Editorial Type:
Article
Article Type:
Research Article

On the Recovery of 3D Spatial Statistics of Particles from 1D Measurements: Implications for Airborne Instruments

Michael L. Larsen Department of Physics and Astronomy, College of Charleston, Charleston, South Carolina

Search for other papers by Michael L. Larsen in
Current site
Google Scholar
PubMed Close
,
Clarissa A. Briner Department of Physics, University of Colorado Boulder, Boulder, Colorado

Search for other papers by Clarissa A. Briner in
Current site
Google Scholar
PubMed Close
, and
Philip Boehner Department of Scientific Computing, Florida State University, Tallahassee, Florida

Search for other papers by Philip Boehner in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Oct 2014
DOI:
https://doi.org/10.1175/JTECH-D-14-00004.1
Page(s):
2078–2087
Restricted access

Abstract

The spatial positions of individual aerosol particles, cloud droplets, or raindrops can be modeled as a point processes in three dimensions. Characterization of three-dimensional point processes often involves the calculation or estimation of the radial distribution function (RDF) and/or the pair-correlation function (PCF) for the system. Sampling these three-dimensional systems is often impractical, however, and, consequently, these three-dimensional systems are directly measured by probing the system along a one-dimensional transect through the volume (e.g., an aircraft-mounted cloud probe measuring a thin horizontal “skewer” through a cloud). The measured RDF and PCF of these one-dimensional transects are related to (but not, in general, equal to) the RDF/PCF of the intrinsic three-dimensional systems from which the sample was taken. Previous work examined the formal mathematical relationship between the statistics of the intrinsic three-dimensional system and the one-dimensional transect; this study extends the previous work within the context of realistic sampling variability. Natural sampling variability is found to constrain substantially the usefulness of applying previous theoretical relationships. Implications for future sampling strategies are discussed.

Corresponding author address: Michael L. Larsen, Dept. of Physics and Astronomy, College of Charleston, 66 George St., Charleston, SC 29424. E-mail: larsenml@cofc.edu

Abstract

The spatial positions of individual aerosol particles, cloud droplets, or raindrops can be modeled as a point processes in three dimensions. Characterization of three-dimensional point processes often involves the calculation or estimation of the radial distribution function (RDF) and/or the pair-correlation function (PCF) for the system. Sampling these three-dimensional systems is often impractical, however, and, consequently, these three-dimensional systems are directly measured by probing the system along a one-dimensional transect through the volume (e.g., an aircraft-mounted cloud probe measuring a thin horizontal “skewer” through a cloud). The measured RDF and PCF of these one-dimensional transects are related to (but not, in general, equal to) the RDF/PCF of the intrinsic three-dimensional systems from which the sample was taken. Previous work examined the formal mathematical relationship between the statistics of the intrinsic three-dimensional system and the one-dimensional transect; this study extends the previous work within the context of realistic sampling variability. Natural sampling variability is found to constrain substantially the usefulness of applying previous theoretical relationships. Implications for future sampling strategies are discussed.

Corresponding author address: Michael L. Larsen, Dept. of Physics and Astronomy, College of Charleston, 66 George St., Charleston, SC 29424. E-mail: larsenml@cofc.edu
Save
Cover Journal of Atmospheric and Oceanic Technology
Journal of Atmospheric and Oceanic Technology

  • Anderson, A., and Kostinski A. B. , 2010: Reversible record breaking and variability: Temperature distributions across the globe. J. Appl. Meteor. Climatol., 49, 1681–1691, doi:10.1175/2010JAMC2407.1.

    • Search Google Scholar
    • Export Citation

  • Baker, B., and Lawson R. , 2010: Analysis of tools used to quantify droplet clustering in clouds. J. Atmos. Sci., 67, 3355–3367, doi:10.1175/2010JAS3409.1.

    • Search Google Scholar
    • Export Citation

  • Balkovsky, E., Falkovich G. , and Fouxon A. , 2001: Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett., 86, 2790–2793, doi:10.1103/PhysRevLett.86.2790.

    • Search Google Scholar
    • Export Citation

  • Bateson, C. P., and Aliseda A. , 2012: Wind tunnel measurements of the preferential concentration of inertial droplets in homogeneous isotropic turbulence. Exp. Fluids, 52, 1373–1387, doi:10.1007/s00348-011-1252-6.

    • Search Google Scholar
    • Export Citation

  • Borovoi, A. G., 2002: On the extinction of radiation by a homogeneous but spatially correlated random medium. J. Opt. Soc. Amer., 19A, 2517–2520, doi:10.1364/JOSAA.19.002517.

    • Search Google Scholar
    • Export Citation

  • Brenguier, J.-L., Bourrianne T. , Coelho A. A. , Isbert J. , Peytavi R. , Travarin D. , and Weschler P. , 1998: Improvements of droplet size distribution measurements with the fast-FSSP (forward scattering spectrometer probe). J. Atmos. Oceanic Technol., 15, 1077–1090, doi:10.1175/1520-0426(1998)015<1077:IODSDM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Cramér, H., and Leadbetter M. R. , 2004: Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications. Dover Books on Mathematics, Dover Publications, 368 pp.

    • Search Google Scholar
    • Export Citation

  • Devenish, B. J., and Coauthors, 2012: Droplet growth in warm turbulent clouds. Quart. J. Roy. Meteor. Soc., 138, 1401–1429, doi:10.1002/qj.1897.

    • Search Google Scholar
    • Export Citation

  • Feller, W., 1966: An Introduction to Probability Theory and Its Applications. Vol. 2, Wiley, 669 pp.

  • Fugal, J. P., and Shaw R. A. , 2009: Cloud particle size distributions measured with an airborne digital in-line holographic instrument. Atmos. Meas. Tech., 2, 259–271, doi:10.5194/amt-2-259-2009.

    • Search Google Scholar
    • Export Citation

  • Fugal, J. P., Shaw R. A. , Saw E.-W. , and Sergeyev A. V. , 2004: Airborne digital holographic system for cloud particle measurements. Appl. Opt., 43, 5987–5995, doi:10.1364/AO.43.005987.

    • Search Google Scholar
    • Export Citation

  • Holtzer, G. L., and Collins L. R. , 2002: Relationship between the intrinsic radial distribution function for an isotropic field of particles and lower-dimensional measurements. J. Fluid Mech., 459, 93–102, doi:10.1017/S0022112002008169.

    • Search Google Scholar
    • Export Citation

  • Jameson, A. R., and Kostinski A. B. , 2000: Fluctuation properties of precipitation. Part V: On the distribution of rain rates—Theory and observations in clustered rain. J. Atmos. Sci., 57, 373–388, doi:10.1175/1520-0469(2000)057<0373:FPOPPV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Jameson, A. R., Kostinski A. B. , and Black R. A. , 1998: The texture of clouds. J. Geophys. Res., 103, 6211–6219, doi:10.1029/98JD00081.

    • Search Google Scholar
    • Export Citation

  • Kostinski, A. B., 2001: On the extinction of radiation by a homogeneous but spatially correlated random medium. J. Opt. Soc. Amer., 18A, 1929–1933, doi:10.1364/JOSAA.18.001929.

    • Search Google Scholar
    • Export Citation

  • Kostinski, A. B., 2002: On the extinction of radiation by a homogeneous but spatially correlated random medium: Reply to comment. J. Opt. Soc. Amer., 19A, 2521–2525, doi:10.1364/JOSAA.19.002521.

    • Search Google Scholar
    • Export Citation

  • Kostinski, A. B., and Jameson A. R. , 2000: On the spatial distribution of cloud particles. J. Atmos. Sci., 57, 901–915, doi:10.1175/1520-0469(2000)057<0901:OTSDOC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Larsen, M. L., 2006: Studies of discrete fluctuations in atmospheric phenomena. Ph.D. dissertation, Michigan Technological University, 220 pp.

  • Larsen, M. L., 2012: Scale localization of cloud particle clustering statistics. J. Atmos. Sci., 69, 3277–3289, doi:10.1175/JAS-D-12-02.1.

    • Search Google Scholar
    • Export Citation

  • Larsen, M. L., and Clark A. S. , 2014: On the link between particle size and deviations from the Beer–Lambert–Bouguer law for direct transmission. J. Quant. Spectrosc. Radiat. Transfer, 133, 646–651, doi:10.1016/j.jqsrt.2013.10.001.

    • Search Google Scholar
    • Export Citation

  • Larsen, M. L., Kostinski A. B. , and Tokay A. , 2005: Observations and analysis of uncorrelated rain. J. Atmos. Sci., 62, 4071–4083, doi:10.1175/JAS3583.1.

    • Search Google Scholar
    • Export Citation

  • Martínez, V. J., and Saar E. , 2002: Statistics of the Galaxy Distribution. CRC Press, 432 pp.

  • Matérn, B., 1960: Spatial variation. Medd. Statens Skogsforskningsinst.,49, 1–144.

  • Reade, W. C., and Collins L. R. , 2000: Effect of preferential concentration on turbulent collision rates. Phys. Fluids, 12, 2530–2540, doi:10.1063/1.1288515.

    • Search Google Scholar
    • Export Citation

  • Saw, E.-W., Shaw R. A. , Salazar J. P. L. C. , and Collins L. R. , 2012: Spatial clustering of polydisperse inertial particles in turbulence: II. Comparing simulation with experiment. New J. Phys.,14, 105031, doi:10.1088/1367-2630/14/10/105031.

  • Shaw, R. A., 2003: Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech., 35, 183–227, doi:10.1146/annurev.fluid.35.101101.161125.

    • Search Google Scholar
    • Export Citation

  • Shaw, R. A., Kostinski A. B. , and Larsen M. L. , 2002: Towards quantifying droplet clustering in clouds. Quart. J. Roy. Meteor. Soc., 128, 1043–1057, doi:10.1256/003590002320373193.

    • Search Google Scholar
    • Export Citation

  • Siebert, B., Gerashchenko S. , Gylfason A. , Lehmann K. , Collins L. R. , Shaw R. A. , and Warhaft Z. , 2010: Towards understanding the role of turbulence on droplets in clouds: In situ and laboratory measurements. Atmos. Res., 97, 426–437, doi:10.1016/j.atmosres.2010.05.007.

    • Search Google Scholar
    • Export Citation

  • Stoyan, D., Kendall W. S. , and Mecke J. , 1987: Stochastic Geometry and Its Applications. Wiley Series in Probability and Statistics, Wiley, 345 pp.

    • Search Google Scholar
    • Export Citation

  • Wunsch, C., 1999: The interpretation of short climate records, with comments on the North Atlantic and Southern Oscillations. Bull. Amer. Meteor. Soc., 80, 245–255, doi:10.1175/1520-0477(1999)080<0245:TIOSCR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Young, W. R., Roberts A. J. , and Stuhne G. , 2001: Reproductive pair correlations and the clustering of organisms. Nature, 412, 328–331, doi:10.1038/35085561.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 295 64 6
PDF Downloads 101 32 2