Abstract

This study presents the first-ever complete characterization of random errors in dual-polarimetric spectral observations of meteorological targets by cloud radars. The characterization is given by means of mathematical equations for joint probability density functions (PDF) and error covariance matrices. The derived equations are checked for consistency using real radar measurements. One of the main conclusions of the study is that the convenient representation of spectral polarimetric measurements including differential reflectivity ZDR, correlation coefficient pHV, and differential phase ΦDP is not suited for the proper characterization of the error covariance matrix. This is because the aforementioned quantities are complex, non-linear functions of the radar raw data and thus their error covariance matrix is commonly derived using simplified linear relations and by neglecting the correlation of errors. This study formulates the spectral polarimetric measurements in terms of a different set of quantities that allows for a proper analytic treatment of their error covariance matrix. The results given in this study allow for utilization of spectral polarimetric measurements for advanced meteorological applications, among which are variational retrieval techniques, data assimilation, and sensitivity analysis.

Highlights

  • Cloud radars are a major component of state-of-the-art, ground-based observation platforms (Illingworth et al, 2007; Kollias et al, 2020)

  • The characterization is given by means of mathematical equations for joint probability density functions (PDF) and error covariance matrices

  • 360 concluded that any application of spectral polarimetric measurements which require the estimate of the error covariance matrix should be performed in the space of observations brather than c

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Summary

Introduction

Cloud radars are a major component of state-of-the-art, ground-based observation platforms (Illingworth et al, 2007; Kollias et al, 2020). A number of studies (e.g. Hogan (2007); Cao et al (2013); Yoshikawa et al (2014); Chang et al (2016); Huang et al (2020)) characterize the joint PDF of polarimetric radar measurements by the error covariance matrix. The elements in the main diagonal of the error covariance matrix – variances of random errors – are found using the first-order Taylor approximation following Bringi and Chandrasekar (2001) Conventional polarimetric variables such as differential reflectivity, correlation coefficient, and differential phase are , highly non-linear functions.

Likelihood of elements of the covariance matrix B
Likelihood function in the c–x basis
Likelihood function in the h–v basis
Error covariance matrices
Error covariance matrix of b
Error covariance matrix of c
Processing
Evaluation of fb(ˆb|b, Ns) and fb(ˆc|b, Ns)
Evaluation of Σc
355 5.5 Evaluation of Σb
Summary
Change of variables in a PDF
Likelihood functions for Dcc and Dxx
Likelihood functions for Rcx and Jcx
510 Acknowledgements
Full Text
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