H. Kumagai, and W. C. Boncyk, 1992: A study of rain estimation methods from space using dual-wavelength radar measurements at near-nadir incidence over ocean. J. Atmos. Oceanic Technol., 9, 364382, https://doi.org/10.1175/1520-0426(1992)009<0364:ASOREM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Meyers, M. P., R. L. Walko, J. Y. Harrington, and W. R. Cotton, 1997: New RAMS cloud microphysics. Part II: The two-moment scheme. Atmos. Res., 45, 339, https://doi.org/10.1016/S0169-8095(97)00018-5.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Milbrandt, J. A., and M. K. Yau, 2005: A multimoment bulk microphysics parameterization. Part I: Analysis of the role of the spectral shape parameter. J. Atmos. Sci., 62, 30513064, https://doi.org/10.1175/JAS3534.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., 1996: Use of mass- and area-dimensional power laws for determining precipitation particle terminal velocities. J. Atmos. Sci., 53, 17101723, https://doi.org/10.1175/1520-0469(1996)053<1710:UOMAAD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mitchell, D. L., R. Zhang, and R. L. Pitter, 1990: Mass-dimensional relationships for ice particles and the influence of riming on snowfall rates. J. Appl. Meteor., 29, 153163, https://doi.org/10.1175/1520-0450(1990)029<0153:MDRFIP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Moisseev, D. N., and V. Chandrasekar, 2007: Examination of the μλ relation suggested for drop size distribution parameter. J. Atmos. Oceanic Technol., 24, 847855, https://doi.org/10.1175/JTECH2010.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., and J. A. Milbrandt, 2015: Parameterization of cloud microphysics based on the prediction of bulk ice particle properties. Part I: Scheme description and idealized tests. J. Atmos. Sci., 72, 287311, https://doi.org/10.1175/JAS-D-14-0065.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., G. Thompson, and V. Tatarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Mon. Wea. Rev., 137, 9911007, https://doi.org/10.1175/2008MWR2556.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Munchak, S. J., and A. Tokay, 2008: Retrieval of raindrop size distribution from simulated dual-frequency radar measurements. J. Appl. Meteor. Climatol., 47, 223239, https://doi.org/10.1175/2007JAMC1524.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Naeger, A. R., B. A. Colle, and A. Molthan, 2017: Evaluation of cloud microphysical schemes for a warm frontal snowband during the GPM Cold Season Precipitation Experiment (GCPEx). Mon. Wea. Rev., 145, 46274650, https://doi.org/10.1175/MWR-D-17-0081.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Petersen, W., and M. Schwaller, 2008: Global precipitation mission (GPM) ground validation science implementation plan. National Aeronautics and Space Administration Doc., 41 pp., https://pmm.nasa.gov/sites/default/files/document_files/GPM_GVS_imp_plan_Jul08.pdf.

  • Seto, S., T. Iguchi, and T. Oki, 2013: The basic performance of a precipitation retrieval algorithm for the global precipitation measurement mission’s single/dual-frequency radar measurements. IEEE Trans. Geosci. Remote Sens., 51, 52395251, https://doi.org/10.1109/TGRS.2012.2231686.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Skofronick-Jackson, G., and Coauthors, 2015: Global Precipitation Measurement Cold Season Precipitation Experiment (GCPEX): For measurement’s sake, let it snow. Bull. Amer. Meteor. Soc., 96, 17191741, https://doi.org/10.1175/BAMS-D-13-00262.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Skofronick-Jackson, G., and Coauthors, 2017: The Global Precipitation Measurement (GPM) Mission for science and society. Bull. Amer. Meteor. Soc., 98, 16791695, https://doi.org/10.1175/BAMS-D-15-00306.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tang, G., Y. Wen, J. Gao, D. Long, Y. Ma, W. Wan, and Y. Hong, 2017: Similarities and differences between three coexisting spaceborne radars in global rainfall and snowfall estimation. Water Resour. Res., 53, 38353853, https://doi.org/10.1002/2016WR019961.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Testud, J., S. Oury, R. A. Black, P. Amayenc, and X. Dou, 2001: The concept of “normalized” distribution to describe the raindrop spectra: A tool for cloud physics and remote sensing. J. Appl. Meteor., 40, 11181140, https://doi.org/10.1175/1520-0450(2001)040<1118:TCONDT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, G., P. R. Field, R. M. Rasmussen, and W. D. Hall, 2008: Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization. Mon. Wea. Rev., 136, 50955115, https://doi.org/10.1175/2008MWR2387.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Toyoshima, K., H. Masunaga, and F. A. Furuzawa, 2015: Early evaluation of Ku- and Ka-band sensitivities for the Global Precipitation Measurement (GPM) Dual-Frequency Precipitation Radar (DPR). SOLA, 11, 1417, https://doi.org/10.2151/sola.2015-004.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ulbrich, C., 1983: Natural variations in the analytical form of the raindrop size distribution. J. Climate Appl. Meteor., 22, 17641775, https://doi.org/10.1175/1520-0450(1983)022<1764:NVITAF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Williams, C. R., and Coauthors, 2014: Describing the shape of raindrop size distributions using uncorrelated raindrop mass spectrum parameters. J. Appl. Meteor. Climatol., 53, 12821296, https://doi.org/10.1175/JAMC-D-13-076.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wu, W., and G. M. McFarquhar, 2018: Statistical theory on the functional form of cloud particle size distributions. J. Atmos. Sci., 75, 28012814, https://doi.org/10.1175/JAS-D-17-0164.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, G., J. Vivekanandan, and E. A. Brandes, 2001: A method for estimating rain rate and drop size distribution from polarimetric radar measurements. IEEE Trans. Geosci. Remote Sens., 39, 830841, https://doi.org/10.1109/36.917906.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, G., J. Vivekanandan, E. A. Brandes, R. Meneghini, and R. Kozu, 2003: The shape-slope relation in observed gamma raindrop size distributions: Statistical error or useful information? J. Atmos. Oceanic Technol., 20, 11061119, https://doi.org/10.1175/1520-0426(2003)020<1106:TSRIOG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, G., J. Sun, and E. A. Brandes, 2006: Improving parameterization of rain microphysics with disdrometer and radar observations. J. Atmos. Sci., 63, 12731290, https://doi.org/10.1175/JAS3680.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save