• Anagnostou, M. N., , E. N. Anagnostou, , and J. Vivekanandan, 2006: Correction for rain path specific and differential attenuation of X-band dual-polarization observations. IEEE Trans. Geosci. Remote Sens., 44, 24702480.

    • Search Google Scholar
    • Export Citation
  • Barber, P., , and C. Yeh, 1975: Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies. Appl. Opt., 14, 28642872.

  • Bechini, R., , L. Baldini, , R. Cremonini, , and E. Gorgucci, 2008: Differential reflectivity calibration for operational radars. J. Atmos. Oceanic Technol., 25, 15421555.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., , and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar. Cambridge University Press, 636 pp.

  • Bringi, V. N., , V. Chandrasekar, , N. Balakrishnan, , and D. Zrnić, 1990: An examination of propagation effects in rainfall on radar measurements at microwave frequencies. J. Atmos. Oceanic Technol., 7, 829840.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., , T. D. Keenan, , and V. Chandrasekar, 2001: Correcting C-band radar reflectivity and differential reflectivity data for rain attenuation: A self-consistent method with constraints. IEEE Trans. Geosci. Remote Sens., 39, 19061915.

    • Search Google Scholar
    • Export Citation
  • Carey, L. D., , S. A. Rutledge, , D. A. Ahijevych, , and T. D. Keenan, 2000: Correcting propagation effects in C-band polarimetric radar observations of tropical convection using differential propagation phase. J. Appl. Meteor., 39, 14051433.

    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., , S. Lim, , and E. Gorgucci, 2006: Simulation of X-band rainfall observations from S-band radar data. J. Atmos. Oceanic Technol., 23, 11951205.

    • Search Google Scholar
    • Export Citation
  • Desroziers, G., , L. Berre, , B. Chapnik, , and P. Poli, 2005: Diagnosis of observation, background and analysis error statistics in observation space. Quart. J. Roy. Meteor. Soc., 131, 33853396.

    • Search Google Scholar
    • Export Citation
  • Doelling, I. G., , J. Joss, , and J. Riedl, 1998: Systematic variations of Z–R relationships from drop size distributions measured in northern Germany during seven years. Atmos. Res., 47–48, 635649.

    • Search Google Scholar
    • Export Citation
  • Furness, G., 2005: Using optimal estimation theory for improved rainfall rates from polarization radar. M.S. dissertation, Dept. of Mathematics, University of Reading, 66 pp.

  • Gorgucci, E., , and V. Chandrasekar, 2005: Evaluation of attenuation correction methodology for dual-polarization radars: Application to X-band systems. J. Atmos. Oceanic Technol., 22, 11951206.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., , V. Chandrasekar, , and L. Baldini, 2006: Correction of X-band radar observation for propagation effects based on the self-consistency principle. J. Atmos. Oceanic Technol., 23, 16681681.

    • Search Google Scholar
    • Export Citation
  • Gu, J.-Y., , A. Ryzhkov, , P. Zhang, , P. Neilley, , M. Knight, , B. Wolf, , and D.-I. Lee, 2011: Polarimetric attenuation correction in heavy rain at C band. J. Appl. Meteor. Climatol., 50, 3958.

    • Search Google Scholar
    • Export Citation
  • Hitschfeld, W., , and J. Bordan, 1954: Errors inherent in the radar measurement of rainfall at attenuating wavelengths. J. Meteor., 11, 5867.

    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., 2007: A variational scheme for retrieving rainfall rate and hail reflectivity fraction from polarization radar. J. Appl. Meteor. Climatol., 46, 15441564.

    • Search Google Scholar
    • Export Citation
  • Hubbert, J., , and V. N. Bringi, 1995: An iterative filtering technique for the analysis of copolar differential phase and dual-frequency radar measurements. J. Atmos. Oceanic Technol., 12, 643648.

    • Search Google Scholar
    • Export Citation
  • Hubbert, J., , S. M. Ellis, , M. Dixon, , and G. Meymaris, 2010a: Modeling, error analysis, and evaluation of dual-polarization variables obtained from simultaneous horizontal and vertical polarization transmit radar. Part I: Modeling and antenna errors. J. Atmos. Oceanic Technol., 27, 15831598.

    • Search Google Scholar
    • Export Citation
  • Hubbert, J., , S. M. Ellis, , M. Dixon, , and G. Meymaris, 2010b: Modeling, error analysis, and evaluation of dual-polarization variables obtained from simultaneous horizontal and vertical polarization transmit radar. Part II: Experimental data. J. Atmos. Oceanic Technol., 27, 15991607.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., 1991: Polarization radar measurements in rain at 5 and 9 GHz. J. Appl. Meteor., 30, 15001513.

  • Jou, B. J.-D., , W.-C. Lee, , and R. H. Johnson, 2011: An overview of SoWMEX/TiMREX and its operation. The Global Monsoon System: Research and Forecast, C.-P. Chang, Ed., 2nd ed., World Scientific, 303–318.

  • Li, H., , E. Kalnay, , and T. Miyoshi, 2009: Simultaneous estimation of covariance inflation and observations errors within an ensemble Kalman filter. Quart. J. Roy. Meteor. Soc., 135, 523533.

    • Search Google Scholar
    • Export Citation
  • Lim, S., , V. Chandrasekar, , P. Lee, , and A. P. Jayasumana, 2011: Real-time implementation of a network-based attenuation correction in the CASA IP1 testbed. J. Atmos. Oceanic Technol., 28, 197209.

    • Search Google Scholar
    • Export Citation
  • Park, S. G., , V. N. Bringi, , V. Chandrasekar, , M. Maki, , and K. Iwanami, 2005a: Correction of radar reflectivity and differential reflectivity for rain attenuation at X band. Part I: Theoretical and empirical basis. J. Atmos. Oceanic Technol., 22, 16211632.

    • Search Google Scholar
    • Export Citation
  • Park, S. G., , M. Maki, , K. Iwanami, , V. N. Bringi, , and V. Chandrasekar, 2005b: Correction of radar reflectivity and differential reflectivity for rain attenuation at X band. Part II: Evaluation and application. J. Atmos. Oceanic Technol., 22, 16331655.

    • Search Google Scholar
    • Export Citation
  • Rodgers, C. D., 2000: Inverse Methods for Atmospheric Soundings: Theory and Practice. World Scientific, 238 pp.

  • Ryzhkov, A. V., 2007: The impact of beam broadening on the quality of radar polarimetric data. J. Atmos. Oceanic Technol., 24, 729744.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., , and D. S. Zrnić, 1996: Assessment of rainfall measurement that uses specific differential phase. J. Appl. Meteor., 35, 20802090.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., , V. B. Zhuravlyov, , and N. A. Rybakova, 1994: Preliminary results of X-band polarization radar studies of clouds and precipitation. J. Atmos. Oceanic Technol., 11, 132139.

    • Search Google Scholar
    • Export Citation
  • Sachidananda, M., , and D. S. Zrnic, 1986: Differential propagation phase shift and rainfall rate estimation. Radio Sci., 21, 235247.

  • Scarchilli, G., , E. Gorgucci, , V. Chandrasekar, , and T. A. Seliga, 1993: Rainfall estimation using polarimetric techniques at C-band frequencies. J. Appl. Meteor., 32, 11501160.

    • Search Google Scholar
    • Export Citation
  • Schönhuber, M., , H. E. Urban, , J. P. V. Poiares Baptista, , W. L. Randeu, , and W. Riedler, 1997: Weather radar versus 2D-video-distrometer data. Weather Radar Technology for Water Resources Management, B. Braga Jr. and O. Massambani, Eds., UNESCO Press, 159–171.

  • Smyth, T. J., , and A. J. Illingworth, 1998: Correction for attenuation of radar reflectivity using polarization data. Quart. J. Roy. Meteor. Soc., 124A, 23932415.

    • Search Google Scholar
    • Export Citation
  • Snyder, J. C., , H. B. Bluestein, , G. Zhang, , and S. J. Frasier, 2010: Attenuation correction and hydrometeor classification of high-resolution, X-band, dual-polarized mobile radar measurements in severe convective storms. J. Atmos. Oceanic Technol., 27, 19792001.

    • Search Google Scholar
    • Export Citation
  • Steiner, M., , and R. A. Houze Jr., 1997: Sensitivity of the estimated monthly convective rain fraction to the choice of ZR relation. J. Appl. Meteor., 36, 452462.

    • Search Google Scholar
    • Export Citation
  • Steiner, M., , and J. A. Smith, 2000: Reflectivity, rain rate, and kinetic energy flux relationships based on raindrop spectra. J. Appl. Meteor., 39, 19231940.

    • Search Google Scholar
    • Export Citation
  • Steiner, M., , J. A. Smith, , and R. Uijlenhoet, 2004: A microphysical interpretation of radar reflectivity–rain rate relationships. J. Atmos. Sci., 61, 11141131.

    • Search Google Scholar
    • Export Citation
  • Testud, J., , E. Le Bouar, , E. Obligis, , and M. Ali-Mehenni, 2000: The rain profiling algorithm applied to polarimetric weather radar. J. Atmos. Oceanic Technol., 17, 332356.

    • Search Google Scholar
    • Export Citation
  • Ulbrich, C. W., 1983: Natural variations in the analytical form of the raindrop size distribution. J. Climate Appl. Meteor., 22, 17641775.

    • Search Google Scholar
    • Export Citation
  • Vivekanandan, J., , W. M. Adams, , and V. N. Bringi, 1991: Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions. J. Appl. Meteor., 30, 10531063.

    • Search Google Scholar
    • Export Citation
  • Vivekanandan, J., , S. M. Ellis, , R. Oye, , D. S. Zrnić, , A. V. Ryzhkov, , and J. Straka, 1999: Cloud microphysics retrieval using S-band dual-polarization radar measurements. Bull. Amer. Meteor. Soc., 80, 381388.

    • Search Google Scholar
    • Export Citation
  • Vivekanandan, J., , G. Zhang, , S. M. Ellis, , D. Rajopadhyaya, , and S. K. Avery, 2003: Radar reflectivity calibration using differential propagation phase measurement. Radio Sci., 38, 8049, doi:10.1029/2002RS002676.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    The X-band forward models based on a simplified electromagnetic scattering and propagation model: (a) ZDR, (b) KDP/ZHH, (c) AH/ZHH, and (d) ADP/ZHH, all vs log(ZHH/R). Gamma raindrop size distributions (RSDs) and measured video disdrometer DSDs were used as input into the electromagnetic scattering and propagation model. The black lines represent the forward models calculated using a gamma RSD with μ = 5 at 20°C. The gray dots represent the rigorous scattering model simulation corresponding to discretely measured DSDs based on the 5 yr of data from NCU 2DVD in northern Taiwan.

  • View in gallery

    Flowchart describing various numbered steps (at the top-right box corners) in the variational attenuation correction method. The dashed box represents a complete loop of the Gauss–Newton minimization process. See section 2a for a detailed description.

  • View in gallery

    The simulated X-band polarimetric radar observables for measured DSDs: (a) ZHH and AH, (b) ZDR and Ahv, and (c) ΦDP, R, and a, all vs distance. The DSDs were obtained from 5 yr of NCU 2DVD measurements. The modeled polarimetric observations without observation error were arranged along a radial with a gate resolution of 0.125 km. The total attenuation (Ah) and total differential attenuation (Ahv) for this 50-km beam are 12.07 and 1.44 dB, respectively.

  • View in gallery

    (a) RMSEs of attenuation (Ah) and differential attenuation (Ahv) derived from variational (var.) and ΦDP-based methods vs distance. (b) As in (a), but for dBR; the values of dBR are defined as 10 log10(R).

  • View in gallery

    The estimated values of the standard deviation (σ) of the measurement error of ZDR (blue solid line) and ΦDP (red solid line) as a function of the iteration number. The true values are shown as blue and red dashed lines, respectively.

  • View in gallery

    (a) The relation between the estimated standard deviation of the measurement error of ZDRest.ZDR) obtained from step 8 in Fig. A1 and the true value (σtrueZDR) shown as a red line with the dashed line being a 1:1 line. (b) As in (a), but for ΦDP.

  • View in gallery

    As in Fig. 4, but vs the measurement error in ZDR.

  • View in gallery

    Radar data at a 3.19°-elevation angle from TEAM-R and S-Pol radar data: (a) ZHH, (b) ZDR, and (c) ΦDP, all vs distance. The blue lines represent the measured raw values from TEAM-R and the red lines represent the attenuation-corrected ZHH and ZDR, and predicted ΦDP based on the variational scheme. The black lines in (a) and (b) represent averaged measurements from the S-Pol radar. The distance between the TEAM-R and NCAR S-Pol radars was 50 m.

  • View in gallery

    RHI scans of (left) ZHH and (right) ZDR measurements from NCAR S-Pol and TEAM-R. (a)–(d) Intrinsic data measured during TiMREX 2008 with attenuation correction for S-Pol and system bias correction methods for TEAM-R applied; a threshold of ρHV < 0.7 was applied to the X-band measurements, and no threshold was applied to the S-band measurements. Further, the intrinsic difference in the ZDR observations between X-band and S-band wavelengths was removed by converting S-band ZDR to X-band ZDR (Chandrasekar et al. 2006). (e),(f) The variational attenuation-corrected ZHH and ZDR from TEAM-R, respectively. The solid lines represent the beams with elevation angles of 5.5° and 8.1°, respectively; the dashed line represents a beam of 3.19°.

  • View in gallery

    (a) The predicted ΦDP from the variational method and (b) the measured ΦDP. The solid lines represent beam elevation angles of 5.5° and 8.1°, respectively, and the dashed lines represent beams of 3.19°.

  • View in gallery

    The ΦDP attenuation correction with coefficients α = 0.374 and β = 0.052 for attenuation (AH: black solid) and differential attenuation (ADP: black dashed) vs KDP. The gray dots represent the variational algorithm derived attenuation (AH), differential attenuation (ADP), and KDP. The corresponding AH (ADP) and KDP with ZDR values between 0–1 dB and between 3–4 dB are shown by red and blue dots, respectively.

  • View in gallery

    (a) The estimated optimal standard deviations of ZDR (from steps 4–6 of Fig. A1) as a function of elevation angle. (b),(c) As in (a), but for ΦDP and the optimal background error covariance matrices (), respectively.

  • View in gallery

    (a) The diagnostic standard error of ZDR (derived from step 8 of Fig. A1) vs distance obtained by using the statistical diagnostic method. (b) As in (a), but for ΦDP. The solid lines represent elevation angles of 5.5° and 8.1°, respectively.

  • View in gallery

    Flowchart describing the various numbered steps (numbers at the top-right box corners) in the discrete approximation of and diagnostic of ZDR and ΦDP. See appendix A for a detailed description.

  • View in gallery

    The cumulative density function of total attenuation (Ah), total differential attenuation (Ahv), and differential phase shift (ΦDP) of 600 modeled polarimetric observation beams.

  • View in gallery

    (a) Mean absolute errors as a function of the parameter a. The x axis represents the coefficient a from 20 to 650. The y axis represents the absolute difference on a logarithmic scale between the forward model predicted parameters (here ZDR and ΦDP) and the actual measurements of a radar beam. TEAM-R measurements were used. (b) An example of the optimal initial a values derived for various elevation angles in RHI scan data. The red and blue lines represent the optimal initial a values from ZDR and ΦDP, respectively.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 70 70 13
PDF Downloads 41 41 0

Estimation of X-Band Polarimetric Radar Attenuation and Measurement Uncertainty Using a Variational Method

View More View Less
  • 1 Institute of Atmospheric Physics, National Central University, Jhongli City, Taiwan, and Advanced Study Program, National Center for Atmospheric Research,* Boulder, Colorado
  • | 2 Earth Observing Laboratory, National Center for Atmospheric Research, Boulder, Colorado
  • | 3 Institute of Atmospheric Physics, National Central University, Jhongli City, Taiwan
© Get Permissions
Full access

Abstract

A variational algorithm for estimating measurement error covariance and the attenuation of X-band polarimetric radar measurements is described. It concurrently uses both the differential reflectivity ZDR and propagation phase ΦDP. The majority of the current attenuation estimation techniques use only ΦDP. A few of the ΦDP-based methods use ZDR as a constraint for verifying estimated attenuation. In this paper, a detailed observing system simulation experiment was used for evaluating the performance of the variational algorithm. The results were compared with a single-coefficient ΦDP-based method. Retrieved attenuation from the variational method is more accurate than the results from a single coefficient ΦDP-based method. Moreover, the variational method is less sensitive to measurement noise in radar observations. The variational method requires an accurate description of error covariance matrices. Relative weights between measurements and background values (i.e., mean value based on long-term DSD measurements in the variational method) are determined by their respective error covariances. Instead of using ad hoc values, error covariance matrices of background and radar measurement are statistically estimated and their spatial characteristics are studied. The estimated error covariance shows higher values in convective regions than in stratiform regions, as expected. The practical utility of the variational attenuation correction method is demonstrated using radar field measurements from the Taiwan Experimental Atmospheric Mobile-Radar (TEAM-R) during 2008’s Southwest Monsoon Experiment/Terrain-Influenced Monsoon Rainfall Experiment (SoWMEX/TiMREX). The accuracy of attenuation-corrected X-band radar measurements is evaluated by comparing them with collocated S-band radar measurements.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Dr. Jothiram Vivekanandan, National Center for Atmospheric Research, 3450 Mitchell Ln., Boulder, CO 80301. E-mail: vivek@ucar.edu

Abstract

A variational algorithm for estimating measurement error covariance and the attenuation of X-band polarimetric radar measurements is described. It concurrently uses both the differential reflectivity ZDR and propagation phase ΦDP. The majority of the current attenuation estimation techniques use only ΦDP. A few of the ΦDP-based methods use ZDR as a constraint for verifying estimated attenuation. In this paper, a detailed observing system simulation experiment was used for evaluating the performance of the variational algorithm. The results were compared with a single-coefficient ΦDP-based method. Retrieved attenuation from the variational method is more accurate than the results from a single coefficient ΦDP-based method. Moreover, the variational method is less sensitive to measurement noise in radar observations. The variational method requires an accurate description of error covariance matrices. Relative weights between measurements and background values (i.e., mean value based on long-term DSD measurements in the variational method) are determined by their respective error covariances. Instead of using ad hoc values, error covariance matrices of background and radar measurement are statistically estimated and their spatial characteristics are studied. The estimated error covariance shows higher values in convective regions than in stratiform regions, as expected. The practical utility of the variational attenuation correction method is demonstrated using radar field measurements from the Taiwan Experimental Atmospheric Mobile-Radar (TEAM-R) during 2008’s Southwest Monsoon Experiment/Terrain-Influenced Monsoon Rainfall Experiment (SoWMEX/TiMREX). The accuracy of attenuation-corrected X-band radar measurements is evaluated by comparing them with collocated S-band radar measurements.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Dr. Jothiram Vivekanandan, National Center for Atmospheric Research, 3450 Mitchell Ln., Boulder, CO 80301. E-mail: vivek@ucar.edu

1. Introduction

The C- and X-band radars are preferred choices for mobile ground-based and airborne deployments because the radar antenna is much smaller in size than the S-band antenna for a specified beamwidth. The C- and X-band radars are more sensitive when detecting lighter precipitation than is the S-band radar. However, measurements with the higher-frequency radars are more susceptible to attenuation. The amount of attenuation is proportional to the intensity of the precipitation. For example, horizontal X-band copolarization reflectivity ZHH and differential reflectivity ZDR are usually underestimated due to attenuation of the radar signal as it propagates through precipitation. Therefore, attenuated X-band radar measurements must be corrected for attenuation effects before retrieving rain-rate and microphysical information from them.

In the case of single-polarization radar, an empirical relation between the one-way specific attenuation of the horizontally polarized signal AH and the reflectivity is used for attenuation correction (Hitschfeld and Bordan 1954). The attenuation correction based on the AHZHH empirical relation is unstable and sensitive to calibration error in ZHH. On the other hand, the accuracy and stability of the attenuation correction scheme is vastly improved when dual-polarization observations are used (Bringi et al. 1990).

In the case of rain, the specific differential propagation phase (KDP; ° km−1) derived from the gradient of differential phase shift (ΦDP; °) along the beam is proportional to rain intensity (Sachidananda and Zrnić 1986; Ryzhkov and Zrnić 1996; Scarchilli et al. 1993). One-way specific attenuation (AH; dB km−1) and differential attenuation (ADP; dB km−1) are nearly linearly proportional to KDP (Bringi et al. 1990; Jameson 1991). As KDP is unaffected by attenuation, radar system bias due to changes in transmit power, and antenna and receiver gain factors, it is more commonly used for estimating attenuation. Noting that ΔΦDP is twice the sum of KDP over a specified range, the corresponding two-way total attenuation (Ah; dB) and differential attenuation (Ahv; dB) can be estimated in the ΦDP-based method (Bringi et al. 1990; Anagnostou et al. 2006) to be approximately
e1
e2
The coefficients α and β relate ΔΦDP to Ah and Ahv, respectively. However, the dynamic range of coefficients of α and β is too large to be accurately derived (Bringi and Chandrasekar 2001). Variations in α and β are caused by natural variations in the drop size distribution (DSD), drop shape oscillation, and temperature, and also they are insensitive to the ΦDP of drizzle and cloud droplets in the rain volume at these wavelengths.

Polarization radar measurements, namely differential reflectivity, provide additional information about precipitation and allow for better microphysical characterization of hydrometeors (Vivekanandan et al. 1999; Chandrasekar et al. 2006). The ZDR provides information about the mean drop shape and mean size of DSD. Therefore, the inclusion of ZDR in the attenuation correction scheme has the potential to reduce the uncertainty in estimated attenuation.

The majority of attenuation correction techniques use the AHKDP or Ah–ΔΦDP relations, and ZDR is used as a constraint for estimating the differential attenuation. Testud et al. (2000) modified the rain-profiling algorithms of the spaceborne Tropical Rainfall Measuring Mission rain radar and proposed the ZPHI algorithm, using ΦDP as an external constraint. Bringi et al. (2001) further improved the ZPHI algorithm and demonstrated its applicability to C-band polarimetric radar attenuation correction. This technique effectively recalibrates the attenuation-corrected ZDR to the expected 0-dB value in the regions of drizzle (Smyth and Illingworth 1998). It uses an optimization approach to determine linear coefficients that maximize the consistency between estimated path-integrated attenuation and ΔΦDP profiles. Gorgucci and Chandrasekar (2005) adopted the method for the X band and evaluated the error structure of the attenuation correction methodologies using the differential phase measurement. Lim et al. (2011) implemented the algorithm to a network of X-band radars.

Park et al. (2005a,b) adopted the algorithm from Bringi et al. (2001) for X-band radar measurements and provided a detailed description of the AHKDP and AHADP nonlinear relations for X band based on T-matrix scattering calculations (Barber and Yeh 1975). The relations are derived as a power-law expression where the proportionality constant is sensitive to DSD, temperature, and drop shape. The exponent in the relation is mostly sensitive to the temperature of the raindrop. The proportionality constant in the AHADP relation is less sensitive to DSD and the exponent in the relation is sensitive to temperature, as in the AHKDP relation. Snyder et al. (2010) demonstrated that the attenuation correction based on Park et al. (2005b) had better levels of agreement with S-band radar in a tornadic supercell case. Gorgucci et al. (2006) developed an alternative method based on the self-consistency principle for estimating both attenuation and differential attenuation with less than 10% bias and 15% error.

A variational algorithm proposed by Hogan (2007) considers not only the beam-to-beam dependency of α and β for attenuation correction but also the gate-to-gate dependent variability of DSD inferred from ZDR. The variational method for S-band radar observations (Furness 2005; Hogan 2007) demonstrated the usefulness of the method for detecting rain–hail regions and the estimation of attenuation and rain rate from S-band polarization radar. In this paper, a modified version of the variational technique for estimating attenuation in the X band that uses both ΦDP and ZDR is presented. The major objective of this paper is to investigate if relatively large values of attenuation at X band can be estimated by the variational method.

The variational algorithm uses a 1D retrieval method along the radar beam. The scheme is based on a linear estimation theory in which the background information and the observations are weighed proportional to the inverse of their respective error covariances. It is not uncommon to use predetermined error covariance matrices (Hogan 2007), but they fail to characterize the errors of a specific rain event. The performance of the variational scheme depends on the accuracy of the error covariance matrices of background terms and the observations.

Nevertheless, the quality of the polarimetric measurements is influenced by a range of factors under various measurement conditions. Ryzhkov (2007) shows that significant uncertainty or error in ZDR up to ±0.7 dB, and in ΦDP up to ±10°, can be introduced in the presence of nonuniform beam filling. Hubbert et al. (2010a,b) demonstrate in the case of simultaneous horizontally and vertically transmitted mode orientations that scatters cause cross coupling between horizontally and vertically polarized signals. Since all of the above-described errors vary spatiotemporally, it is necessary to use a case-dependent measurement error covariance matrix rather than a predetermined value as in the case of a standard variational algorithm.

The mathematical formulation of the variational method and estimation of the error covariance matrices are described in section 2. In section 3, an observing system simulation experiment (OSSE) evaluation of the variational algorithm is used to demonstrate the feasibility of the variational method at X band. The attenuation correction technique was applied to X-band radar measurements and the corrected X-band radar measurements were compared to collocated S-band radar measurements in section 4. An analysis of error characteristics of ZDR and ΦDP measurements is presented in section 5. Results are summarized in section 6.

2. Variational method and error covariance matrices

a. Variational method

The variational method consists of two major components: 1) the minimization of a cost function J and 2) a forward model . The essence of the variational algorithm is to derive an optimal set of state variables with the lowest sums of discrepancy between observations and forward model predictions. The forward model predicts radar observations as a function of reflectivity and rain intensity (Hogan 2007). The forward model for X-band radar is derived using a rigorous electromagnetic scattering and propagation model that is capable of predicting a suite of radar observables (ZDR and KDP) and the specific attenuation (AH and ADP) for a specified rain DSD (Vivekanandan et al. 1991; Bringi and Chandrasekar 2001). The forward model assumes a gamma DSD model (Ulbrich 1983) of the form
e3
The gamma parameters, namely, N0 (intercept parameter; mm−1 m−3), μ (shape parameter; dimensionless), and D0 (median volume diameter; mm), are the inputs into the electromagnetic scattering and wave propagation model for computing ZHH, ZDR, KDP, AH, and ADP and for estimating the rainfall rate R. The values of gamma DSD parameters are varied for simulating the natural variability of DSD based on observational statistics from the 2D-video disdrometer (2DVD; Schönhuber et al. 1997) of National Central University (NCU) in Taiwan. The D0 varied from 0.1 to 6 mm with an interval of 0.01 mm and the values of log10(N0) varied from 1.0 to 16.0 with an interval of 1.0; meanwhile, μ is fixed at 5.

In this paper, the forward model is simplified using the results from rigorous scattering model calculation. In a power-law relation between reflectivity and rainfall rate (R), Z = aRb, where parameters a and b represent variations in the DSD (Steiner et al. 2004; Steiner and Smith 2000). Since the temperature dependency of a and b is relatively minor, it is not considered in this research. Large μ values (μ = 5) reduce the sensitivity of the exponent b to DSD in the ZR relation (Steiner et al. 2004). Further, the multiplicative parameter a is much more sensitive to DSD variations than is the exponent parameter b (Bringi and Chandrasekar 2001; Hogan 2007). Hogan (2007) also indicated the variability of parameter b is small; therefore, the exponent b is fixed and it is equal to 1.5 (Steiner and Smith 2000; Doelling et al. 1998; Hogan 2007) based on observational statistics from NCU 2DVD. This particular type of ZR relation represents rain DSDs controlled by variations in both number concentration and characteristic size (Steiner et al. 2004; Steiner and Houze 1997). Thus, the forward models use the ratio Z/R as a proxy for variation in DSD. Parameter a of ZR is the target variable since the exponent b is fixed. Typically, the natural logarithm of a [i.e., ln(a)] is estimated to avoid retrieving an unrealistic negative value of a. Hereinafter, the symbol ã represents ln(a).

The forward models predict ZDR, KDP, AH, and ADP as a function of Z/R, which are then smoothed by curve fitting and organized into a precalculated lookup table for values of Z/R between 100.6588 and 104.3045, as shown in Fig. 1. The error in the gamma DSD-based forward model is examined by calculating the root-mean-square error (RMSE) defined as
e4
In this case, y represents the parameters of ZDR, KDP, AH, and ADP. The superscript true represents the parameters derived by rigorous electromagnetic scattering and propagation model simulation from 5 yr of NCU 2DVD-measured DSDs. The m represents the number of DSDs. The superscript est. represents the parameters derived from the forward model with Z/R as input. The values of RMSE (σ) are fairly low for all parameters, as shown in Table 1. Nevertheless, the actual uncertainty in the forward model should be larger, since the rain microphysics, namely, the DSD model, axis ratio of raindrops, temperature, and maximum size of raindrop, exhibit natural variability and they can only be approximated as mean values in the forward model. The uncertainty in the forward model due to natural variability in rain microphysics is considered to be part of the observation error in this research.
Fig. 1.
Fig. 1.

The X-band forward models based on a simplified electromagnetic scattering and propagation model: (a) ZDR, (b) KDP/ZHH, (c) AH/ZHH, and (d) ADP/ZHH, all vs log(ZHH/R). Gamma raindrop size distributions (RSDs) and measured video disdrometer DSDs were used as input into the electromagnetic scattering and propagation model. The black lines represent the forward models calculated using a gamma RSD with μ = 5 at 20°C. The gray dots represent the rigorous scattering model simulation corresponding to discretely measured DSDs based on the 5 yr of data from NCU 2DVD in northern Taiwan.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

Table 1.

RMSE (σ) of the forward model. The RMSE of ΦDP is derived using 400 gates with 0.125-km gate spacing.

Table 1.
The cost function (J) is composed of three components, namely, , as shown:
e5
They are functions of ZDR, ΦDP, and an optimal target variable xk along the ray at gate k. The contribution to the cost function from each component is weighted by the corresponding variance (σ2). An array of radar measurements, n gates, along a radar beam is processed one at a time. The background term ãbackground is included in the cost function as an “a priori” so that an appropriate value of ã is obtained in the regions of light rain where ZDR and KDP are too small or where no useful polarimetric radar measurements are available. These regions may have nonnegligible cumulative attenuation.
In the presence of attenuation, the intrinsic ZHH and ZDR (dB) are related to the observed reflectivity and differential reflectivity by the following equations:
e6
e7
The Ah and Ahv are the two-way path-integrated AH and ADP, respectively. The two-way propagation phase ΦDP in degrees at range r2 can be derived from the ΦDP at range r1 and integration of the KDP between the range r1 and r2 as
e8
An optimal estimation of the target variable ã was obtained via minimizing the cost function. The Gauss–Newton iteration method is used (Rodgers 2000; Hogan 2007). The matrix form of the iterative solution is
e9
where
e10
and
e11
In the above equations, xi represents the target variable ã at the ith iteration, y represents the observations ZDR and ΦDP, the subscript “background” represents the background values of ã, the and matrices represent the error covariance matrices of the observation (ZDR and ΦDP) and the background value, respectively. The and are diagonal matrices with fixed values. The matrix represents the forward model and the represents the Jacobian matrix, a matrix containing the partial derivative of each observation with respect to ã. The derivation of can be found in Hogan (2007).

The flowchart in Fig. 2 shows various steps in the variational attenuation retrieval scheme. After removing the system bias in ZHH and ZDR, as well as the nonmeteorological signal, the minimization process starts with a first guess of ã and the AH is predicted via the forward model. As in Hogan (2007), the measured ZHH is corrected cumulatively using the AH (step 3). Using the corrected ZHH, the forward model predicts intrinsic ZDR, ADP, and KDP (step 4). The intrinsic ZDR is reduced by the amount of ADP. By integrating KDP over the range, we are able to obtain ΦDP (step 5). The corresponding and adjusted ã for the next iteration are calculated in steps 7 and 8. The above-described steps are repeated until the solutions converge satisfactorily (step 6). A χ2 test is used to determine convergence. It usually requires 4–10 iterations for convergence. All of the computations are performed at range gate resolution and the measured data are not filtered or averaged over a range.

Fig. 2.
Fig. 2.

Flowchart describing various numbered steps (at the top-right box corners) in the variational attenuation correction method. The dashed box represents a complete loop of the Gauss–Newton minimization process. See section 2a for a detailed description.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

Predetermined diagonal matrices with fixed variance (σ2) values of and were used in Hogan (2007). However, the values of and are known only approximately as uncertainties in radar measurements and the natural variabilities of DSDs are unknown in practice. In X-band radar measurements with higher path-integrated attenuation, appropriate and have to be specified objectively to obtain satisfactory results by the variational method. In the following two subsections, procedures for estimating and are described.

b. Estimating the background error covariance matrix

In practice, the background term is usually obtained from the climatological mean value of ã (Hogan 2007). Hence, the corresponding fixed value of the diagonal error covariance matrix of the background term () is determined from its respective variance from the background term. In this research, the value of the background term ãbackground is 5.5 and the variance is 0.57. These values were obtained from NCU 2DVD data in Taiwan. However, actual values of the background term and the associated background error covariance can change from event to event as a function of sampling volume due to the natural variability of the DSDs.

For estimating an optimal , predicted and measured radar variables are compared using a discrete approximation method. In this method the variational algorithm is performed for a range of values between 0.03 and 1.26. The range of includes the mean theoretical value of 0.57 inferred from climatological data. A lower value of weights the background term more strongly than the observations and vice versa. To obtain the best predictions of ZDR and ΦDP compared to the radar observations, the “optimal” is determined by the lowest sum of the first two terms of the cost function (i.e., and ).

There are three benefits of varying over a wide range: 1) the assumption of an ad hoc background error covariance () is avoided, 2) a reasonable outcome is guaranteed in cases of either poor observational data or if the forward model fails to represent the particular DSD, and 3) it compares relative weights between the background and measurements in all of the expected measurement scenarios.

c. Estimating the observation error covariance matrices

Instead of using a predetermined (Hogan 2007), a statistical diagnostic method is used for estimating the values of of ZDR and ΦDP (Desroziers et al. 2005). The diagnostic approach combines observation, analysis, and background information for tuning observation and background errors adaptively using innovation vectors. Innovation vectors of the background and analysis from observation are defined as
e12
and
e13
In the above equations, x represents the ã of ZR. The superscripts ana., bg., and obs. represent the analyses (results from variational algorithm), background terms (i.e., climatological ã of the ZR relation), and the observation from radar, respectively. As described in section 2a, is the forward operator. Variable y represents the ZDR and ΦDP measurements. The observation, background, and analysis errors are εobs., εbg., and εana., respectively. As the variational method processes radar beams sequentially, the innovation vectors of the background and analysis of ZDR for each beam are
e14
and
e15
and vice versa for ΦDP, where k is the available gates along a radar beam.
In general, background and analysis errors of observations are unbiased and mutually uncorrelated to each other statistically. Then, the mean value of the diagonal terms of the cross product of the innovation vector of background and analysis is the observation error variance (σ2):
e16
Therefore, diagonal terms of for each beam are obtained by calculating of ZDR and ΦDP. The is then obtained as a 2k × 2k matrix:
e17

There are two advantages to using the statistical diagnostic method: 1) predetermined observation error covariance matrices are no longer needed and 2) the true spatial distribution characteristics of radar measurement error are represented in . The observation error matrix takes into account uncertainty in radar measurements due to Mie scattering, the backscatter phase, and artifacts caused by gradients in reflectivity and nonuniform beam filling.

As suggested by Li et al. (2009), the proper values of and have to be derived simultaneously. The procedure shown in Fig. 2 for attenuation estimation is modified for estimating and adaptively. The modified procedure includes the discrete approximation and statistical diagnostic method is shown in appendix A.

3. The observing system simulation experiments

The OSSE is designed to demonstrate the feasibility of the variational algorithm for estimating rain attenuation and rainfall rate at X band. The intrinsic polarimetric observations, ZHH, ZDR, ΦDP, the attenuation, AH, ADP, and rainfall rate were simulated rigorously from 6-min-averaged DSDs of 5 yr of NCU 2DVD data. There are 6880 averaged DSDs available for OSSE studies. The procedures for modeling polarimetric observation beam are introduced in appendix B.

A set of DSDs with a maximum rainfall rate of 70.0 mm h−1 and that have a wide range in the coefficient a of ZR were selected to demonstrate the simulated radar measurements of a representative precipitation system with natural variations in DSD, as shown in Fig. 3. The Ah and Ahv along the beam were 12.07 and 1.44 dB, respectively. The ΦDP was obtained by integrating KDP along the range (Fig. 3c). The OSSE was performed using two types of data: 1) assuming no error in measurements and 2) assuming observation error in ZHH, ZDR, and ΦDP. The configurations of sensitivity studies are summarized in Table 2.

Fig. 3.
Fig. 3.

The simulated X-band polarimetric radar observables for measured DSDs: (a) ZHH and AH, (b) ZDR and Ahv, and (c) ΦDP, R, and a, all vs distance. The DSDs were obtained from 5 yr of NCU 2DVD measurements. The modeled polarimetric observations without observation error were arranged along a radial with a gate resolution of 0.125 km. The total attenuation (Ah) and total differential attenuation (Ahv) for this 50-km beam are 12.07 and 1.44 dB, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

Table 2.

Configurations of the OSSEs. Each simulation contains 600 modeled polarimetric observation beams with the following values of standard errors (σ) of ZHH, ZDR, and ΦDP.

Table 2.

a. The OSSE without observation error

The variational algorithm was applied to the modeled X-band polarimetric data, as “no error” in Table 2. No background term was used in this case since the simulated data have no observational error. In theory, contains radar measurement error and forward model error. In the absence of observation error, the for ZDR and ΦDP are diagonal with forward model error values of 0.05 dB and 0.20°, respectively (see section 2a; Table 1). Performance of the variational algorithm was examined as a function of range with gradually increasing Ah. The purpose of this sensitivity study is to investigate the performance of the variational algorithm for a range of Ah.

The mean RMSE as defined in Eq. (4) was calculated for Ah, Ahv, and the logarithm of rainfall rate (dBR), for both variational and ΦDP-based attenuation correction schemes, from 600 modeled polarimetric radar beams. The rainfall rate of the variational algorithm was derived from ZR using the estimated a and fixed b. The rainfall rate from the ΦDP-based attenuation correction scheme was derived from RZHHZDR as
e18
Even though regression coefficients α and β for attenuation estimation, and a1 and b1 for rainfall rate estimation were specifically tuned to the DSD data, RMSEs of dBR, Ah, and Ahv obtained from the ΦDP-based schemes are higher than of variational algorithm-based results as shown in Fig. 4. The RMSEs in the ΦDP-based technique are larger as expected because DSD fluctuations were not characterized. The RMSE from the ΦDP-based technique deteriorates as the maximum range increases due to accumulated DSD variations. The values of RMSE increase from 0.7 to 2.8 dB for dBR, from 0.08 to 0.21 dB for Ah, and from 0.02 to 0.05 dB for Ahv.
Fig. 4.
Fig. 4.

(a) RMSEs of attenuation (Ah) and differential attenuation (Ahv) derived from variational (var.) and ΦDP-based methods vs distance. (b) As in (a), but for dBR; the values of dBR are defined as 10 log10(R).

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

As no error was added to the observations, the RMSE of the variational algorithm is primarily caused by residual differences between the forward model based on simplified gamma DSD and measured DSD, as shown in Fig. 1. Despite the use of a simplified forward model, the lower RMSEs from variational-based algorithms indicate the simplified forward model has the capability to predict the polarimetric measurements and attenuation with sufficient accuracy.

b. The OSSE with observation error

One of the advantages of the variational algorithm is that it is relatively less sensitive to observation error (Hogan 2007). The background term in the cost function could be used to constrain the result in the presence of observation error. Two sensitivity studies, “variable error in ZDR” and “variable error in ΦDP” in Table 2, investigate the impacts of ZDR and ΦDP observation error on the variational algorithm. Specified amounts of Gaussian standard errors (σ) were added to the simulated intrinsic polarization observables. The configurations of sensitivity studies with various values of standard errors (σ) of ZHH, ZDR, and ΦDP are summarized in Table 2.

The procedure described in appendix A was used for estimating optimal and . The discrete approximation was first performed with prescribed observation error covariance matrices of ZDR and ΦDP for estimating the first-guess . Even though ZDR and ΦDP error are varied between 0.1 and 1.5 dB and 1.7° and 2.7°, respectively, the initial values of of ZDR and ΦDP were assumed to be 0.1 dB and 1.0° respectively to verify whether the statistical diagnostic method is able to recover the proper values of . An example demonstrating the progression of convergence to known errors of ZDR and ΦDP with standard deviations (σ) of 1.1 dB and 2.5° is shown in Fig. 5. The diagnosed of ZDR and ΦDP converge in four iterations to the specified error values.

Fig. 5.
Fig. 5.

The estimated values of the standard deviation (σ) of the measurement error of ZDR (blue solid line) and ΦDP (red solid line) as a function of the iteration number. The true values are shown as blue and red dashed lines, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

In the case of the variable error in ZDR, optimal values of ZDR are greatly improved from the initial assumption of 0.1 dB, despite the ZDR error being varied between 0.1 and 1.5 dB, as shown in Fig. 6a. The diagnosed of ZDR are close to the 1:1 line, as desired. Similarly, the diagnosed of ΦDP error of variable error in ΦDP case are improved from the initial assumption of 1.0° and they are comparable to true ΦDP error in Fig. 6b. These results indicate that the statistical diagnostic method estimates the optimal values of of ZDR and ΦDP fairly well. This methodology is used in the analysis of measurements, as variances of measurements are unknown.

Fig. 6.
Fig. 6.

(a) The relation between the estimated standard deviation of the measurement error of ZDRest.ZDR) obtained from step 8 in Fig. A1 and the true value (σtrueZDR) shown as a red line with the dashed line being a 1:1 line. (b) As in (a), but for ΦDP.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

In Fig. 7a, the values of the RMSEs of Ah and Ahv from the variational and ΦDP-based algorithms are shown as a function of ZDR observation error from the variable error in ZDR case. Since the ΦDP-based algorithm does not include ZDR, the RMSEs of Ah and Ahv from the ΦDP-based algorithm remain nearly constant as expected with values of 0.5 and 0.08 dB, respectively. While the corresponding values for the RMSEs of the variational algorithm are 0.1 and 0.02 dB. The RMSEs of dBR are also lower for the variational algorithm, as shown in Fig. 7b. Even in the presence of typical observation errors in ZHH, ZDR, and ΦDP, the variational algorithm provides better estimations of dBR, AH, and ADP.

Fig. 7.
Fig. 7.

As in Fig. 4, but vs the measurement error in ZDR.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

The OSSE studies have shown that the variational method can provide more accurate estimations of attenuation and rainfall rate, even in the presence of observational error in ZHH, ZDR, and ΦDP. Moreover, errors in ZDR and ΦDP can be estimated accurately using the statistical diagnostic method. In the next section, the variational method will be applied to field measurements from an X-band radar.

4. Applications of the attenuation correction scheme to field measurements

The variational scheme was applied to actual X-band field measurements during the Southwest Monsoon Experiment/Terrain-Influenced Monsoon Rainfall Experiment (SoWMEX/TiMREX; Jou et al. 2011). The selected X-band polarimetric radar measurements collected by the Taiwan Experimental Atmospheric Mobile-Radar (TEAM-R) were examined. The specifications of TEAM-R are listed in Table 3. To verify the performance of the variational attenuation correction at X band, the collocated measurements from a synchronized RHI scan of the TEAM-R and the National Center for Atmospheric Research (NCAR) S-band dual-polarization Doppler radar (S-Pol) were analyzed.

Table 3.

Engineering specifications of TEAM-R.

Table 3.

Before applying the variational scheme, the raw TEAM-R radar data were quality controlled by filtering ground clutter and mixed-phase precipitation using ρHV (copolar correlation coefficient). Usually regions with ρHV < 0.7 correspond to either a mixed phase or ground clutter (Ryzhkov et al. 1994). System biases in ZHH estimated by Vivekanandan et al. (2003) and in ZDR estimated by vertical-pointing data were also removed before applying the variational correction. The system bias in ZHH and ZDR should be known within accuracies of ±0.5 and ±0.2 dB, respectively, to ensure the variational method outperforms the ΦDP-based method according to OSSE studies (not shown). The melting level was about 4.6 km MSL based on sounding data. Therefore, radar data above 3.5 km MSL were not processed in order to avoid ice-phase and mixed-phase precipitation, as they are not accounted for in the forward model used in this study. The elevation angle dependence effect on polarimetric measurements ZDR, KDP, and ADP was considered. The elevation angle dependency correction suggested by Bechini et al. (2008) was applied to the forward model.

Convergence of the target variable ã to an optimal value is achieved using the Gauss–Newton iteration method. Since attenuation at X band is much higher than at S band, a poor first guess of ã leads to physically unrealistic attenuation correction for ZHH and causes numerical error. It is important to avoid a poor first guess of ã. The procedure for deriving a proper first guess of a in ZR is described in appendix C.

An initial prescribed is required in the statistical diagnostic method for estimating optimal as discussed in section 2c. The values are determined by calculating the standard deviation (σ) of actual radar measurements. The σ of ZDR was derived from the vertical-pointing data in a stratiform light rain. The σ of the ΦDP observation error was computed using measurements from a light stratiform rain region where there was nearly no ΦDP increment as a function of range. The values of σ for ZDR and ΦDP are 0.18 dB and 2.4°, as shown in Table 4.

Table 4.

The standard deviation (σ) of radar measurement errors estimated using statistical analysis and statistical diagnostic methods.

Table 4.

a. Analysis of a single radar beam

Performance of the attenuation correction method for a single beam is examined by comparing corrected TEAM-R and S-Pol measurements. Figure 8 shows ZHH, ZDR, and ΦDP for a radar beam at a 3.19° elevation angle from X- and S-band measurements. Higher values of ZHH and ZDR, as well as a large gradient in ΦDP measurements (in about 30° increments), indicate convective precipitation between 5 and 13 km from radar. A stratiform precipitation system with flat ΦDP exists between the ranges of 15 and 35 km. The raw X-band ZDR values in the stratiform system varied between −2 and −1 dB due to cumulative differential attenuation.

Fig. 8.
Fig. 8.

Radar data at a 3.19°-elevation angle from TEAM-R and S-Pol radar data: (a) ZHH, (b) ZDR, and (c) ΦDP, all vs distance. The blue lines represent the measured raw values from TEAM-R and the red lines represent the attenuation-corrected ZHH and ZDR, and predicted ΦDP based on the variational scheme. The black lines in (a) and (b) represent averaged measurements from the S-Pol radar. The distance between the TEAM-R and NCAR S-Pol radars was 50 m.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

For this radar beam, the variational scheme estimated Ah and Ahv as 11.1 and 1.5 dB, respectively. The intrinsic difference in ZDR observations between X- and S-band wavelengths was removed by converting S-band ZDR to X-band ZDR (Chandrasekar et al. 2006). The attenuation-corrected ZHH and ZDR from TEAM-R are in good agreement with the ZHH and ZDR measurements from S-Pol. Good agreement among the ZDR results in stratiform regions confirms that the differential attenuation was estimated correctly. In the convective region around 11-km range the significantly attenuated TEAM-R ZHH and ZDR are satisfactorily recovered as they compare favorably with the S-Pol measurements.

Figure 8c shows ΦDP as measured by TEAM-R and predicted ΦDP from the variational scheme. The agreement between the predicted and measured ΦDP confirms the minimization procedure converged and the ΦDP was predicted correctly, but there are differences at individual range gates. The measured ΦDP exhibited significant fluctuations in both the convective and stratiform regions. In the convective region, the fluctuation was caused by the strong reflectivity gradient (Ryzhkov 2007) and backscattering Mie phase shift (δ). In the stratiform region, the fluctuation was caused by a low signal-to-noise ratio (SNR). Significant fluctuations in measured ΦDP lead to poor estimation of attenuation by the ΦDP-based algorithm. Due to the strong reflectivity gradient, the measured ΦDP shows a localized minimum between 9 and 9.5 km. However, the predicted ΦDP does not exhibit the local minimum. The variational method predicted a smooth ΦDP with no fluctuation. This predicted ΦDP from the variational scheme was derived from a physical approach rather than a mathematical approach, namely, filtering or smoothing of measured ΦDP (Hubbert and Bringi 1995). The smoothing technique degrades the spatial resolution and gradient in ΦDP. Since the predicted ΦDP is free from measurement noise and backscattering Mie phase shift (δ), it is straightforward to estimate KDP from predicted ΦDP without losing any spatial resolution.

b. Analysis of an RHI scan

The measurements of ZHH and ZDR from synchronized RHI scans of S-Pol and TEAM-R are shown in Fig. 9. Figure 9b shows the equivalent intrinsic X-band ZDR derived from the measured S-band ZDR (Chandrasekar et al. 2006). Figures 9c and 9d show that the measurements from TEAM-R were severely underestimated due to the attenuation effect. The attenuation-corrected TEAM-R measurements, ZHH and ZDR in Figs. 9e and 9f, are in good agreement with the S-Pol measurements in Figs. 9a and 9b. The strength of the rear part of the convection system (10–12 km) and stratiform precipitation (15–35 km) were enhanced by 10 dB. The negative ZDR values at the stratiform region between 15 and 35 km were brought back to 0 dB, which is more appropriate for a drizzle region. In the convective precipitation region, minor discrepancies between attenuation-corrected X-band ZHH and ZDR and S-band radar measurements can be noticed. Considering the beamwidth of TEAM-R is 1.25°, this discrepancy is due to three possible reasons: 1) biases in ZHH and ZDR in the regions of large gradients due to partial beam filling (Ryzhkov 2007) limited the performance of the variational algorithm in estimating the attenuation accurately, 2) the differences in sample volumes between the TEAM-R and S-Pol radars amplified the ZHH and ZDR differences between the X- and S-band radar measurements, and 3) the rapidly evolving convective precipitation system might have also contributed to the majority of the difference in the ZDR fields as the X- and S-band radar measurements were offset by 10 s.

Fig. 9.
Fig. 9.

RHI scans of (left) ZHH and (right) ZDR measurements from NCAR S-Pol and TEAM-R. (a)–(d) Intrinsic data measured during TiMREX 2008 with attenuation correction for S-Pol and system bias correction methods for TEAM-R applied; a threshold of ρHV < 0.7 was applied to the X-band measurements, and no threshold was applied to the S-band measurements. Further, the intrinsic difference in the ZDR observations between X-band and S-band wavelengths was removed by converting S-band ZDR to X-band ZDR (Chandrasekar et al. 2006). (e),(f) The variational attenuation-corrected ZHH and ZDR from TEAM-R, respectively. The solid lines represent the beams with elevation angles of 5.5° and 8.1°, respectively; the dashed line represents a beam of 3.19°.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

The predicted ΦDP and measured ΦDP of the variational method are in good agreement, as shown in Fig. 10. The fluctuation in predicted ΦDP is lower than in measured ΦDP. The variational scheme underestimates ΦDP at the lowest elevation angle when compared to the measured values. The elimination of the ZHH and ZDR measurements at low-elevation beams with poor data quality due to ground clutter contamination underestimated the predicted ΦDP. In the case of noisy measurements, the variational method had to rely on the background term. However, the background term (i.e., climatology-based ã) is not good enough to represent local DSDs in this case. Underestimation of predicted ΦDP can also be noticed between 5.5° and 8.1° in Fig. 10a, and also as stripes in the corrected Figs. 9e and 9f in the stratiform precipitation region.

Fig. 10.
Fig. 10.

(a) The predicted ΦDP from the variational method and (b) the measured ΦDP. The solid lines represent beam elevation angles of 5.5° and 8.1°, respectively, and the dashed lines represent beams of 3.19°.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

To further quantify the differences between measured S-Pol ZDR and attenuation-corrected ZDR measurements, average values in convective and stratiform regions were computed. As shown in Table 5, the values from the variational method are closer to those of S-Pol than those of the ΦDP-based method in the convective and stratiform regions. In this case the ΦDP-based method used the value of β as 0.036, which was computed from SoWMEX/TiMREX disdrometer measurements. A value of 0 dB for ZDR in stratiform is widely used as a physical constraint to reduce the uncertainty in the attenuation estimation by the ΦDP-based method (Smyth and Illingworth 1998; Bringi et al. 2001; Park et al. 2005a,b). Nevertheless, the ZDR value was not zero in the stratiform region based on S-Pol measurements in this case. The satisfactory performance of the variational method in this case reinforces its advantage, as no such constraint is required for the variational method.

Table 5.

Mean values of S-Pol and attenuation-corrected ZDR from TEAM-R are listed. Mean values of ZDR were computed for convective and stratiform regions.

Table 5.

c. KDP–AH and KDP–ADP relations

A scatterplot showing attenuation (AH and ADP) and KDP is presented in Fig. 11. Attenuation values as a function of KDP were obtained using the variational scheme. The values of AH and ADP estimated by the variational scheme showed broader scatter compared to the ΦDP-based method represented by solid and dash lines in Fig. 11. The data were categorized into two subgroups depending on the associated ZDR values. For a specified KDP, AH and ADP are higher when ZDR is larger. This feature is consistent with the results from the electromagnetic scattering and wave propagation simulation (Carey et al. 2000). The results indicate that the variational algorithm can reproduce not only the broader distribution of the AH and ADP, but also the proper values of the AH and ADP that are consistent with the natural DSD. The improved accuracy in the variational method is mainly due to the usage of both ΦDP and ZDR measurements for the estimation of AH and ADP rather than using only ΦDP.

Fig. 11.
Fig. 11.

The ΦDP attenuation correction with coefficients α = 0.374 and β = 0.052 for attenuation (AH: black solid) and differential attenuation (ADP: black dashed) vs KDP. The gray dots represent the variational algorithm derived attenuation (AH), differential attenuation (ADP), and KDP. The corresponding AH (ADP) and KDP with ZDR values between 0–1 dB and between 3–4 dB are shown by red and blue dots, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

5. Characteristics of ZDR and ΦDP measurement error

The results in previous sections show the variational algorithm produces better results than the ΦDP-based algorithm. The major reason for this is that the variational method uses both ZDR, ΦDP and climatological values of a in ZR. Moreover, the error covariance matrices of background term () and measurements of ZDR and ΦDP () were obtained objectively by the discrete approximation and statistical diagnostic method for each beam, rather than using the predetermined or a priori error covariance matrices.

It is interesting to note that the standard deviations of the ΦDP and ZDR observations vary as a function of elevation angle, as shown in Fig. 12. These estimated values are consistently higher for elevation angles < 0.5° due to ground clutter contamination, as expected. The corresponding values of optimal are shown in Fig. 12c for evaluating how the analysis weighted the measurements and background term. The values of are relatively lower between elevation angles 5.5° and 8.1°, indicating the variational algorithm relied more on the background term than on observations of ZDR and ΦDP. This is consistent with higher values of for ZDR and ΦDP between 5.0 and 8.0 km and for elevation angles between 5.5° and 8.1°, as shown in Fig. 13. Unlike Eq. (16), which is used for deriving the standard error of measurements for each beam, the for each range gate is estimated as
e19
Higher values of , ZDR and ΦDP further confirm the pronounced stripes in the predicted ΦDP, as well as the corrected ZHH and ZDR in Figs. 9 and 10 as the variational method failed to provide a reasonable estimation.
Fig. 12.
Fig. 12.

(a) The estimated optimal standard deviations of ZDR (from steps 4–6 of Fig. A1) as a function of elevation angle. (b),(c) As in (a), but for ΦDP and the optimal background error covariance matrices (), respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

Fig. 13.
Fig. 13.

(a) The diagnostic standard error of ZDR (derived from step 8 of Fig. A1) vs distance obtained by using the statistical diagnostic method. (b) As in (a), but for ΦDP. The solid lines represent elevation angles of 5.5° and 8.1°, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

The mean values of for ZDR and ΦDP in convective (distances between 5 and 13 km) and in stratiform (distances between 15 and 34 km) regions are summarized in Table 4. The mean values of diagnosed of 0.27-dB for ZDR and 2.2° for ΦDP in the stratiform region are in good agreement with the prescribed values of the observation errors of the estimated 0.18-dB for ZDR and 2.4° for ΦDP in the stratiform region, as shown in Table 4. However, the mean values of diagnosed for ZDR and ΦDP in the convective region are 0.38 dB and 2.88°, respectively, and are higher than the corresponding values in stratiform regions. The results indicate the statistical diagnostic method not only estimates accurate values of the error covariance of ZDR and ΦDP, but also characterizes the spatial distribution of the error covariance of the same.

6. Summary

A variational method originally designed for S-band radar has been modified for X-band radar data. Several modifications have been implemented in the original algorithm to overcome the difficulties caused by the high attenuation at X band. First, a discrete approximation method for estimating optimal initial values of a for each beam is used for smooth convergence of the Gauss–Newton minimization method. The results showed beam-dependent optimal initial values of a significantly improved estimation.

Second, instead of using fixed a priori values for error covariances, discrete approximation and statistical diagnostic methods were used for characterizing the temporal and spatial covariance matrices of the background term () and observations (). The OSSE studies demonstrated that error covariance matrix observations () can be estimated with sufficient accuracy, even though rather underestimated prescribed values were used initially.

A modified variational method in this research has demonstrated several advantages compared to the ΦDP-based algorithm: 1) more accurately corrected ZHH and ZDR compared to the ΦDP-based algorithm, 2) physically derived ΦDP for estimating KDP without any loss of spatial resolution, 3) an estimated background error covariance () for identifying the region with poor quality of measurement that failed to provide useful information to the variational algorithm, and 4) an estimated observation error covariance matrix () for characterizing the spatiotemporal error characteristics of polarimetric measurement ZDR and ΦDP.

The variational method uses both the measurements of ΦDP and ZDR from X-band radar. The inclusion of ZDR in the variational method improved the accuracy of the retrieved attenuation. The variational method yielded more accurate results than did the ΦDP-based method even though the later method used retrieval coefficients specifically tuned to the particular disdrometer dataset. Since, DSD can change from event to event and even within the same storm, it is difficult to tune retrieval coefficients of the ΦDP-based algorithm by not having a priori DSD information. The variational algorithm requires no knowledge about prior coefficients, yet it can estimate attenuation (AH and ADP) more accurately. Sensitivities of the variational correction algorithm to observation noise in ZHH, ZDR, and ΦDP were also investigated using OSSEs. The results suggested that the variational method is relatively immune to random measurement errors when compared to the ΦDP-based method.

Application of the method was demonstrated using radar measurements collected during the SoWMEX/TiMREX field program in Taiwan. Attenuation-corrected X-band measurements from TEAM-R using the variational method agreed with collocated S-band radar measurements. Moreover, the variational algorithm was able to reconstruct not only a broader, but also a more accurate distribution of AH and ADP for a specified KDP. Inclusion of DSD information via ZDR in the variational method significantly improved the accuracy of the retrieved attenuation.

In the process of estimating attenuation by the variational method, ΦDP is predicted as a by-product. It is an alternative approach to a standard mathematical filtering technique for estimating smooth ΦDP and KDP that are free of observation noise and backscattering Mie phase shift (δ).

The error covariance matrices and of real radar measurements play a critical role in the variational attenuation correction method at X band. The discrete approximation ensured that is consistent with the data quality of the radar measurement. Higher weighting of the background term (i.e., lower value of ) was used in the variational method when noise in the radar measurements was high. The error covariance matrix () of the ZDR and ΦDP measurements obtained from the statistical diagnostic method are used for characterizing the dynamic nature of the spatial and temporal error characteristics instead of fixed a priori values. The value of optimal obtained by the statistical diagnostic method is consistent with the standard deviation (σ) of the actual radar measurements in a stratiform light rain. The results of diagnosed indicate the statistical diagnostic method cannot only estimates accurate values of error covariance of ZDR and ΦDP but can also characterize the spatial distribution of the error covariance of the same. The results show the observation errors were higher in a convective region than in a stratiform region. The spatial distributions of the measurement error further help to explain why the variational algorithm failed to retrieve reliable attenuation in the regions of noisy radar measurements.

It should be noted that the variational algorithm is designed for attenuation correction in rain only as the forward models are based on rain DSDs. The presence of hail can cause significant attenuation (Gu et al. 2011) in practice. The presence of hail (featured with low ZDR and high ZHH) can be identified by high values of for ZDR; thus, the ΦDP-based algorithm can be used as an alternative to the variation method. The simplified background term of ã is used in this research (i.e., mean value of ã) and shows fairly promising results. It is recommended the background terms of ã be modified in correspondence with rain DSD climatology for favorable attenuation estimation.

Acknowledgments

We thank Dr. Wen-Chau Lee of NCAR and Prof. Ben Jou of the National Taiwan University for sharing the use of S-Pol data and Robert Rilling and Scott Ellis for assistance with the S-Pol data quality and calibration. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.

APPENDIX A

Implementation of the Discrete Approximation and Statistical Diagnostic Method

As shown in Fig. A1, the first guess of is determined by discrete approximation using a prescribed at step 3 (as described in section 2b). The prescribed is determined approximately based on the sampling time of the radar measurements and radar hardware characteristics. Using the first guess of , the procedures shown in step 4–6 diagnose the optimal by replacing it with iteratively. The optimal is obtained when the values of converge. Another discrete approximation for is performed again at step 7, since the first-guess at step 3 is obtained using the prescribed rather than an optimal . This procedure ensures that an optimal is derived with an optimal in step 8 (Li et al. 2009). The modified procedure determines the spatiotemporal characteristics of the error covariance matrices of individual radar measurements.

Fig. A1.
Fig. A1.

Flowchart describing the various numbered steps (numbers at the top-right box corners) in the discrete approximation of and diagnostic of ZDR and ΦDP. See appendix A for a detailed description.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

APPENDIX B

Modeled Polarimetric Observation Beam Data

Six hundred polarimetric observation beams are modeled evenly from 5 yr of the NCU 2DVD dataset with different combinations of rainfall intensity to cover the natural variability of the DSD. Each beam contains 400 gates of intrinsic ZHH and ZDR with gate resolution of 0.125 km according to the configuration of most X-band radars. The total attenuation effects (Ah and Ahv) were included in intrinsic ZHH and ZDR cumulatively along the beam. The ΦDP is obtained from integrating KDP along the beam. The values of Ah, Ahv, and ΦDP of these 600 modeled polarimetric observation beams are summarized in Fig. B1. Half of the modeled polarimetric observation beams have Ah > 8.1 dB, Ahv > 0.87 dB, and ΦDP > 28°. For OSSE studies with observation error, Gaussian random errors of the standard distribution (σ) were added to the modeled polarization observables accordingly, as shown in Table 2.

Fig. B1.
Fig. B1.

The cumulative density function of total attenuation (Ah), total differential attenuation (Ahv), and differential phase shift (ΦDP) of 600 modeled polarimetric observation beams.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

APPENDIX C

Optimal First Guess of a of ZR

A discrete approximation method is proposed to derive the optimal first guess of a of ZR. Figure C1a shows that parameter a can vary between 20 and 650 as a function of rain intensity and natural variation in DSD. For a range of values of a, the predicted attenuated ZDR and ΦDP are derived. The total absolute difference between the forward model predicted parameters and the actual measurement is calculated. The value of a that corresponds to the minimum total absolute difference is chosen as the optimal initial value.

Fig. C1.
Fig. C1.

(a) Mean absolute errors as a function of the parameter a. The x axis represents the coefficient a from 20 to 650. The y axis represents the absolute difference on a logarithmic scale between the forward model predicted parameters (here ZDR and ΦDP) and the actual measurements of a radar beam. TEAM-R measurements were used. (b) An example of the optimal initial a values derived for various elevation angles in RHI scan data. The red and blue lines represent the optimal initial a values from ZDR and ΦDP, respectively.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0191.1

The initial value of a could change from one precipitation event to the next and even within an individual precipitation event. In Fig. C1b initial values of a for the range of elevation angles in a RHI scan shown in Fig. 9 are plotted. Initial values of a vary from one elevation to another due to the variability of the DSD. When there are differences in the optimal initial a based on ZDR and ΦDP, an average of these values is taken as the optimal initial a.

REFERENCES

  • Anagnostou, M. N., , E. N. Anagnostou, , and J. Vivekanandan, 2006: Correction for rain path specific and differential attenuation of X-band dual-polarization observations. IEEE Trans. Geosci. Remote Sens., 44, 24702480.

    • Search Google Scholar
    • Export Citation
  • Barber, P., , and C. Yeh, 1975: Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies. Appl. Opt., 14, 28642872.

  • Bechini, R., , L. Baldini, , R. Cremonini, , and E. Gorgucci, 2008: Differential reflectivity calibration for operational radars. J. Atmos. Oceanic Technol., 25, 15421555.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., , and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar. Cambridge University Press, 636 pp.

  • Bringi, V. N., , V. Chandrasekar, , N. Balakrishnan, , and D. Zrnić, 1990: An examination of propagation effects in rainfall on radar measurements at microwave frequencies. J. Atmos. Oceanic Technol., 7, 829840.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., , T. D. Keenan, , and V. Chandrasekar, 2001: Correcting C-band radar reflectivity and differential reflectivity data for rain attenuation: A self-consistent method with constraints. IEEE Trans. Geosci. Remote Sens., 39, 19061915.

    • Search Google Scholar
    • Export Citation
  • Carey, L. D., , S. A. Rutledge, , D. A. Ahijevych, , and T. D. Keenan, 2000: Correcting propagation effects in C-band polarimetric radar observations of tropical convection using differential propagation phase. J. Appl. Meteor., 39, 14051433.

    • Search Google Scholar
    • Export Citation
  • Chandrasekar, V., , S. Lim, , and E. Gorgucci, 2006: Simulation of X-band rainfall observations from S-band radar data. J. Atmos. Oceanic Technol., 23, 11951205.

    • Search Google Scholar
    • Export Citation
  • Desroziers, G., , L. Berre, , B. Chapnik, , and P. Poli, 2005: Diagnosis of observation, background and analysis error statistics in observation space. Quart. J. Roy. Meteor. Soc., 131, 33853396.

    • Search Google Scholar
    • Export Citation
  • Doelling, I. G., , J. Joss, , and J. Riedl, 1998: Systematic variations of Z–R relationships from drop size distributions measured in northern Germany during seven years. Atmos. Res., 47–48, 635649.

    • Search Google Scholar
    • Export Citation
  • Furness, G., 2005: Using optimal estimation theory for improved rainfall rates from polarization radar. M.S. dissertation, Dept. of Mathematics, University of Reading, 66 pp.

  • Gorgucci, E., , and V. Chandrasekar, 2005: Evaluation of attenuation correction methodology for dual-polarization radars: Application to X-band systems. J. Atmos. Oceanic Technol., 22, 11951206.

    • Search Google Scholar
    • Export Citation
  • Gorgucci, E., , V. Chandrasekar, , and L. Baldini, 2006: Correction of X-band radar observation for propagation effects based on the self-consistency principle. J. Atmos. Oceanic Technol., 23, 16681681.

    • Search Google Scholar
    • Export Citation
  • Gu, J.-Y., , A. Ryzhkov, , P. Zhang, , P. Neilley, , M. Knight, , B. Wolf, , and D.-I. Lee, 2011: Polarimetric attenuation correction in heavy rain at C band. J. Appl. Meteor. Climatol., 50, 3958.

    • Search Google Scholar
    • Export Citation
  • Hitschfeld, W., , and J. Bordan, 1954: Errors inherent in the radar measurement of rainfall at attenuating wavelengths. J. Meteor., 11, 5867.

    • Search Google Scholar
    • Export Citation
  • Hogan, R. J., 2007: A variational scheme for retrieving rainfall rate and hail reflectivity fraction from polarization radar. J. Appl. Meteor. Climatol., 46, 15441564.

    • Search Google Scholar
    • Export Citation
  • Hubbert, J., , and V. N. Bringi, 1995: An iterative filtering technique for the analysis of copolar differential phase and dual-frequency radar measurements. J. Atmos. Oceanic Technol., 12, 643648.

    • Search Google Scholar
    • Export Citation
  • Hubbert, J., , S. M. Ellis, , M. Dixon, , and G. Meymaris, 2010a: Modeling, error analysis, and evaluation of dual-polarization variables obtained from simultaneous horizontal and vertical polarization transmit radar. Part I: Modeling and antenna errors. J. Atmos. Oceanic Technol., 27, 15831598.

    • Search Google Scholar
    • Export Citation
  • Hubbert, J., , S. M. Ellis, , M. Dixon, , and G. Meymaris, 2010b: Modeling, error analysis, and evaluation of dual-polarization variables obtained from simultaneous horizontal and vertical polarization transmit radar. Part II: Experimental data. J. Atmos. Oceanic Technol., 27, 15991607.

    • Search Google Scholar
    • Export Citation
  • Jameson, A. R., 1991: Polarization radar measurements in rain at 5 and 9 GHz. J. Appl. Meteor., 30, 15001513.

  • Jou, B. J.-D., , W.-C. Lee, , and R. H. Johnson, 2011: An overview of SoWMEX/TiMREX and its operation. The Global Monsoon System: Research and Forecast, C.-P. Chang, Ed., 2nd ed., World Scientific, 303–318.

  • Li, H., , E. Kalnay, , and T. Miyoshi, 2009: Simultaneous estimation of covariance inflation and observations errors within an ensemble Kalman filter. Quart. J. Roy. Meteor. Soc., 135, 523533.

    • Search Google Scholar
    • Export Citation
  • Lim, S., , V. Chandrasekar, , P. Lee, , and A. P. Jayasumana, 2011: Real-time implementation of a network-based attenuation correction in the CASA IP1 testbed. J. Atmos. Oceanic Technol., 28, 197209.

    • Search Google Scholar
    • Export Citation
  • Park, S. G., , V. N. Bringi, , V. Chandrasekar, , M. Maki, , and K. Iwanami, 2005a: Correction of radar reflectivity and differential reflectivity for rain attenuation at X band. Part I: Theoretical and empirical basis. J. Atmos. Oceanic Technol., 22, 16211632.

    • Search Google Scholar
    • Export Citation
  • Park, S. G., , M. Maki, , K. Iwanami, , V. N. Bringi, , and V. Chandrasekar, 2005b: Correction of radar reflectivity and differential reflectivity for rain attenuation at X band. Part II: Evaluation and application. J. Atmos. Oceanic Technol., 22, 16331655.

    • Search Google Scholar
    • Export Citation
  • Rodgers, C. D., 2000: Inverse Methods for Atmospheric Soundings: Theory and Practice. World Scientific, 238 pp.

  • Ryzhkov, A. V., 2007: The impact of beam broadening on the quality of radar polarimetric data. J. Atmos. Oceanic Technol., 24, 729744.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., , and D. S. Zrnić, 1996: Assessment of rainfall measurement that uses specific differential phase. J. Appl. Meteor., 35, 20802090.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., , V. B. Zhuravlyov, , and N. A. Rybakova, 1994: Preliminary results of X-band polarization radar studies of clouds and precipitation. J. Atmos. Oceanic Technol., 11, 132139.

    • Search Google Scholar
    • Export Citation
  • Sachidananda, M., , and D. S. Zrnic, 1986: Differential propagation phase shift and rainfall rate estimation. Radio Sci., 21, 235247.

  • Scarchilli, G., , E. Gorgucci, , V. Chandrasekar, , and T. A. Seliga, 1993: Rainfall estimation using polarimetric techniques at C-band frequencies. J. Appl. Meteor., 32, 11501160.

    • Search Google Scholar
    • Export Citation
  • Schönhuber, M., , H. E. Urban, , J. P. V. Poiares Baptista, , W. L. Randeu, , and W. Riedler, 1997: Weather radar versus 2D-video-distrometer data. Weather Radar Technology for Water Resources Management, B. Braga Jr. and O. Massambani, Eds., UNESCO Press, 159–171.

  • Smyth, T. J., , and A. J. Illingworth, 1998: Correction for attenuation of radar reflectivity using polarization data. Quart. J. Roy. Meteor. Soc., 124A, 23932415.

    • Search Google Scholar
    • Export Citation
  • Snyder, J. C., , H. B. Bluestein, , G. Zhang, , and S. J. Frasier, 2010: Attenuation correction and hydrometeor classification of high-resolution, X-band, dual-polarized mobile radar measurements in severe convective storms. J. Atmos. Oceanic Technol., 27, 19792001.

    • Search Google Scholar
    • Export Citation
  • Steiner, M., , and R. A. Houze Jr., 1997: Sensitivity of the estimated monthly convective rain fraction to the choice of ZR relation. J. Appl. Meteor., 36, 452462.

    • Search Google Scholar
    • Export Citation
  • Steiner, M., , and J. A. Smith, 2000: Reflectivity, rain rate, and kinetic energy flux relationships based on raindrop spectra. J. Appl. Meteor., 39, 19231940.

    • Search Google Scholar
    • Export Citation
  • Steiner, M., , J. A. Smith, , and R. Uijlenhoet, 2004: A microphysical interpretation of radar reflectivity–rain rate relationships. J. Atmos. Sci., 61, 11141131.

    • Search Google Scholar
    • Export Citation
  • Testud, J., , E. Le Bouar, , E. Obligis, , and M. Ali-Mehenni, 2000: The rain profiling algorithm applied to polarimetric weather radar. J. Atmos. Oceanic Technol., 17, 332356.

    • Search Google Scholar
    • Export Citation
  • Ulbrich, C. W., 1983: Natural variations in the analytical form of the raindrop size distribution. J. Climate Appl. Meteor., 22, 17641775.

    • Search Google Scholar
    • Export Citation
  • Vivekanandan, J., , W. M. Adams, , and V. N. Bringi, 1991: Rigorous approach to polarimetric radar modeling of hydrometeor orientation distributions. J. Appl. Meteor., 30, 10531063.

    • Search Google Scholar
    • Export Citation
  • Vivekanandan, J., , S. M. Ellis, , R. Oye, , D. S. Zrnić, , A. V. Ryzhkov, , and J. Straka, 1999: Cloud microphysics retrieval using S-band dual-polarization radar measurements. Bull. Amer. Meteor. Soc., 80, 381388.

    • Search Google Scholar
    • Export Citation
  • Vivekanandan, J., , G. Zhang, , S. M. Ellis, , D. Rajopadhyaya, , and S. K. Avery, 2003: Radar reflectivity calibration using differential propagation phase measurement. Radio Sci., 38, 8049, doi:10.1029/2002RS002676.

    • Search Google Scholar
    • Export Citation
Save