PUBLICATIONS. Water Resources Research. Exploration of discrepancy between radar and gauge rainfall estimates driven by wind fields

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1 PUBLICATIONS RESEARCH ARTICLE Key Points: We present a scheme to adjust windinduced error on radar-gauge comparison Downscaled wind can be used to simulate the movement of raindrop in the air Wind effects on radar adjustment includes wind-drift and windturbulent errors Correspondence to: Q. Dai, q.dai@bristol.ac.uk, dqgis@hotmail.com Citation: Dai, Q., and D. Han (2014), Exploration of discrepancy between radar and gauge rainfall estimates driven by wind fields, Water Resour. Res., 50, , doi: / 2014WR Received 1 MAY 2014 Accepted 15 OCT 2014 Accepted article online 21 OCT 2014 Published online 7 NOV 2014 Exploration of discrepancy between radar and gauge rainfall estimates driven by wind fields Qiang Dai 1 and Dawei Han 1 1 WEMRC, Department of Civil Engineering, University of Bristol, Bristol, UK Abstract Due to the fact that weather radar is prone to several sources of errors, it is acknowledged that adjustment against ground observations such as rain gauges is crucial for radar measurement. Spatial matching of precipitation patterns between radar and rain gauge is a significant premise in radar bias corrections. It is a conventional way to construct radar-gauge pairs based on their vertical locations. However, due to the wind effects, the raindrops observed by the radar do not always fall vertically to the ground, and the raindrops arriving at the ground may not all be caught by the rain gauge. This study proposes a fully formulated scheme to numerically simulate the movement of raindrops in a three-dimensional wind field in order to adjust the wind-induced errors. The Brue catchment (135 km 2 ) in Southwest England covering 28 radar pixels and 49 rain gauges is an experimental catchment, where the radar central beam height varies between 500 and 700 m. The 20 typical events (with durations of 6 36 h) are chosen to assess the correlation between hourly radar and gauge rainfall surfaces. It is found that for most events, the improved rates of correlation coefficients are greater than 10%, and nearly half of the events increase by 20%. With the proposed method, except four events, all the event-averaged correlation values are greater than 0.5. This work is the first study to tackle both wind effects on radar and rain gauges, which could be considered as one of the essential components in processing radar observational data in its hydrometeorological applications. 1. Introduction Modern weather radar products, due to their large areal coverage and high spatial and temporal resolutions, have been widely used to provide rainfall information as input or initial conditions to hydrological models. However, the complicated measurement procedure and intricate atmospheric conditions determine that large uncertainties are associated with the radar rainfall estimates. Traditionally, the primitive radar measurement has to be processed to solve the following fundamental problems: ground clutter and anomalous propagation, signal attenuation (includes radome wetting), beam blockage, vertical variability of the reflectivity [Cluckie et al., 2000; Villarini and Krajewski, 2010]. After these steps, the radar estimates are essentially compared with the ground truth gauge measurement to adjust their systematic biases for hydrological applications. It is a common practice to construct radar-gauge pairs according to their projected vertical positions on the ground. In other words, the rainfall observed by radar aloft is assumed to fall vertically to the ground and should correspond to the surface rainfall directly below the volume sampled by the radar beam. Obviously, this assumption cannot always be satisfied as it ignores a significant fact that the raindrops will drift in the air driven by the wind when they fall. In fact, wind plays a vital role in determining the quality of adjusted radar data using gauge measurement, which is reflected in two aspects. First, wind can cause the raindrop drift to induce an inconsistent spatial correlation of radar and gauge measurements, which is called wind-drift effect. Lack and Fox [2007] have demonstrated that the drift of raindrops could be over considerable distances and cause large errors under certain circumstances (even at a resolution of 2 km). In addition, the wind-induced turbulent airflow surrounding a rain gauge causes the rain gauge to miss some raindrops, which is named as wind-turbulent effect. Nespor and Sevruk [1999] concluded that this effect could lead to an average error (catch loss) of 2% 10% for rain and 10% 50% for snow. In this paper, we are interested in the two wind effects: (i) the wind-drift displacement and (ii) the gauge catch loss due to wind turbulence. In terms of wind-drift effect, there are few studies on the topic until recently. Nicholas [1990] studied the effect of wind drift on warm season precipitations using a wind computational model with the assumption of a monodisperse drop size distribution (DSD). Collier [1999] also DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8571

2 realized this problem, and concluded that the out-of-synch in pixels between radar and rain gauge would be greater with the growth of spatial resolutions. The first quantified method to tackle this problem was proposed by Mittermaier et al. [2004], in which the wind effect was analyzed by simulating the fall streaks of snow. It was found that the order for the displacements could reach as large as km. Later, Lack and Fox [2005, 2007] studied the effect of wind drift using the wind field simulated by radar observation itself to estimate the trajectories of falling drops. Lauri et al. [2012] presented a method to compute the displacement of hydrometeors using an advection scheme. The efforts to tackle the wind-turbulent effect on gauge measurement could be divided into two types. The first one is to establish an empirical relationship between wind-induced error and wind speed as well as rainfall intensity. The key point of this method is to introduce a rainfall measurement mechanism that could avoid the effect of wind. For example, Harrold et al. [1974] improved the quality of rain gauge observations using anemometers and a nearby radiosonde. Seibert and Moren [1999] investigated the relationship between a standard gauge without wind shield and a special weighting gauge. Yang et al. [1998] assessed the catch ratio of rain gauge errors by comparing it with the double-fence intercomparison reference (DFIR). The other type is to estimate the wind-turbulent error using numerical simulation or statistical model. Nespor and Sevruk [1999] proposed a scheme to estimate the wind-turbulent error of gauge measurement through numerically simulating the airflow around the gauge. The calculated flow fields are used for raindrop trajectory simulations and the errors are estimated for the given drop diameter and wind speed. This correction procedure is applied by Habib et al. [1999] at different scenarios to investigate the influence of temporal scales. De Lima et al. [2002] also proposed a numerical model to simulate the movement of individual drops after they were released from the nozzle of a downward-spraying rainfall simulator. Blocken et al. [2005] numerically simulated the wind-driven rainfall distribution over a small-scale topography in space and time based on computational fluid dynamics. Michelson [2004] estimated the wind error caused by the flow distortion about the gauge orifice through a statistical Dynamic Correction Model (DCM). Video imaging technology has also been applied to estimate the wind-induced error [Nespor et al., 2000]. To sum up, the wind-turbulent effect on gauge measurement has been studied by many researchers and some reliable outcomes could be directly applied in future work. However, the study on quantifying the impact of wind-drift effect on radar rainfall is still in an early stage, and no study has yet been proposed to simultaneously address both those two wind effects on radar-gauge comparison. We have attempted to solve the wind-drift problem with the radar rainfall measurement in Dai et al. [2013a]. However, there are two critical problems associated with that study. First, the wind-turbulent effect on rain gauge measurement was not considered. Even though the magnitude of wind-turbulent error may be smaller than the possible uncertainty of radar rainfall, the adjustment of wind-turbulent error should be synchronously performed with the treatment of wind-drift error, otherwise we cannot estimate where the wind-induced discrepancy between radar and gauge measurement comes from. More importantly, it is acknowledged that the quality of a rain gauge measurement at a point is better than a radar rainfall estimate; errors in the rain gauge measurement may bring new uncertainties into radar-gauge adjustments. As a result, the improvement after conducting the proposed method seemed to be small and unstable. In addition, Dai et al. [2013a] computed the Pearson correlation coefficients between radar and gauge rainfall surfaces, without considering the significance of these statistics. Hence, the evaluated processing scheme cannot solve the possible outlier situations, which may contaminate the final results. The linear correlation indicator is unsuitable for reflecting the relationship between radar and gauge measurements [Ciach et al., 2007; Habib et al., 2008]. For this reason, we propose a more comprehensive scheme in this paper which can treat the two wind effects (wind-drift and wind-turbulent errors) and construct a more realistic radar-gauge relationship. This is the first study to tackle both wind effects on radar data, and could be considered as an essential component in processing radar data alongside other fundamental processes, which should be undertaken between bias correction and the aforementioned physical processes. This paper is organized as follows. After the introduction, section 2 illustrates the WRF model, the drop size distribution model, and data sets used. The following section 3 describes the algorithm of adjusting winddrift and wind-turbulent effects. Section 4 presents the outcomes of the simulation and evaluation of the proposed scheme. The section 5 elaborates on three key issues associated with this scheme. Finally, section 6 summarizes the key findings and the future work. DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8572

3 Table 1. The Basic Information for 20 Storm Events a Event ID Storm Start Time (YY-MM-DD:HH) Storm End Time Duration (h) Accumulated Rainfall (mm) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : a The accumulated rainfall is the areal averaged rainfall of the catchment using gauge measurement. 2. Models and Data Sets 2.1. Study Area and Data Sets The Brue catchment (135 km 2 ) in Somerset, Southwest England (51.08 N and 2.58 W) is an experimental catchment used in this study. The study area has a dense rain gauge network of 49 gauges, three radars, and a variety of related meteorological data. The radar from Wardon Hill located around 40 km to the center of the catchment, with 2 km spatial resolution and 5 min temporal resolution, is mainly used [Dai et al., 2013b]. The Marshall-Palmer Z-R relationship [Marshall et al., 1955] is applied in this study to convert radar reflectivity into rainfall intensity, which is commonly used in the UK. The gauge data are from standard Cassella 0.2 mm tipping bucket rain gauges. The radar and gauge data are from October 1993 to April The wind data sets include the global three-dimension wind field and the regional ground wind measurement. The meteorological data set from the ERA-40 global reanalysis data produced by the European Centre for Medium-range Weather Forecasts (ECMWF) is used herein. The second-generation reanalysis data ERA- 40 include many sources of the meteorological observations [Courtier et al., 1998]. The ground wind data are from an automatic weather station (AWS) located within the catchment. It records wind speed and direction, solar irradiance and net radiation, wet and dry bulb temperatures, atmospheric pressure, and rainfall (0.2mm tip) every 15 min. Among them, the wind speed and direction at height 2 m are mainly used in this study. The ERA-40 data cover the period from September 1957 to August 2002 and the automatic weather station is available from September 1993 to February To test the proposed scheme, 20 typical events are chosen from the period when all the above mentioned data sets are available, which are listed in Table 1. It is worth remarking that not all radar pixels (i.e., the spatial grids of radar data) or rain gauges have valid data at the same time. To avoid bringing in new uncertainties, we do not adopt any methods to fill the gap if part of radar pixels or rain gauges have the missing data. For this reason, some storms are discarded even though they have heavy rainfall and strong wind. Table 1 shows the event ID, start time, end time, and duration of the events, together with the accumulated event rainfall. The accumulated values are calculated using gauge measurements. To be consistent with the ECMWF data, the start and end times are set to 00, 06, 12, and 18 UTC The WRF Model The WRF model is a mesoscale forecast model and data assimilation system with advanced dynamics, physics, and numerical schemes [Liu et al., 2012; Skamarock et al., 2005]. It is designed to advance the understanding and prediction of mesoscale weather through compressible, nonhydrostatic, and primitive Euler equations. Its principal component is the WRF software framework (WSF), which can provide the fundamental infrastructure that accommodates the dynamics solvers and physics packages. There are two physical DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8573

4 solvers in the WRF model, namely Advanced Research WRF (ARW) and Nonhydrostatic Mesoscale Model (NMM). The WRF model is chosen in this study mainly because it is developed by a broad community of government and academic researchers, which has many advantages compared with other numerical weather models [Skamarock et al., 2005]. For example, it has better numeric schemes than the Fifth- Generation Penn State/NCAR Mesoscale Model (MM5) as WRF has less diffusive, larger effective resolution and permits longer time steps. Moreover, it is a flexible, state-of-the-art, portable code based on a plug-in architecture, which means a wide range of individuals or groups can supply their codes to improve the WRF model. More importantly, WRF can work efficiently in computing environments ranging from laptops to massively parallel supercomputers [Skamarock et al., 2005]. In this study, we use the WRF ARW solver to downscale (to get higher spatial and temporal resolutions) the reanalysis ECMWF data to obtain the threedimensional instantaneous wind field with high spatial (3.3 km), temporal (1 h), and vertical (28 levels) resolutions. On average, the lowest level is about 0.01 km above the ground. The WRF model achieves this through solving a series of atmospheric and physical equations. The ARW solver uses a C grid staggering scheme for spatial discretization and a time-split integration scheme for temporal discretization. In terms of the vertical discretization, ARW is formulated on a terrain-following hydrostatic-pressure vertical coordinate system. Further information about the WRF model and its derivation of three-dimensional wind field can be found in the WRF user manual and WRF technical manual [Skamarock et al., 2005; Wang et al., 2009]. It is worth remarking that there is also an option to derive the three-dimensional wind data using the radar measurement itself. However, there are two major shortcomings of this method. First, this method can only calculate the wind data at certain altitudes at radar beam heights. We have found that such wind data cannot replace the three-dimensional wind field (see section 4.5). Second, the radar-derived wind velocities lack the spatial resolution as compared with the WRF-derived ones The Normalized Gamma Drop Size Distribution Model To simulate the trajectory of a raindrop in the air, its diameter is a key element. A drop size distribution (DSD) model can provide the frequency distribution of all possible diameters of drops within a unit volume. Although the DSD cannot reveal diameters for all drops in the air (it is actually impossible or impractical to obtain such information), it can be used to estimate the possible relationship between the mean raindrop diameter and rainfall intensity. Islam et al. [2012] have studied the characteristics of DSD using a total of 162,415 1 min raindrop spectra obtained from the Joss-Waldvogel disdrometer for 7 year records within the Chilbolton Observatory, UK region, which is very close to this study area. The experimental results were fitted to a normalized gamma DSD model by drop diameter (D), shape parameter (u), and normalized intercept parameter (N w ), which is expressed as [Bringi et al., 2003; Ulbrich, 1983]: D u ND ð Þ5N w gu ð Þ exp 2ð41uÞ D (1) D m D m where D m is the mass-weighted mean drop diameter and g(u) is a function of the u. This model was parameterized under different seasonal (cold and warm), atmospheric (dry and wet), and rain type (stratiform and convective) conditions. It is found from the experimental results that exponential relation exists between the mass-weighted mean drop diameter and rainfall intensity. This relationship is used in this study to estimate the mean drop diameter in the individual radar pixel. 3. Methodology The task of the proposed scheme is to minimize wind effects on the process of bias correction for radar data by considering wind-drift and wind-turbulent induced errors. There are three major steps to achieve this. First, it is necessary to calculate the missed rainfall by gauges caused by wind-turbulent effect. Then the positions of gauges to obtain correct spatial relationship between radar and gauge measurements should be transformed. Finally, the radar bias should be calculated to adjust radar rainfall estimates. This process can be expressed as: w5r 3 fr; ½ CðG 1 dþš (2) where R, G, and w refer to the radar, gauge measurement, and corrected radar product. f is a function to calculate the radar rainfall bias. For example, radar rainfall bias can be estimated by the ratio between radar DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8574

5 and gauge rainfall measurements. The length of the accumulation period of radar and gauge data is a key issue in calculating the radar rainfall bias. C is a transform scheme to obtain correct radar-gauge match. d is the wind-turbulent error Adjustment of Wind-Drift Error As the Brue catchment is in a moist temperate maritime climate, and the initial heights of rain drops are within m, it can be assumed that the effects of growth (due to condensation) and decay (due to evaporation) of raindrops are ignored in this study. However, dry subcloud will lead to significant evaporation problems, especially for light-moderate rain. Therefore, this assumption may not be satisfied in some climate conditions. Interested readers can refer to Li and Srivastava [2001] to investigate the possible error due to this assumption. Moreover, we assume each raindrop can be regarded as being alone in the air, which means it does not interfere with other particles. This assumption can be accepted in the near ground area because the speed of drops is relatively fast and the collision rate is low compared with the upper air. In general, there are four main factors influencing the magnitude of the wind-drift effect. The first one is three-dimensional wind field from radar beam to the ground. Second, the distance of the study area to the radar can determine the initial spatial drop positions. The wind-drift effect increases with the growth of distances to the radar. Third, the type of hydrometeors is also a significant factor as it determines the terminal velocity of falling hydrometeors. The snowfall surely suffers more serious wind-drift error than rainfall if other conditions are the same. Finally, the radar and gauge rainfall accumulation period is a key issue as well. With the increase of the comparison periods, the wind drift becomes less significant. To make the simulation practical, we divide the space between the ground surface and radar beam into multiple vertical layers with each layer further divided into horizontal squared grids. Thus, the space is composed of a large number of three-dimensional subspaces in which the horizontal resolution (coordinates x and y) and the vertical resolution (coordinate g) are configured to be equal to those of the WRF model. In this way, the individual atmospheric element is uniform within a subspace. Herein the subspace is defined as the volume element (Dx, Dy, Dg) surrounding each grid point (x, y, g) in the three-dimensional grid domain. For the general numerical simulation process, one should configure the boundary condition and initial condition. As we extend several kilometers of the catchment in four directions to make sure the raindrops would not move out the space, there is no need to take special account on simulating the boundary situation. In terms of the initial condition, two essential factors should be configured. The first one is the diameter of raindrop, which can be derived from the drop size distribution (DSD). The study of windturbulent effect on rain gauge also considers the DSD. We use the normalized gamma DSD in both aspects. The DSD actually varies with each storm. However, we can divide it up with two typical situations, namely warm and cold. The events during autumn and winter seasons (23 September to 19 March) are defined as cold, and the events during spring and summer seasons (20 March to 22 September) are defined as warm. The discussion about DSD has been given in section 2.3. The other initial condition is the initial position of raindrop, including the horizontal coordinates and height. For the sake of simplicity, the center point of radar pixel is regarded as the initial horizontal coordinates. The beam height corresponding to this point is configured as the initial height. To improve the accuracy, we divide the original radar pixel equally into four subgrids and employ the center points in these subgrids as initial positions. Thus, we can simulate the rain drop trajectory in a higher spatial resolution. We attempt to further divide the subgrids, but the difference of simulated trajectory is quite small. Therefore, we assume the resolution of four subgrid scheme is sufficient. For the beam height, we estimate it using a simple geometric method from Dai et al. [2013a]. The center beam height can be approximately obtained by: R d 2 H c 5R d 3 sin ðreaþ1 (3) 2 3 IR 3 R e where R d is the distance from the radar site and R e (6371 km) denotes the earth radius, both in km. REA refers to the radar elevation angle in degrees, and IR (1.21) is the refractive index. The simulation process of raindrop trajectory for the mass-weighted drop starts from the radar beam position in the related subspace. The movements of raindrops are simulated in each subspace by solving the particle motion equations [Choi, 1997]. Gravitational force and drag force by the wind are the main contributing factors to determine the movements of raindrops in the wind field. The trajectories of the raindrops in each layer are computed separately with: DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8575

6 m d2 x dx Cd R e 56plD U2 dt2 dt 24 m d2 y dy Cd R e 56plD V2 dt2 dt 24 m d2 g dg Cd R e 56plD W2 dt2 dt 24 2mg 12 q a q w (4) where U, V, and W are the x, y, and g components of the wind field, respectively. For other parameters, m represents the mass of raindrop, R e refers to the Reynolds number, q a and q w refer to the densities of air and water, respectively, l is the air viscosity and D is the diameter of drop. C d denotes the drag coefficient on the raindrop, which is estimated using its corresponding empirical relationship with the diameter of drop [Chow et al., 1988]. The numerical simulation process is performed in each time step. At the end of the step, new positions and velocities of the drops obtained are used for the next step. The displacements in x and y directions for all time steps are accumulated separately. The procedure is iterated until the raindrops move out of the subspace. The drop exit positions and speeds at the boundary are then regarded as the input for the next subspace. The determination of the next subspace provides the raindrop exit position. The displacement in a layer is given as: dx i;k 5 Xtn k t51 dy i;k 5 Xtn k under the condition that the vertical displacement is equal to the layer height: t51 dx i;k;t (5) dy i;k;t (6) X tn k t51 dg i;k;t 5Dg (7) where dx i,k,t, dy i,k,t, and dg i,k,t are the displacements in the x, y, and g directions, respectively, for the grid i, layer k, and time step t. tn k is the time step for the layer k, which is obtained by solving equations (4) and (7). The simulation is performed in the new subspace. This process is repeated until the raindrops reach the ground. These points on the ground are named as revised raindrop points (RRPs), whose coordinates (X i,r, Y i,r ) for grid i are obtained using: X i;r 5X i;o 1 Xkn k51 Y i;r 5Y i;o 1 Xkn k51 dx i;k (8) dy i;k (9) where (X i,o, Y i,o ) is the original horizontal coordinate of raindrops, which is called the original raindrop point (ORP). kn denotes the layer number. Thus, we obtain the final positions on the ground of the observed raindrops by radar. To evaluate the proposed scheme, we need to find the matching radar pixel corresponding to each rain gauge. In other words, we need to reversely derive the corresponding radar pixel for a given rain gauge. Traditionally, we only check the radar pixel and the matching rain gauge according to their spatial positions. Herein, for each rain gauge, we search the surrounding RRPs and choose the nearest one to that gauge. The radar pixel with the final position of RRP and the matching rain gauge is regarded as an adjusted radar-gauge pair. This process is shown in Figure Adjustment of Wind-Turbulence Error at Gauge/Ground Level Gauge measurement is normally assumed to be the ground truth in adjusting radar rainfall estimates. However, it is also subjected to the wind induced error, which has been discussed in section 1. The adjustment of wind-turbulent error should be synchronously performed with the treatment of wind-drift error, otherwise we cannot estimate where the wind-induced discrepancy between radar and gauge DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8576

7 Altitude PDF D Beam top Beam centre H Y Wind speed X Z Beam bottom Displacement Figure 1. Schematic diagram for wind-drift effect on radar-gauge comparison. There are four key elements in this process: (1) initial spatial positions of drops (within radar beam); (2) drop size distribution; (3) wind field; (4) height of the radar center beam above the ground (H). The image on the top right corner denotes the drop size distribution with variable D as the drop diameter. The grids refer to the radar pixels, while the orange point on it represents the projective location of the raindrop observed by radar, and the blue point is the displaced location of the raindrop on the ground [Dai et al., 2013a]. measurement comes from. More importantly, the outcomes may be smudged and inconsistent if we only consider one aspect of the wind effect. The catch ratio (ratio of the number of drops that ended up in the rain gauge to those which are supposed to be caught by the gauge without wind effect) of raindrops for a given rain gauge is determined by three factors: wind speed, drop size distribution, and dimension of the rain gauge. Two types of simulated methods are adopted in this study to estimate the catch ratio, namely numerical-based scheme and empiricalbased scheme. Under the same assumption in wind-drift adjusted method, the numerical-based scheme aims to simulate the trajectories of rain droplets. This is divided into three main steps. 1. Simulate the turbulent flow around the given rain gauge numerically based on the measured wind field. 2. The trajectories of drops within the airflow are simulated and the error for a given wind speed and drop diameter is obtained. 3. The total catch rate of the rain gauge is estimated. The numerical simulation of ground airflow is a general topic in meteorology and many results have been achieved in the previous studies. Herein, we utilize the results given by Nespor and Sevruk [1999]. Three rain gauges are studied and compared in Nespor and Sevruk [1999] and they showed a similar behavior in estimating the wind-turbulent error. We have observed that the standard tipping bucket Cassella used in this study has the similar dimension toward the automatic station tipping bucket ASTA used there, especially for the diameters of the funnel, as a pivotal factor that influences the catch rate of gauge, they are exactly the same (160 mm) for both gauges. We use the same parameters to estimate the wind-turbulent error e num for a given wind speed w and drop diameter D with the equation [Nespor and Sevruk, 1999]: e num ðw; DÞ5b n1 ðwþd b n2ðw Þ e 2b n3ðwþ (10) where b n1, b n2, b n3 are the coefficients depending merely on the wind speed. This error is integrated over a drop size distribution: d num ðw; D m Þ5 p 6 ð 1 0 e num ðw; DÞD 3 ND; ð D m ÞvD ð ÞdD (11) where d num represents the total missing rainfall due to wind effects in mm. The adjusted rainfall is the sum of the original measurement and total missing rainfall. D m is the mass-weighted drop diameter, which is DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8577

8 determined by rain intensity. v refers to the terminal fall speed of drops, which is estimated as a function of drop diameter. N(D, D m ) is the drop size distribution. The empirical-based scheme is a regressed method using the long-term historical data sets of gauge measurements subjected and not subjected to the wind-induced error. Yang et al. [1998] used the double-fence intercomparison reference of true rainfall to explore a relation between wind speed and wind-induced error. The regression equation for unshielded gauge is expressed as: d emp ðw; RÞ5 12exp b e1 2b e2 w b e3 =100 R (12) where R is the rain intensity, and b e1, b e2, b e3 are regressed coefficients. Until now, there are no detailed comparative studies between these two methods, so both methods are implemented to obtain comparative results in this study. Another issue associated with wind-turbulent adjustment is the determination of time scales. In fact, there should be an optimal sampling scale for the given rain gauge. This issue is outside of the scope of this study, and the hourly data are chosen as the default value to be consistent with radar data. We will investigate the influence of time scale on the wind-turbulent error in future studies Verification Method The goal of this study is to assess and match the differences between rainfall surfaces described by radar and gauge. The best result is achieved when two rainfall surfaces (i.e., two-dimensional distribution of rainfall) exactly overlap, indicating that the radar measurement achieves the same quality compared to the wind-free gauge rainfall. As we have 49 rain gauges within the area of only 135 km 2,therainfall surface estimated by gauge rainfall can easily be obtained with reliable accuracy by either simple linear regression or complicated interpolation techniques. On the other hand, there are two challenges associated with rainfall surface estimated by radar measurement. First, there are only 30 radar pixels (with a resolution of 2 km) within the study area. More importantly, the rainfall measured by radar is discrete area-based rainfall. The former one can be addressed by extending the study area to obtain more radar pixels. We only obtain two or three additional pixels in each direction because the radar rainfall quality has a close relationship with the distance to the radar center. The latter issue can be addressed through interpolationwithanassumptionthatthecenterpoints of pixels have the pixel rainfall values. The interpolation of radar rainfall has been widely used in merging radar and gauge rainfall through interpolation algorithms [Haberlandt, 2007; Kalinga and Gan, 2006; Schuurmans et al., 2007; Seo et al., 1990a, 1990b; Velasco-Forero et al.,2009].thisisbecausethenumberofraingauges is not enough for interpolation in some catchments, while radar measurement is a useful option to provide additional information. Herein, we also interpolate radar rainfall to obtain a finer radar sampling data. The Bicubic Spline interpolation, with high stability and calculation efficiency, is adopted in this study. The core part of this algorithm is to divide the study area into a number of subdomains and each subdomain is fitted with a third-degree polynomial, which should satisfy the commonly required conditions, such as function gradient continuity. To evaluate the proposed scheme, we calculate the differences between two rainfall surfaces before and after considering the wind effect. The differences can be expressed by the mean error in radar-gauge pairs, such as mean absolute error (MAE) or root mean square error (RMSE). However, there is an overall bias of radar rainfall compared with gauge rainfall. We find the mean error is very sensitivity to the overall bias. The efforts to calculate this bias will definitely introduce new uncertainty to the evaluation process. For this reason, the Pearson correlation coefficient between hourly radar rainfall and gauge rainfall vectors is calculated to evaluate the proposed scheme. In fact, there are also some problems associated with the Pearson correlation indicator. Primarily, this correlation coefficient only measures the trend relationship between two rainfall surfaces. For example, if radar rainfalls are always half of those of gauges, the correlation between them would be 1, which gives a wrong impression that two surfaces are exactly same. However, our purpose is to assess if the proposed scheme improves the goodness of fit between the shapes of the two rainfall surfaces. If the correlation coefficient improves, even there may be an overall bias, it can still prove that the proposed scheme achieves a positive effect. As mentioned above, the proposed scheme is just a premise step before radar bias correction. Furthermore, the Pearson correlation coefficient can only describe the linear correlation of two vectors, while the radar-gauge pairs sometimes show nonlinear trend. This is the reason why some studies present that there exist conditional biases DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8578

9 Table 2. Major WRF Model Settings Items Parameters/Schemes References Dynamics solvers ARW Data interval (s) 21,600 s Latitude of center Longitude of center Levels of eta coordinate system 28 Microphysics scheme WRF single-moment three class Hong et al. [2004] Cumulus parameterization scheme Kain-Fritsch Kain [2004] Planetary boundary scheme Yonsei University scheme Hong et al. [2006] Land surface model Noah model Chen and Dudhia [2001a, 2001b] Shortwave radiation scheme Dudhia model Dudhia [1989] Longwave radiation scheme Rapid radiative transfer model Mlawer et al. [1997] between radar and gauge measurements, which means the bias changes with the rain intensity [Ciach et al., 2000; Habib et al., 2004]. To make the evaluated process more credible, we also investigate the Spearman rank correlation that can deal with nonlinear situation between radar and gauge rainfall surfaces. As we have limited radar and gauge pairs to compute the correlation coefficient, the significance of correlation should be carried out to ensure the correlation coefficient is reliable. The p value is used to test the hypothesis of zero correlation. Each p value is the probability of getting a correlation as large as the observed value by random chance, when the true correlation is zero. We calculate the p value to test the significance of a correlation coefficient by employing the t distribution. If the p value is small, say less than 0.05, the corresponding correlation is statistically significant. 4. Model Computations and Results 4.1. Calculation of Three-Dimensional Wind Field by WRF The WRF model is used to downscale the ERA-40 wind data into higher spatial and temporal resolutions (3 km and 1 h, respectively). Three nesting domains are configured with spatial resolutions of 3.3, 10, and 30 km in the numerical experiments. The center coordinates of domain 3 are set to Nand22.47 W, which are at the center of the study catchment. To obtain detailed vertical variation of the wind field, 28 vertical layers are chosen on a hydrostatic-pressure vertical coordinate system (the lowest 6 layers are mainly considered in the study domain). The major physical parameterizations such as microphysics scheme, planetary boundary, together with their references, are shown in Table 2. As the goal to derive the three-dimensional wind field is a fundamental processing of the WRF model, we adopt the most extensively used parameterizations [Chen and Dudhia, 2001a, 2001b; Dudhia, 1989; Hong et al., 2004, 2006; Kain,2004;Mlawer et al., 1997]. The WRF model is set up to run for all 20 events. The wind data in domain 3 are then used in this study Simulation of Rain Drops Trajectory and Displacement The first step in numerical simulation of the raindrop trajectory is to calculate the initial conditions of the model. The area covering these 30 pixels is extended by adding 4 km along its four boundaries. This can ensure the movements of raindrops are constrained within the computation area and there is no need to consider the boundary condition. The relationship between rainfall intensity and mass-weighted drop diameter is parameterized and displayed in Figure 2 for the warm and cold situations, respectively. In the chart, the crosses and circles represent the observed data, which are parameterized to exponential equations, respectively. With these conditions, the trajectories of raindrops are numerically simulated and their displacements on the ground are estimated. The time step of the numerical calculation is set to 0.5 s. There is no need to shorten the computation time as it is found that the velocities of drops change very little at higher temporal resolutions. Figure 3 illustrates the simulated results for Event 1 (detailed information about this event is listed in Table 1). In these figures, the orange dots are the locations of rain gauges, and the background grids correspond to the real radar pixels (with pixel size of 2 km). The x and y axes are the easting and northing coordinates based on the British National Grid coordinate system. The red dots are the locations that the rain gauge should be compared. It can be seen from Figure 3 that there are totally different patterns of drift distance and direction in each time step. For example, in Figures 3f and 3g, the drift distances are quite small, and the raindrops almost stay in the original radar pixels. However, as shown in DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8579

10 Mass weighted diameter (mm) Cold input Warm input y=0.970x y=0.966x Radar rainfall (mm/h) Figure 2. The relationships between radar rainfall and mass-weighted rain drop diameter for cold and warm situations, respectively. The input data are from the Chilbolton Observatory experiment [Islam et al., 2012]. The observed data sets are fitted into the exponential equations. Figures 3b, 3c, and 3e, most raindrops move to the neighbored pixels, whose traveling distances even cover two radar pixels. In addition, we can observer that the vector lengths in the north part of the study area are longer than those in the south part, indicating the spatial variation of drift distances. Tables 3 and 4 show the detailed drift distances and directions, together with ground wind speeds and directions for Events 1 and 12, respectively. The comparison between the ground wind direction and drift direction reveals there is no clear relation between them.wecanobservetotally opposite directions in some cases for Event 1. For example, the observed ground wind direction is 86 at 01 05:20, while its corresponding drift direction is 238. The case with the observed wind direction of only 77 at 01 05:22 surprisingly drifts to the direction of 255. These facts indicate that the ground wind direction simply cannot represent the variation of three-dimensional wind field.in terms of ground wind speed and drift distance, we can observe a positive correlation between them. For example, the maximum ground wind speed (4.2 m/s) in Event 1 (at time 01 05:12) corresponds to the largest drift distance (3.0 km). In addition, in Table 4, the ground wind speed at the last two time steps are minimum values (1.9 and 1.7 m/s, respectively) during this event, and their corresponding drift distances are also the smallest (1.3 and 1.4 km, respectively). In section 4.5, the further exploration of the possible relationship between the ground observed wind field and simulated wind-drift field is given Adjustment of Wind-Turbulence Error The wind-turbulent-induced error of rain gauge is calculated using both the numerical-based method and empirical-based method. The results are shown in Tables 3 and 4 for Events 1 and 12, respectively. The parameters of the numerical equations are from Nespor and Sevruk [1999]. One can see from these tables that for the numerical method, the ground wind speed plays a major role in determining the wind-turbulent error. For instance, the gauge rainfall at time 01-05:06 (2.57 mm) is larger than that at time 01-05:07 (2.13 mm), while its wind speed (2.8 m/s) is weaker than the latter (3.2 m/s). It is observed that the wind-induced error calculated using the numerical method follows the trend of wind speed as the former error is smaller than the latter one (0.08 mm compared with 0.10 mm). On the contrary, gauge rainfall intensity is more dominating than wind speed for the empirical method. In the above case, the wind-turbulent error based on the empirical method of the former one (0.28 mm) is larger than the latter (0.24 mm). In addition, it can be seen from Tables 3 and 4 that during the light rain, the estimated results by two methods are quite similar. This can be proven by the last five time steps in Table 4, when the gauge rainfall intensities are all less than 1 mm. The wind-turbulent errors estimated by the two methods are almost the same for low values. However, with the rainfall intensity grows, the changes of the estimated error by the numerical method are quite small. In other words, the numerical scheme is insensitive to the rainfall rate. This is because the rainfall rate changes the DSD in the numerical method, while only huge differences of rainfall rates will generate quite different DSDs. Based on this fact, the estimated results of the empirical method are more reasonable Wind-Drift Effects in All Events and Evaluation of the Method The radar data are first fitted to a finer rainfall surface to better evaluate the effects of the proposed scheme. We apply the Bicubic Spline interpolation with grid sizes ranging from 0.1 to 1 km and the evaluated results DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8580

11 142 (a) 1994/01/05 01: (b) 1994/01/05 05: (c) 1994/01/05 06: (d) 1994/01/05 07: (e) 1994/01/05 12: (f) 1994/01/05 20: Figure 3. The estimated displacements of raindrops under the effects of wind drift for Event 1 ( ). The background grids correspond to the real radar pixels. The orange dots are the locations of rain gauges, while the red ones are their adjusted positions. It is worth remarking that the adjusted rain gauge should compare to the interpolated radar grid at the resolution of 0.1 km. are found to vary slightly around 0.1 km. A default value of 0.1 km grid size is adopted for all the evaluations. The rank correlation coefficients of the original radar-gauge pairs and improved ones for Events 1 and 12 are shown in Tables 5 and 6, respectively. To focus on the credible results, we only display the time steps with significant correlations (with p values smaller than 0.05). It is worth remarking that significant correlation does not indicate high correlation coefficient. Except several cases, the correlation coefficients increase DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8581

12 142 (g) 1994/01/05 21: (h) 1994/01/05 22: Figure 3. (continued) in most time steps (see Figure 4). The correlation coefficients of the original radar-gauge pairs are sometimes very poor, and even show negative correlation in some situations (e.g., at time 01-05:06). After carrying out the proposed scheme, the coefficients are significantly improved and more acceptable for hydrological application. For example, in Table 6, only 25% correlation coefficients are larger than 0.5, while this rate grows to 58% for the improved radar-gauge pairs. The summary of the evaluation results for all 20 events is listed in Table 7. The rank correlation coefficients are the event-averaged values. To visually illustrate the comparison between the original correlation (OC) and improved correlation (IC), we calculate the improved rates using the equation of (IC-OC)/OC. Herein, we use the improved rate instead of the differences between the original and improved correlation coefficients as the low primitive correlation events attracts more attention. In fact, the unstable quality of the radar product is a major limitation for its applications in hydrological studies. This is mainly due to the issues that radar rainfall product sometimes is of good quality, but sometimes deviates greatly from the reference gauge rainfall. It has been a great challenge for meteorological radar researchers to find out what causes the radar to perform badly in some situations. Wind effects as what we discussed in this study are a significant error source. In Table 7, it is observed that for most events, the improved rates are greater than 10%, and nearly half of the events are more than 20%. After the improvement, except four events, all the eventaveraged correlations are greater than 0.5. A high agreement between radar and gauge measurements makes the radar product more acceptable and trustworthy in hydrological and meteorological applications Relationship Between Wind and Wind-Induced Error This study simulates the movement of raindrops using the three-dimensional wind field. However, one may wonder if there is any possible relationship between the ground wind field and wind-induced error. It should be much easier to implement the proposed scheme if we can replace the three-dimensional wind field with Table 3. The Information of Ground Measurements and Estimated Results of Wind Drift and Wind Turbulence for Event 1 ( ) a Time (MM-DD:HH) Gauge Rainfall (mm) Ground Wind Direction ( ) Ground Wind Speed (m/s) Drift Direction ( ) Drift Distance (km) WT Error-Num (mm) WT Error-Emp (mm) 01-05: : : : : : : : a All statistics are averaged values for the study area. The ground winds represent ones at 2 m height. WT error-num refers to the result of the numerical simulation of wind-turbulence effect, while WT error-emp is from the empirical simulation. DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8582

13 Table 4. The Same as Table 3 But for Event 12 ( ) Time (MM-DD:HH) Gauge Rainfall (mm) Ground Wind Direction ( ) Ground Wind Speed (m/s) Drift Direction ( ) Drift Distance (km) WT Error-Num (mm) WT Error-Emp (mm) 09-06: : : : : : : : : : : : the ground wind observation. For this reason, we investigate the relations between the ground wind speed and drift distance, as well as ground wind direction and drift direction, which are shown in Figures 5 and 6. The observed wind speeds, together with their corresponding simulated wind-drift distances for all time steps of the 20 events are plotted in Figure 5. In general, there is a visual agreement between wind speed and drift distance. For better comparison, they are fitted into a simple linear model. The fitted model coefficients and the error indicators are also shown in Figure 5. We use the R square (R 2 ) and RMSE to evaluate the goodness of fit. With only of R 2 and as high as of RMSE, the regressed effect is unacceptable for practical application. As shown in Figure 5, the fitted error can reach approximate 2.5 km (e.g., the dot with drift distance of around 5.5 km). A stricter model can improve the fitted effect and a cross validation can be carried out if there are more simulated results. However, it should be admitted that uncertainty will inevitably bring in the proposed scheme with a consideration of large dispersion of dots in Figure 5. In terms of the drift direction, we also explore its connection with the wind direction. For a better visual comparison, the pairs of the ground direction and drift direction are divided into four main orientations (see Figure 6). It is interesting to find that the correlations of the northeast (0 90 ) and southwest ( ) seem to be better than the other two orientations. With the southwest case as an example (see Figure 6c), the drift directions are almost within the southwest domain. On the contrary, it can be found from Figure 6d that there is no drift direction greater than 270 in the northwest case ( ). Based on these facts, it is concluded that there are large differences between the ground wind direction and wind-drift direction. In other words, it is unreasonable to use the ground wind data to replace the three-dimensional wind field in studying the wind-induced error for radar-gauge comparisons. In midlatitudes precipitation is commonly associated with extratropical cyclones. This may be the reason that in most cases ground winds provide poor information for wind drift estimation. Although Tables 3 and 4 and Figure 5 do not reflect the impact of wind shear in general, they quantitatively demonstrate that the vertical wind shear invalidates the use of ground winds in wind drift estimation (at least in England) even when the radar beam is at the height of only m above the ground. Table 5. The Correlation Coefficients Between Radar and Gauge Rainfalls for Event 1 ( ) a Time (MM-DD:HH) Original Improved Significance Difference 01-05: : : : : : : : Average a Significance refers to the p value that tests the hypothesis of no correlation. 5. Discussion In this study, we have proposed a fully formulated scheme to tackle the wind effects on radar-gauge comparison. The evaluation results show the proposed method has a good performance in improving the agreement between radar and gauge rainfall surfaces. However, there are still some key concerns that readers should be aware of. First, it is still debatable in the hydrological and meteorology community whether it is essential to adjust wind induced error. DAI AND HAN VC American Geophysical Union. All Rights Reserved. 8583