## 1. Introduction

Recent technological and scientific advances in the field of rainfall estimation using weather radars present unprecedented opportunities for providing accurate and timely rainfall information. Weather radars provide many advantages over traditional gauge observations especially in terms of their near real-time extensive spatial coverage and relatively high temporal and spatial resolutions, which can make them highly advantageous for a variety of hydrological and meteorological applications. However, radar-rainfall estimates are uncertain due to a variety of effects such as: hardware calibration, non-uniqueness in the relationship between radar-measured reflectivity and rainfall rate, beam overshooting and partial beam filling, anomalous propagation of the radar beam, and non-uniformity in vertical profiles of reflectivity (see Villarini and Krajewski, 2010b for a recent review). Implications of such uncertainties have been well recognized in several hydrological applications that use radar-rainfall estimates (e.g., Sharif *et al.*, 2002; Gourley and Vieux, 2005; Habib *et al.*, 2008). While research on uncertainties in radar-rainfall estimates has been going on for many years, quantitative knowledge on the statistical characteristics and the full structure of the estimation error is still at an early stage. Ciach and Krajewski (1999) proposed a framework known as the Error Variance Separation (EVS) method, which focused on the estimation of one aspect (variance) of the error distribution (Young *et al.*, 2000; Habib *et al.*, 2002; Zhang *et al.*, 2007; Mandapaka *et al.*, 2009). While EVS is limited to the estimation of the error variance only, Habib *et al.* (2004) developed a more general approach that filters out rain gauge errors and retrieves the bi-variate distribution of radar estimates and the corresponding unknown true surface rainfall. Building on such efforts, and motivated by the need for practical methods for modelling radar-rainfall uncertainties, Ciach *et al.* (2007) proposed a product-driven, empirically-based model (referred to herein as PED) which focuses on modelling the combined sources of uncertainties in radar-rainfall estimates. Similar approaches for modelling the combined radar uncertainties (or total estimation error) have been reported in Germann *et al.* (2009).

The essence of the PED method is based on empirical modelling of the relationship between radar-rainfall estimates and the corresponding true surface rainfall (or its approximation from gauge observations) *via* explicit separation of the radar error into two components: deterministic and random. Both of these components are modelled as a function of the radar-rainfall estimate. In its first application, Ciach *et al.* (2007) demonstrated the implementation of the method for the National Weather Service (NWS) Next Generation Weather Radar (NEXRAD) Digital Precipitation Array (PDA) hourly 4 × 4 km^{2} product in Oklahoma, the United States. Villarini and Krajewski (2009) implemented the PED method for a 2 × 2 km^{2} product from a C-band radar in Great Britain and investigated its performance at different time scales (5–180 min). Other applications of the PED method included development of an ensemble generator of probable true surface rainfall fields (Villarini *et al.*, 2009a), analysis of the impact of radar-rainfall uncertainties on rainfall-runoff modelling (Habib *et al.*, 2008) flash-flood forecasting (Villarini *et al.*, 2010), statistical validation of satellite-based precipitation estimates (Villarini *et al.* (2009b), and scaling properties of rainfall (Mandapaka *et al.*, 2010). While the PED methodology provides a promising mechanism for characterizing and modelling radar-rainfall uncertainties, its assumptions, parameter estimation, and transferability to other geographic regions and radar setups warrant further investigations. For example, Villarini and Krajewski (2010a) investigated the sensitivity of the PED method to the selection of the reflectivity-to-rainfall (Z-R) relationship and to an algorithm to discriminate between meteorological and non-meteorological returns. They also suggested an additive formulation of the error in addition to its originally proposed multiplicative form. The current study follows on these efforts focusing on several PED methodological aspects such as: (1) application of the PED method to a widely-used operational radar-based multi-sensor estimation algorithm (MPE), (2) sensitivity of the method to different processing procedures and products within the MPE algorithm, (3) transferability of the PED formulation and functional relationships to other geographical regions different from those in earlier applications, and (4) investigation of the sampling-related effects on the estimation of the PED parameters and relationships. Analysis of these aspects will provide further insight into the generality and sampling requirements of the PED methodology and its underlying assumptions, and more importantly, will guide the development and further enhancements of this and other future methods on uncertainty modelling of radar-rainfall products.