Analytic characterization of random errors in spectral dual-polarized cloud radar observations

Atmos. Meas. Tech., 15, 1333–1354, 2022 https://doi.org/10.5194/amt-15-1333-2022 © Author(s) 2022. This work is distributed under the Creative Commons Attribution 4.0 License. Analytic characterization of random errors in spectral dual-polarized cloud radar observations Alexander Myagkov1 and Davide Ori2 1 Radiometer 2 Institute Physics GmbH, Meckenheim, Germany for Geophysics and Meteorology, University of Cologne, Cologne, Germany Correspondence: Alexander Myagkov () Received: 24 July 2021 – Discussion started: 18 August 2021 Revised: 8 January 2022 – Accepted: 28 January 2022 – Published: 14 March 2022 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 (PDFs) 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 ρH V , and differential phase 8DP 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. 1 Introduction Cloud radars are a major component of state-of-the-art, ground-based observation platforms (Illingworth et al., 2007; Kollias et al., 2020). Their unique capabilities make these instruments extremely valuable for cloud and precipitation research. First, these radars have Doppler capabilities; i.e., they can independently characterize hydrometeors coexisting in the same volume but moving with different speeds relative to the radar (Kollias et al., 2007). Second, the high sensitivity and vast dynamic range make cloud radars capable of measuring return signals from a wide range of particle sizes, which is a challenging task for other instruments like lidars (Bühl et al., 2013). Third, due to relatively low attenuation of microwave signals by liquid water, cloud radars profile clouds up to the top even in the presence of light to moderate rain. These capabilities promote cloud radars for investigation of different formation and development processes throughout the life cycle of clouds. For instance, cloud radars help to characterize initial ice formation and development in mixed-phase clouds (Bühl et al., 2019a, b), improve characterization of pure liquid clouds (Rusli et al., 2017; Acquistapace et al., 2017), estimate rates of aggregation (Kneifel et al., 2015, 2016) and riming (Kalesse et al., 2016; Moisseev et al., 2017; Kneifel and Moisseev, 2020), and quantitatively analyze solid and liquid precipitation (Matrosov, 2005; Matrosov et al., 2006, 2008; Tridon and Battaglia, 2015; Tridon et al., 2017, 2019). Many cloud radars have dual-polarization capabilities. An interest in polarimetry-based methods in the cloud radar community has been growing, which is indicated by a number of studies during the last decade (Matrosov et al., 2012; Oue et al., 2015; Lu et al., 2015; Myagkov et al., 2016a, b; Matrosov et al., 2017; Oue et al., 2018; Myagkov et al., 2020). Vertically pointed cloud radars often operate in the LDR (linear depolarization ratio) mode; i.e., they transmit a linearly polarized wave (either horizontally or vertically) and receive co- and cross-polarized components of the backscattered signal (e.g., Görsdorf et al., 2015). The LDR mode Published by Copernicus Publications on behalf of the European Geosciences Union. 1334 A. Myagkov and D. Ori: Analytic characterization of random errors is efficient for clutter removal and detection of the melting layer and columnar-shaped ice particles. As shown by Matrosov et al. (2001), however, the applicability of the LDR mode at low elevation angles might be limited due to its high sensitivity to the orientation of cloud particles. Therefore, scanning polarimetric cloud radars often have polarimetric modes which are less sensitive to the orientation. One such mode is the hybrid mode (also denoted as the STSR (simultaneous transmission and simultaneous reception) or STAR (simultaneous transmission and reception) mode in the literature). Radars with the hybrid mode emit the horizontal and vertical components of the transmitted wave simultaneously (Myagkov et al., 2015; Bringi and Chandrasekar, 2001, Sect. 4.7). Cloud radars with the hybrid mode allow for adoption of polarimetry-based methods developed during the last several decades for centimeter-wavelength meteorological radars (further denoted as precipitation radars). Operational precipitation radars are used by weather services to continuously scan the atmosphere, providing polarimetric variables integrated for a scattering volume. In addition to the integrated quantities, cloud radars with the hybrid mode enable spectrally resolved polarimetric observations and, therefore, can provide the same set of polarimetric variables for different types of cloud particles coexisting in the same resolution volume (Oue et al., 2015; Myagkov et al., 2016b, 2020). Spectral observations are in general possible with precipitation radars (Spek et al., 2008; Dufournet and Russchenberg, 2011; Pfitzenmaier et al., 2018). Such measurements, however, are not performed by operational radars due to fast azimuth scanning. Spectral polarimetry can be used for a development of advanced retrieval methods. For example variational retrievals developed for dual-frequency spectra (Tridon and Battaglia, 2015; Tridon et al., 2017) could be applied also to spectral polarimetry. Moisseev and Chandrasekar (2007) presented first attempts to retrieve profiles of raindrop size distributions using polarimetric spectra from a precipitation radar. This approach, however, has not yet been explored in polarimetric cloud radars. Recent review studies (Zhang et al., 2019; Morrison et al., 2020; Ryzhkov et al., 2020) demonstrate that polarimetric observations from precipitation radar networks are highly beneficial for the evaluation and development of numerical weather prediction and cloud resolving models. The high value of polarimetric observations is given by their sensitivity to microphysical properties of cloud and precipitation particles such as size, shape, numbe (...truncated)