### Abstract

Ageneralized expression is derived for the correlation function of signals backscattered from hydrometeors observed with a vertically pointed radar beam in which particle size distribution, turbulence, and mean wind are not homogeneous. This study extends the previous work of Fang and Doviak by including the effects of particle size distribution on the measured Doppler spectrum. It shows the measured Doppler spectrum to be the volumetric mean of the weighted convolution of the normalized Doppler spectra associated with turbulence, mean wind, particle oscillation/wobble, and terminal velocity. Without particle oscillation/wobble, mean wind, and turbulence, the Doppler spectrum is the mirror image of the terminal velocity spectrum under the condition that the second-order effect of finite beamwidth can be ignored. This generalized Doppler spectrum reduces further to a previously derived expression if the particle size distribution, or equivalently reflectivity, is uniform. Provided there is a unique relationship between the particle's terminal velocity and its effective diameter, the derived equations can be applied to scatterers consisting of ice particles as well as water droplets. This study derives the analytical expression for the Doppler spectrum of mean wind and also shows that if stationary homogeneous turbulence is the only contributor to spectral broadening, then the average of a large number of radar-measured Doppler spectra will be equal to the velocity probability density function of turbulence independent of the angular, range, reflectivity, and weighting functions.

Original language | English (US) |
---|---|

Pages (from-to) | 500-509 |

Number of pages | 10 |

Journal | Journal of Atmospheric and Oceanic Technology |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - 2012 |

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### ASJC Scopus subject areas

- Atmospheric Science
- Ocean Engineering

### Cite this

*Journal of Atmospheric and Oceanic Technology*,

*29*(4), 500-509. https://doi.org/10.1175/JTECH-D-11-00005.1

**Analytical Expressions for Doppler Spectra of Scatter from Hydrometeors Observed : With a Vertically Directed Radar Beam.** / Fang, Ming; Doviak, Richard J.; Albrecht, Bruce A.

Research output: Contribution to journal › Article

*Journal of Atmospheric and Oceanic Technology*, vol. 29, no. 4, pp. 500-509. https://doi.org/10.1175/JTECH-D-11-00005.1

}

TY - JOUR

T1 - Analytical Expressions for Doppler Spectra of Scatter from Hydrometeors Observed

T2 - With a Vertically Directed Radar Beam

AU - Fang, Ming

AU - Doviak, Richard J.

AU - Albrecht, Bruce A

PY - 2012

Y1 - 2012

N2 - Ageneralized expression is derived for the correlation function of signals backscattered from hydrometeors observed with a vertically pointed radar beam in which particle size distribution, turbulence, and mean wind are not homogeneous. This study extends the previous work of Fang and Doviak by including the effects of particle size distribution on the measured Doppler spectrum. It shows the measured Doppler spectrum to be the volumetric mean of the weighted convolution of the normalized Doppler spectra associated with turbulence, mean wind, particle oscillation/wobble, and terminal velocity. Without particle oscillation/wobble, mean wind, and turbulence, the Doppler spectrum is the mirror image of the terminal velocity spectrum under the condition that the second-order effect of finite beamwidth can be ignored. This generalized Doppler spectrum reduces further to a previously derived expression if the particle size distribution, or equivalently reflectivity, is uniform. Provided there is a unique relationship between the particle's terminal velocity and its effective diameter, the derived equations can be applied to scatterers consisting of ice particles as well as water droplets. This study derives the analytical expression for the Doppler spectrum of mean wind and also shows that if stationary homogeneous turbulence is the only contributor to spectral broadening, then the average of a large number of radar-measured Doppler spectra will be equal to the velocity probability density function of turbulence independent of the angular, range, reflectivity, and weighting functions.

AB - Ageneralized expression is derived for the correlation function of signals backscattered from hydrometeors observed with a vertically pointed radar beam in which particle size distribution, turbulence, and mean wind are not homogeneous. This study extends the previous work of Fang and Doviak by including the effects of particle size distribution on the measured Doppler spectrum. It shows the measured Doppler spectrum to be the volumetric mean of the weighted convolution of the normalized Doppler spectra associated with turbulence, mean wind, particle oscillation/wobble, and terminal velocity. Without particle oscillation/wobble, mean wind, and turbulence, the Doppler spectrum is the mirror image of the terminal velocity spectrum under the condition that the second-order effect of finite beamwidth can be ignored. This generalized Doppler spectrum reduces further to a previously derived expression if the particle size distribution, or equivalently reflectivity, is uniform. Provided there is a unique relationship between the particle's terminal velocity and its effective diameter, the derived equations can be applied to scatterers consisting of ice particles as well as water droplets. This study derives the analytical expression for the Doppler spectrum of mean wind and also shows that if stationary homogeneous turbulence is the only contributor to spectral broadening, then the average of a large number of radar-measured Doppler spectra will be equal to the velocity probability density function of turbulence independent of the angular, range, reflectivity, and weighting functions.

UR - http://www.scopus.com/inward/record.url?scp=84864762534&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84864762534&partnerID=8YFLogxK

U2 - 10.1175/JTECH-D-11-00005.1

DO - 10.1175/JTECH-D-11-00005.1

M3 - Article

AN - SCOPUS:84864762534

VL - 29

SP - 500

EP - 509

JO - Journal of Atmospheric and Oceanic Technology

JF - Journal of Atmospheric and Oceanic Technology

SN - 0739-0572

IS - 4

ER -