Correcting bias in radar Z–R relationships due to uncertainty in point rain gauge networks

Highlights

• Uncertainties in the rain gauge network at radar pixel resolution were evaluated.

• Extent of parameter bias present in the rainfall–reflectivity relationship as a result point gauge rainfall was estimated.

• Results indicate that gauge rainfall uncertainty affects average radar rainfall estimates.

Summary

One of the key challenges in hydrology is to accurately measure and predict the spatial and temporal distribution of rainfall. Rain gauges measuring at point locations are often considered as the “ground truth” for grid based radar rainfall calibration. Usually, no consideration is given to the uncertainty in the measurement that varies depending on the number of rain gauges that fall within each grid cell. If this uncertainty in the rain gauge network measurements is ignored, the Z–R relationship used to convert reflectivity (Z) to rainfall (R) will be biased. We investigate the effects of point gauge rainfall uncertainty on parameter bias in the Z–R relationship. An error model is developed to compute point gauge rainfall uncertainty at the radar grid resolution. This error model has two components: (1) the error in the gauge measurement itself, and (2) the error introduced by the gauge not capturing the spatial variability within a radar pixel. The Simulation Extrapolation method (SIMEX) is used to determine the extent of parameter bias present in the rainfall–reflectivity relationship as a result of this uncertainty. When considering the point gauge rainfall uncertainty a 4% decrease in the average radar rainfall estimates is found.

Introduction

Rainfall is one of the most important inputs to hydrological analysis and modelling. Any error in this input will propagate through the model and will introduce uncertainty in subsequent predictions (Hossain et al., 2004, Morin et al., 2005, Pessoa et al., 1993, Sharif et al., 2002, Vieux and Bedient, 1998, Vivoni et al., 2007, Zhu et al., 2013). One source of uncertainty and possible errors is the temporal and spatial variability of rainfall (AghaKouchak et al., 2010, Faurès et al., 1995, Goodrich et al., 1995, Shah et al., 1996). Therefore, accurately measuring and predicting the spatial and temporal distribution of rainfall is an important challenge in hydrology.

Remote sensing provides an increasingly important source for spatial estimation of precipitation. Using measurements of reflectivity, weather radars can produce estimates of precipitation over large geographic areas, and therefore provide information about rainfall patterns at high temporal and spatial resolution. As a result, the potential applications of weather radars for hydrologic modelling have been the subject of extensive research (Bonnifait et al., 2009, Cole and Moore, 2008, Cole and Moore, 2009, Collier, 2009, Keblouti et al., 2013, Looper and Vieux, 2012, Viviroli et al., 2009).

Traditionally, radar reflectivities Z (mm⁶/m³) are converted to ground rainfall rates R (mm/h) using a power–law relationship (Z = AR^b) known as the Z–R relationship. The relationship between radar reflectivity and rainfall rate depends on the nature of rainfall, and in particular the drop size distribution (DSD) (Chumchean et al., 2006a, Chumchean et al., 2008, Islam et al., 2012, Uijlenhoet and Pomeroy, 2001). One way to calibrate the Z–R relationship is through the use of disdrometers which can explicitly measure the DSD and therefore the relationship with the radar reflectivity (Ochou et al., 2011, Verrier et al., 2013). However in many parts of the world, including Australia, gauge based calibration of the Z–R relationships is routinely used (Bringi et al., 2011, Rendon et al., 2013). Furthermore, in developing countries where disdrometer data is not available and even sub-daily rainfall measurements are infrequent, daily rainfall gauges are the sole source of information on ground rainfall estimates (Mapiam et al., 2009). In these cases gauge densities are also likely to be very low compared to developed countries such as the United States of America and Europe. Thus errors in the recorded gauge rainfall can bias the Z–R relationship, resulting in errors in the radar rainfall estimates. A biased Z–R relationship may be due to an inadequate consideration of the spatial subgrid scale gauge rainfall variability and its representation through the handful of gauges that are available for use (Ciach et al., 2007, Jordan et al., 2003) and therefore methods that take account of gauge uncertainty can be very useful.

The performance of the radar rainfall estimates is evaluated by comparing them with point gauge measurement at the ground (Anagnostou et al., 1999, Habib and Krajewski, 2002). These point gauge measurements are therefore considered as the “ground truth” (Lebel and Amani, 1999, Wolff et al., 2005). However, there is a spatial discrepancy between the two data sources, since radar rainfall estimates are provided as spatial averages with a resolution 1–4 km² (Krajewski and Smith, 2002). In contrast, whilst rain gauges measure precipitation at fixed point locations, these are often too sparse to properly represent the spatial variability and areal structure of rainfall (Morrissey et al., 1995, Rodríguez-Iturbe and Mejía, 1974). Furthermore, the uncertainty associated with spatial averages over a single radar grid cell is considerable, and is a function of the number of gauges that are within the cell being considered. In addition, rain gauge measurements can contain a variety of errors including wind effects, evaporation and mechanical tipping bucket errors (Groisman and Legates, 1994). An important question is whether the parameters estimated for the Z–R relationship would remain the same if consideration was given to the nature of the errors associated with spatial averaging of the gauge rainfall over the radar grid scale. We argue that if these errors are significant (and vary with space as they will if the rainfall and the gauge density are not uniform across the network), the estimated parameters will have a bias with respect to the true parameters that ought to be used.

This research aims to address whether the A parameter in the radar Z–R relationship needs to change to account for uncertainty in the point gauge rainfall network. It has been reported that the parameter A carries most of the variability in the Z–R relationship, whereas the uncertainty in b can be seen as second-order (Chumchean et al., 2003, Chumchean et al., 2006b, Steiner et al., 1999). Marshall and Palmer (1948) proposed the power law relation, Z = 200 R^1.6 with specified values for A equal to 200 and b equal to 1.6. Since then, several studies have been conducted to find appropriate A and b parameter values in different settings. Given the need for consistent estimates of rainfall, for operational purposes the Australian Bureau of Meteorology (BOM) specifies a fixed value for b equal to 1.53 for most of the weather radars that form its rainfall measuring network. Fixing b for operational networks has also previously been used by other researchers (Steiner and Smith, 2004, Verrier et al., 2013). We use the Simulation Extrapolation method (SIMEX) (Cook and Stefanski, 1994) to investigate the significance of point gauge uncertainty on the Z–R relationship parameter A. The SIMEX method involves developing a relationship between the gauge uncertainty distribution and input gauge rainfall, which can then be used to assess the bias in the parameters of the Z–R relationship.

The paper is organised in seven sections. Section 2 outlines the logic behind the SIMEX approach. Section 3 describes the radar and gauge data used in this study, whilst Section 4 presents the development of an error model for the lowest radar pixel resolution (1 km²). The application of SIMEX method on radar Z–R relationship is presented in Section 5 followed by results and discussion in Section 6. The main findings are summarised in Section 7.