Editorial Type:
Article
Article Type:
Research Article

Spatially Variable Advection Correction of Radar Data. Part I: Theoretical Considerations

Alan Shapiro School of Meteorology, and Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma

Search for other papers by Alan Shapiro in
Current site
Google Scholar
PubMed Close
,
Katherine M. Willingham Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, and NOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma

Search for other papers by Katherine M. Willingham in
Current site
Google Scholar
PubMed Close
, and
Corey K. Potvin School of Meteorology, and Center for Analysis and Prediction of Storms, and Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma

Search for other papers by Corey K. Potvin in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Nov 2010
DOI:
https://doi.org/10.1175/2010JAS3465.1
Page(s):
3445–3456
Restricted access

Abstract

Radar data–based analysis products, such as accumulated rainfall maps, dual-Doppler wind syntheses, and thermodynamic retrievals, are prone to substantial error if the temporal sampling interval is too coarse. Techniques to mitigate these errors typically make use of advection-correction procedures (space-to-time conversions) in which the analyzed radial velocity or reflectivity field is idealized as a pattern of unchanging form that translates horizontally at constant speed. The present study is concerned with an advection-correction procedure for the reflectivity field in which the pattern-advection components vary spatially. The analysis is phrased as a variational problem in which errors in the frozen-turbulence constraint are minimized subject to smoothness constraints. The Euler–Lagrange equations for this problem are derived and a solution is proposed in which the trajectories, pattern-advection fields, and reflectivity field are analyzed simultaneously using a combined analytical and numerical procedure. The potential for solution nonuniqueness is explored.

Corresponding author address: Alan Shapiro, School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Room 5900, Norman, OK 73072. Email: ashapiro@ou.edu

Abstract

Radar data–based analysis products, such as accumulated rainfall maps, dual-Doppler wind syntheses, and thermodynamic retrievals, are prone to substantial error if the temporal sampling interval is too coarse. Techniques to mitigate these errors typically make use of advection-correction procedures (space-to-time conversions) in which the analyzed radial velocity or reflectivity field is idealized as a pattern of unchanging form that translates horizontally at constant speed. The present study is concerned with an advection-correction procedure for the reflectivity field in which the pattern-advection components vary spatially. The analysis is phrased as a variational problem in which errors in the frozen-turbulence constraint are minimized subject to smoothness constraints. The Euler–Lagrange equations for this problem are derived and a solution is proposed in which the trajectories, pattern-advection fields, and reflectivity field are analyzed simultaneously using a combined analytical and numerical procedure. The potential for solution nonuniqueness is explored.

Corresponding author address: Alan Shapiro, School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Room 5900, Norman, OK 73072. Email: ashapiro@ou.edu

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Anagnostou, E. N., and W. F. Krajewski, 1999: Real-time radar rainfall estimation. Part I: Algorithm formulation. J. Atmos. Oceanic Technol., 16 , 189–197.

    • Search Google Scholar
    • Export Citation

  • Andrejczuk, M., S. Moszkowicz, K. Haman, and T. Szoplik, 2003: Radar-echo tracking by use of invariant moments. Appl. Opt., 42 , 5891–5896.

    • Search Google Scholar
    • Export Citation

  • Austin, G. L., and A. Bellon, 1974: The use of digital weather radar records for short-term precipitation forecasting. Quart. J. Roy. Meteor. Soc., 100 , 658–664.

    • Search Google Scholar
    • Export Citation

  • Austin, P. M., 1987: Relation between measured radar reflectivity and surface rainfall. Mon. Wea. Rev., 115 , 1053–1070.

  • Biggerstaff, M. I., and Coauthors, 2005: The Shared Mobile Atmospheric Research and Teaching Radar: A collaboration to enhance research and teaching. Bull. Amer. Meteor. Soc., 86 , 1263–1274.

    • Search Google Scholar
    • Export Citation

  • Caillault, K., and Y. Lemaitre, 1999: Retrieval of three-dimensional wind fields corrected for the time-induced advection problem. J. Atmos. Oceanic Technol., 16 , 708–722.

    • Search Google Scholar
    • Export Citation

  • Carbone, R. E., 1982: A severe frontal rainband. Part I: Stormwide hydrodynamic structure. J. Atmos. Sci., 39 , 258–279.

  • Caya, A., S. Laroche, I. Zawadzki, and T. Montmerle, 2002: Using single-Doppler data to obtain a mesoscale environmental field. J. Atmos. Oceanic Technol., 19 , 21–36.

    • Search Google Scholar
    • Export Citation

  • Chong, M., J. Testud, and F. Roux, 1983: Three-dimensional wind field analysis from dual-Doppler radar data. Part II: Minimizing the error due to temporal variation. J. Climate Appl. Meteor., 22 , 1216–1226.

    • Search Google Scholar
    • Export Citation

  • Clarke, T. L., F. I. Harris, and C. G. Mohr, 1980: Errors in wind fields derived from multiple-Dopper radars: Random errors and temporal errors associated with advection and evolution. J. Appl. Meteor., 19 , 1273–1284.

    • Search Google Scholar
    • Export Citation

  • Courant, R., and D. Hilbert, 1953: Methods of Mathematical Physics. Vol. 1. Wiley-Interscience, 560 pp.

  • Crook, A., 1994: Numerical simulations initialized with radar-derived winds. Part I: Simulated data experiments. Mon. Wea. Rev., 122 , 1189–1203.

    • Search Google Scholar
    • Export Citation

  • Crum, T. D., and R. L. Alberty, 1993: The WSR-88D and the WSR-88D Operational Support Facility. Bull. Amer. Meteor. Soc., 74 , 1669–1687.

    • Search Google Scholar
    • Export Citation

  • Dixon, M., and G. Wiener, 1993: TITAN: Thunderstorm Identification, Tracking Analysis and Nowcasting—A radar-based methodology. J. Atmos. Oceanic Technol., 10 , 785–797.

    • Search Google Scholar
    • Export Citation

  • Dowell, D. C., and H. B. Bluestein, 1997: The Arcadia, Oklahoma, storm of 17 May 1981: Analysis of a supercell during tornadogenesis. Mon. Wea. Rev., 125 , 2562–2582.

    • Search Google Scholar
    • Export Citation

  • Dowell, D. C., and H. B. Bluestein, 2002: The 8 June 1995 McLean, Texas, Storm. Part I: Observations of cyclic tornadogenesis. Mon. Wea. Rev., 130 , 2626–2648.

    • Search Google Scholar
    • Export Citation

  • Fabry, F., A. Bellon, M. R. Duncan, and G. L. Austin, 1994: High resolution rainfall measurements by radar for very small basins: The sampling problem reexamined. J. Hydrol., 161 , 415–428.

    • Search Google Scholar
    • Export Citation

  • Flanigan, F. J., 1983: Complex Variables. Dover, 353 pp.

  • Fulton, R., J. Breidenbach, D-J. Seo, D. Miller, and T. O’Bannon, 1998: The WSR-88D rainfall algorithm. Wea. Forecasting, 13 , 377–395.

    • Search Google Scholar
    • Export Citation

  • Gal-Chen, T., 1978: A method for the initialization of the anelastic equations: Implications for matching models with observations. Mon. Wea. Rev., 106 , 587–606.

    • Search Google Scholar
    • Export Citation

  • Gal-Chen, T., 1982: Errors in fixed and moving frame of references: Applications for conventional and Doppler radar analysis. J. Atmos. Sci., 39 , 2279–2300.

    • Search Google Scholar
    • Export Citation

  • Gao, J., M. Xue, A. Shapiro, Q. Xu, and K. K. Droegemeier, 2001: Three-dimensional simple adjoint velocity retrievals from single Doppler radar. J. Atmos. Oceanic Technol., 18 , 26–38.

    • Search Google Scholar
    • Export Citation

  • Germann, U., and I. Zawadzki, 2002: Scale-dependence of the predictability of precipitation from continental radar images. Part I: Description of the methodology. Mon. Wea. Rev., 130 , 2859–2873.

    • Search Google Scholar
    • Export Citation

  • Gerstner, E-M., and G. Heinemann, 2008: Real-time areal precipitation determination from radar by means of statistical objective analysis. J. Hydrol., 352 , 296–308.

    • Search Google Scholar
    • Export Citation

  • Hane, C. E., 1993: Storm-motion estimates derived from dynamic-retrieval calculations. Mon. Wea. Rev., 121 , 431–443.

  • Hane, C. E., R. B. Wilhelmson, and T. Gal-Chen, 1981: Retrieval of thermodynamic variables within deep convective clouds: Experiments in three dimensions. Mon. Wea. Rev., 109 , 564–576.

    • Search Google Scholar
    • Export Citation

  • Heymsfield, G. M., 1978: Kinematic and dynamic aspects of the Harrah tornadic storm analyzed from dual-Doppler radar data. Mon. Wea. Rev., 106 , 233–254.

    • Search Google Scholar
    • Export Citation

  • Islam, M. R., and P. F. Rasmussen, 2008: Improved high-resolution radar-based rainfall estimation. J. Hydrol. Eng., 13 , 910–913.

  • Johnson, J., P. MacKeen, A. Witt, E. Mitchell, G. Stumpf, M. Eilts, and K. Thomas, 1998: The storm cell identification and tracking algorithm: An enhanced WSR-88D algorithm. Wea. Forecasting, 13 , 263–276.

    • Search Google Scholar
    • Export Citation

  • Klazura, G. E., and D. A. Imy, 1993: A description of the initial set of analysis products available from the NEXRAD WSR-88D system. Bull. Amer. Meteor. Soc., 74 , 1293–1311.

    • Search Google Scholar
    • Export Citation

  • Lakshmanan, V., R. Rabin, and V. DeBrunner, 2003: Multiscale storm identification and forecast. Atmos. Res., 67–68 , 367–380.

  • Lakshmanan, V., T. Smith, K. Hondl, G. J. Stumpf, and A. Witt, 2006: A real-time, three-dimensional, rapidly updating, heterogeneous radar merger technique for reflectivity, velocity, and derived products. Wea. Forecasting, 21 , 802–823.

    • Search Google Scholar
    • Export Citation

  • Lanczos, C., 1986: The Variational Principles of Mechanics. Dover, 418 pp.

  • Langston, C., J. Zhang, and K. Howard, 2007: Four-dimensional dynamic radar mosaic. J. Atmos. Oceanic Technol., 24 , 776–790.

  • Laroche, S., and I. Zawadzki, 1995: Retrievals of horizontal winds from single-Doppler clear-air data by methods of cross correlation and variational analysis. J. Atmos. Oceanic Technol., 12 , 721–738.

    • Search Google Scholar
    • Export Citation

  • Lazarus, S., A. Shapiro, and K. Droegemeier, 2001: Application of the Zhang–Gal-Chen single-Doppler velocity retrieval to a deep convective storm. J. Atmos. Sci., 58 , 998–1016.

    • Search Google Scholar
    • Export Citation

  • Liou, Y-C., 1999: Single radar recovery of cross-beam wind components using a modified moving frame of reference technique. J. Atmos. Oceanic Technol., 16 , 1003–1016.

    • Search Google Scholar
    • Export Citation

  • Liou, Y-C., 2001: The derivation of absolute potential temperature perturbations and pressure gradients from wind measurements in three-dimensional space. J. Atmos. Oceanic Technol., 18 , 577–590.

    • Search Google Scholar
    • Export Citation

  • Liou, Y-C., and I-S. Luo, 2001: An investigation of the moving-frame single-Doppler wind retrieval technique using Taiwan Area Mesoscale Experiment low-level data. J. Appl. Meteor., 40 , 1900–1917.

    • Search Google Scholar
    • Export Citation

  • Liu, C., and W. F. Krajewski, 1996: A comparison of methods for calculation of radar-rainfall hourly accumulations. Water Resour. Bull., 32 , 305–315.

    • Search Google Scholar
    • Export Citation

  • Liu, S., C. Qiu, Q. Xu, and P. Zhang, 2004: An improved time interpolation for three-dimensional Doppler wind analysis. J. Appl. Meteor., 43 , 1379–1391.

    • Search Google Scholar
    • Export Citation

  • Lu, H., and Q. Xu, 2009: Trade-offs between measurement accuracy and resolutions in configuring phased-array radar velocity scans for ensemble-based storm-scale data assimilation. J. Appl. Meteor. Climatol., 48 , 1230–1244.

    • Search Google Scholar
    • Export Citation

  • Matejka, T., 2002: Estimating the most steady frame of reference from Doppler radar data. J. Atmos. Oceanic Technol., 19 , 1035–1048.

  • Michelson, M., W. W. Shrader, and J. G. Wieler, 1990: Terminal Doppler weather radar. Microwave J., 33 , 139–148.

  • Miller, L. J., and R. A. Kropfli, 1980: The multiple Doppler radar workshop, November 1979. Part II: Experimental design and procedures. Bull. Amer. Meteor. Soc., 61 , 1173–1177.

    • Search Google Scholar
    • Export Citation

  • Mohr, C. G., L. J. Miller, R. L. Vaughan, and H. W. Frank, 1986: The merger of mesoscale datasets into a common Cartesian format for efficient and systematic analyses. J. Atmos. Oceanic Technol., 3 , 143–161.

    • Search Google Scholar
    • Export Citation

  • NOAA/NWS, 2005: TDWR interface control and specifications documentation for the NWS supplemental product generator, version 4.3, 42 pp. [Available online at http://wdtb.noaa.gov/buildTraining/TDWR/TDWR_SPG_ICD_v43.pdf].

    • Search Google Scholar
    • Export Citation

  • Parsons, D. B., C. G. Mohr, and T. Gal-Chen, 1987: A severe frontal rainband. Part III: Derived thermodynamic structure. J. Atmos. Sci., 44 , 1615–1631.

    • Search Google Scholar
    • Export Citation

  • Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992: Numerical Recipes in FORTRAN: The Art of Scientific Computing. 2nd ed. Cambridge University Press, 963 pp.

    • Search Google Scholar
    • Export Citation

  • Protat, A., and I. Zawadzki, 1999: A variational method for real-time retrieval of three-dimensional wind field from multiple-Doppler bistatic radar network data. J. Atmos. Oceanic Technol., 16 , 432–449.

    • Search Google Scholar
    • Export Citation

  • Ray, P. S., M. Gilet, and K. W. Johnson, 1980: The multiple Doppler radar workshop, November 1979. Part IV: Motion field synthesis and radar placement. Bull. Amer. Meteor. Soc., 61 , 1184–1189.

    • Search Google Scholar
    • Export Citation

  • Rosenfeld, D., 1987: Objective method for analysis and tracking of convective cells as seen by radar. J. Atmos. Oceanic Technol., 4 , 422–434.

    • Search Google Scholar
    • Export Citation

  • Roux, F., 1985: Retrieval of thermodynamic fields from multiple-Doppler radar data using the equations of motion and the thermodynamic equation. Mon. Wea. Rev., 113 , 2142–2157.

    • Search Google Scholar
    • Export Citation

  • Sasaki, Y., 1970: Some basic formalisms in numerical variational analysis. Mon. Wea. Rev., 98 , 875–883.

  • Sasaki, Y., 1971: A theoretical interpretation of anisotropically weighted smoothing on the basis of numerical variational analysis. Mon. Wea. Rev., 99 , 698–707.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., S. Ellis, and J. Shaw, 1995: Single-Doppler velocity retrievals with Phoenix II data: Clear air and microburst wind retrievals in the planetary boundary layer. J. Atmos. Sci., 52 , 1265–1287.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., P. Robinson, J. Wurman, and J. Gao, 2003: Single-Doppler velocity retrieval with rapid scan radar data. J. Atmos. Oceanic Technol., 20 , 1758–1775.

    • Search Google Scholar
    • Export Citation

  • Shapiro, A., K. M. Willingham, and C. K. Potvin, 2010: Spatially variable advection correction of radar data. Part II: Test results. J. Atmos. Sci., 67 , 3457–3470.

    • Search Google Scholar
    • Export Citation

  • Sun, J., and R. A. Houze Jr., 1992: Validation of a thermodynamic retrieval technique by application to a simulated squall line with trailing stratiform precipitation. Mon. Wea. Rev., 120 , 1003–1018.

    • Search Google Scholar
    • Export Citation

  • Sun, J., and N. A. Crook, 1994: Wind and thermodynamic retrieval from single-Doppler measurements of a gust front observed during Phoenix II. Mon. Wea. Rev., 122 , 1075–1091.

    • Search Google Scholar
    • Export Citation

  • Sun, J., and N. A. Crook, 2001: Real-time low-level wind and temperature analysis using single WSR-88D data. Wea. Forecasting, 16 , 117–132.

    • Search Google Scholar
    • Export Citation

  • Tabary, P., 2007: The new French operational radar rainfall product. Part I: Methodology. Wea. Forecasting, 22 , 393–408.

  • Taylor, G. I., 1938: The spectrum of turbulence. Proc. Roy. Soc. London, 164A , 476–490.

  • Tuttle, J. D., and G. B. Foote, 1990: Determination of the boundary layer airflow from a single Doppler radar. J. Atmos. Oceanic Technol., 7 , 218–232.

    • Search Google Scholar
    • Export Citation

  • Vasiloff, S. V., 2001: Improving tornado warnings with the Federal Aviation Administration’s terminal Doppler weather radar. Bull. Amer. Meteor. Soc., 82 , 861–874.

    • Search Google Scholar
    • Export Citation

  • Villarini, G., and W. F. Krajewski, 2010: Review of the different sources of uncertainty in single-polarization radar-based estimates of rainfall. Surv. Geophys., 31 , 107–129.

    • Search Google Scholar
    • Export Citation

  • Wurman, J., J. Straka, E. Rasmussen, M. Randall, and A. Zahrai, 1997: Design and deployment of a portable, pencil-beam, pulsed, 3-cm Doppler radar. J. Atmos. Oceanic Technol., 14 , 1502–1512.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., H. Gu, and C. Qiu, 2001a: Simple adjoint retrievals of wet-microburst winds and gust-front winds from single-Doppler radar data. J. Appl. Meteor., 40 , 1485–1499.

    • Search Google Scholar
    • Export Citation

  • Xu, Q., H. Gu, and S. Yang, 2001b: Simple adjoint method for three-dimensional wind retrievals from single-Doppler radar. Quart. J. Roy. Meteor. Soc., 127 , 1053–1067.

    • Search Google Scholar
    • Export Citation

  • Yang, H., J. Zhang, and C. Langston, 2009: Synchronization of radar observations with multi-scale storm tracking. Adv. Atmos. Sci., 26 , 78–86.

    • Search Google Scholar
    • Export Citation

  • Zawadzki, I. I., 1973: Statistical properties of precipitation patterns. J. Appl. Meteor., 12 , 459–472.

  • Zhang, J., and T. Gal-Chen, 1996: Single-Doppler wind retrieval in the moving frame of reference. J. Atmos. Sci., 53 , 2609–2623.

  • Zrnic, D. S., and Coauthors, 2007: Agile-beam phased array radar for weather observations. Bull. Amer. Meteor. Soc., 88 , 1753–1766.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 402 59 6
PDF Downloads 293 51 7