Measuring Second- through Fourth-Order Moments in Noisy Data

Donald H. Lenschow National Center for Atmospheric Research, Boulder, Colorado *

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Volker Wulfmeyer National Center for Atmospheric Research, Boulder, Colorado *

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Christoph Senff Cooperative Institute for Research in Environmental Sciences, University of Colorado, and NOAA/Environmental Technology Laboratory, Boulder, Colorado

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Abstract

The authors derive expressions for correcting second- through fourth-order moments of measured variables that are contaminated by random uncorrelated noise. These expressions are then tested by applying them to an artificially produced time series as well as measurements from two upward-pointing ground-based lidar systems:a differential absorption lidar that measures water vapor density and a high-resolution Doppler lidar that measures vertical wind velocity. Both sets of measurements were obtained in a convective boundary layer, and contain sufficient noise to significantly affect measurements of second- and fourth-order moments (as well as integral scale and skewness) throughout the boundary layer. It is shown that the corrections derived here can be used to obtain useful measurements of these moments from instruments such as lidars, which are inherently noisy. The authors also obtain information on higher-order moments of the noise as well as the correlation between noise and atmospheric measurements.

Corresponding author address: Dr. Donald H. Lenschow, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.

Email: lenschow@ucar.edu

Abstract

The authors derive expressions for correcting second- through fourth-order moments of measured variables that are contaminated by random uncorrelated noise. These expressions are then tested by applying them to an artificially produced time series as well as measurements from two upward-pointing ground-based lidar systems:a differential absorption lidar that measures water vapor density and a high-resolution Doppler lidar that measures vertical wind velocity. Both sets of measurements were obtained in a convective boundary layer, and contain sufficient noise to significantly affect measurements of second- and fourth-order moments (as well as integral scale and skewness) throughout the boundary layer. It is shown that the corrections derived here can be used to obtain useful measurements of these moments from instruments such as lidars, which are inherently noisy. The authors also obtain information on higher-order moments of the noise as well as the correlation between noise and atmospheric measurements.

Corresponding author address: Dr. Donald H. Lenschow, NCAR, P.O. Box 3000, Boulder, CO 80307-3000.

Email: lenschow@ucar.edu

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  • Bevington, P. R., 1969: Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill, 336 pp.

  • Bösenberg, J., 1998: Ground-based differential absorption lidar for water-vapor and temperature profiling: Methodology. Appl. Opt.,37, 3845–3860.

    • Crossref
    • Export Citation
  • Cohn, S. A., S. D. Mayor, C. J. Grund, T. M. Weckwerth, and C. Senff, 1998: The Lidars in Flat Terrain (LIFT) experiment. Bull. Amer. Meteor. Soc.,79, 1329–1343.

    • Crossref
    • Export Citation
  • Frehlich, R., 1997: Effects of wind turbulence on coherent Doppler lidar performance. J. Atmos. Oceanic Technol.,14, 54–75.

    • Crossref
    • Export Citation
  • Frisch, U., 1995: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press, 296 pp.

    • Crossref
    • Export Citation
  • Grund, C. J., 1996: High resolution Doppler lidar measurements of wind and turbulence. Advances in Atmospheric Remote Sensing with Lidar, Springer-Verlag, 235–238.

    • Crossref
    • Export Citation
  • Kaimal, J. C., and J. A. Businger, 1970: Case studies of a convective plume and a dust devil. J. Appl. Meteor.,9, 612–620.

    • Crossref
    • Export Citation
  • ——, J. C. Wyngaard, D. A. Haugan, O. R. Coté, and Y. Izumi, 1976:Turbulence structure in the convective boundary layer. J. Atmos. Sci.,33, 2152–2169.

    • Crossref
    • Export Citation
  • Lenschow, D. H., 1970: Airplane measurements of planetary boundary layer structure. J. Appl. Meteor.,9, 874–884.

    • Crossref
    • Export Citation
  • ——, and L. Kristensen, 1985: Uncorrelated noise in turbulence measurements. J. Atmos. Oceanic Technol.,2, 68–81.

    • Crossref
    • Export Citation
  • ——, and B. B. Stankov, 1986: Length scales in the convective boundary layer. J. Atmos. Sci.,43, 1198–1208.

    • Crossref
    • Export Citation
  • ——, J. C. Wyngaard, and W. T. Pennell, 1980: Mean-field and second-moments budgets in a baroclinic, convective boundary layer. J. Atmos. Sci.,37, 1313–1326.

    • Crossref
    • Export Citation
  • ——, J. Mann, and L. Kristensen, 1994: How long is long enough when measuring fluxes and other turbulence statistics? J. Atmos. Oceanic Technol.,11, 661–673.

    • Crossref
    • Export Citation
  • Lumley, J. L., and H. A. Panofsky, 1964: The Structure of Atmospheric Turbulence. Interscience Publishers, 239 pp.

  • Mahrt, L., 1991: Boundary-layer moisture regimes. Quart. J. Roy. Meteor. Soc.,117, 151–176.

    • Crossref
    • Export Citation
  • Mann, J., D. H. Lenschow, and L. Kristensen, 1995: Comments on“A definite approach to turbulence statistical studies in planetary boundary layers.” J. Atmos. Sci.,52, 3194–3196.

    • Crossref
    • Export Citation
  • Moeng, C.-H., and J. C. Wyngaard, 1984: Statistics of conservative scalars in the convective boundary layer. J. Atmos. Sci.,41, 3161–3169.

    • Crossref
    • Export Citation
  • ——, and ——, 1989: Evaluation of turbulent transport and dissipation closures in second-order modeling. J. Atmos. Sci.,46, 2311–2330.

    • Crossref
    • Export Citation
  • ——, and R. Rotunno, 1990: Vertical-velocity skewness in the buoyancy-driven boundary layer. J. Atmos. Sci.,47, 1149–1162.

    • Crossref
    • Export Citation
  • Monin, A. S., and A. M. Yaglom, 1979: Statistical Fluid Mechanics, Vol. 2. MIT Press, 874 pp.

  • Senff, C., J. Bösenberg, and G. Peters, 1994: Measurement of water vapor flux profiles in the convective boundary layer with lidar and Radar–RASS. J. Atmos. Oceanic Technol.,11, 85–93.

    • Crossref
    • Export Citation
  • ——, ——, ——, and T. Schaberl, 1996: Remote sensing of turbulent ozone fluxes and the ozone budget in the convective boundary layer with DIAL and Radar–RASS: A case study. Contrib. Atmos. Phys.,69, 161–176.

  • Sullivan, P. P., C.-H. Moeng, B. Stevens, D. H. Lenschow, and S. D. Mayor, 1998: Structure of the entrainment zone capping the convective atmospheric boundary layer. J. Atmos. Sci.,55, 3042–3064.

    • Crossref
    • Export Citation
  • Wulfmeyer, V., 1998: Ground-based differential absorption lidar for water-vapor and temperature profiling: Development and specifications of a high-performance laser transmitter. Appl. Opt.,37, 3804–3824.

    • Crossref
    • Export Citation
  • ——, 1999a: Investigation of turbulent processes in the lower troposphere with water vapor DIAL and Radar–RASS. J. Atmos. Sci.,56, 1055–1076.

    • Crossref
    • Export Citation
  • ——, 1999b: Investigations of humidity skewness and variance profiles in the convective boundary layer and comparison of the latter with large eddy simulation results. J. Atmos. Sci.,56, 1077–1087.

    • Crossref
    • Export Citation
  • ——, and J. Bösenberg, 1998: Ground-based differential absorption lidar for water-vapor profiling: Assessment of accuracy, resolution, and meteorological applications. Appl. Opt.,37, 3825–3844.

    • Crossref
    • Export Citation
  • ——, and Coauthors, 1998: Performance and applications of the NOAA 2 μm High Resolution Doppler Lidar. 19th Int. Laser Radar Conf. Annapolis, MD, NASA, 573–576.

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