## 1. Introduction

Weather radar estimates are typically processed and stored in polar coordinates. Because all radars have a finite nonzero beamwidth, the radar sampling volume increases with distance from the radar, while the pulse length is constant. This causes sampling volume dimensions in the azimuthal direction to differ by more than an order of magnitude from the fixed radial dimension. For example, for a beamwidth of 1° and radial resolution of 1 km, radar sampling volumes are less than 0.1 km^{3} near the radar and much greater than 4.0 km^{3} at far ranges.

Native radar data are recorded in spherical-polar coordinates centered on the radar site. However, most applications of weather radar products require conversion to geographical or other planar coordinate systems on a Cartesian grid. In addition, effective weather warning often integrates meteorological information from many sensors such as radars, satellites, and weather stations (Stumpf et al. 2002) typically sampled in different coordinate systems. In other applications, radar data are integrated with geospatial data that are geolocated using a Cartesian coordinate system such as the universal transverse Mercator (UTM) projection. These applications necessitate smoothing and/or filtering of radar data and remapping. Remapping of radar data is also needed when data from multiple radars need to be integrated or compared (e.g., Brandes et al. 1999). Other applications that require radar data remapping include assimilation of radar estimates into numerical weather prediction models, multisensor rainfall estimation algorithms, input of radar-rainfall estimates to hydrologic models, and verification of quantitative precipitation forecasts. Precipitation estimates from the U.S. National Weather Service, Weather Surveillance Radar-1988 Doppler (WSR-88D; http://www.roc.noaa.gov/), for instance, are usually remapped into the Hydrologic Rainfall Analysis Project (HRAP) coordinate system, a polar stereographic map projection using a spherical Earth datum and a nominal grid spacing of 4 × 4 km^{2} (Greene and Hudlow 1982; Fulton 1998) and different temporal resolutions (e.g., 1- and 3-h accumulations). Moreover, in many hydrologic applications, radar-rainfall estimates are remapped onto much smaller grid sizes, such as for the 1 × 1 km^{2} grids.

Numerous methods have been used for remapping of radar data from polar to Cartesian format (e.g., Zhang et al. 2005; Trapp and Doswell 2000; Henja and Michelson 1999). Remapping requires projection of a variable (e.g., rain rate) onto a Cartesian grid based on the values of the variable in all polar grids that lie completely or partially within the Cartesian grid. The most commonly used remapping approach is linear averaging (Greene and Hudlow 1982). Linear averaging calculates the arithmetic average of all polar bins whose centers fall within a Cartesian grid, without consideration of the partial area contribution of each polar bin. This is the method currently used to prepare gridded rainfall products by the U.S. National Weather Service (Fulton 1998). The National Center for Atmospheric Research (NCAR) uses the same approach in addition to others that combine different methods (e.g., Wurman and Gill 2000). Other methods include the nearest-neighbor approach (Jorgensen et al. 1983); the distance-weighting approach (e.g., Harrison et al. 2009; Henja and Michelson 1999); and Cressman weighting, which is a sophisticated type of the distance-weighting approach (e.g., Bousquet and Chong 1998; Zhang et al. 2005; Weygandt et al. 2002). More sophisticated approaches have also been suggested (e.g., Julier and Uhlmann 1997).

Remapping methods such as those mentioned above are approximate methods, at least in the context of radar-rainfall fields. Henja and Michelson (1999) stated, “An optimal 3-D interpolation algorithm, in terms of representativity of its results, would select all polar bins which are defined as influencing a given Cartesian pixel and derive weights based on the proportion of each individual bin’s volume in the sample volume. In practice, this strategy is too computationally demanding to use with contemporary radar computing environments. Instead, this algorithm can be successfully approximated by defining each polar bin as a point in its geometric centre, and then using it wherever it lies within a sample volume.”

In this paper, we describe the development of a method that exactly represents the precise remapping (from polar to Cartesian) approach described above by Henja and Michelson (1999). The computational limitations on this method described by Henja and Michelson (1999) are no longer a factor on contemporary computing hardware. The sensitivity of interpolated radar-rainfall rates and accumulations to the remapping method and interpolation grid size is quantified by examining the case of remapping of two-dimensional rainfall fields computed from radar reflectivity estimates. We derive the equations that quantify the difference in interpolated rainfall between precise remapping and approximate methods that rely on averaging rainfall values of the radar bins whose centers fall within a certain distance from the Cartesian grid center. Radar observations were used to illustrate the difference in interpolated grid-averaged rainfall rates and total event accumulations using approximate and precise remapping approaches. Remapped rainfall estimates from two weather radars with different resolutions were used. We also examined the sensitivity of high-resolution radar-rainfall estimates to the interpolation grid resolution.

## 2. Methods

In this study, gridded rain rates were computed from frequently used approximate remapping methods and compared to precise remapping. Below is a brief description of the approximate remapping methods used, followed by a detailed description of the precise remapping method developed in this study.

### a. Approximate methods

#### 1) Nearest neighbor

The nearest-neighbor method (Jorgensen et al. 1983) is the simplest approach that uses the value of the polar bin whose center is nearest to that of the Cartesian grid cell. However, it has many shortcomings. For example, horizontal advection of rainfall is uncertain, and because radar observations are typically made well above ground level, identifying the nearest bin becomes challenging since precipitation particles will be displaced as they fall. This a problem for all remapping methods but is more serious when only one bin is considered. In addition, a single radar bin estimate is likely to be affected by more measurement noise. These factors make the average of radar estimates over a certain area around the grid center more appealing.

#### 2) Linear averaging

In linear averaging (e.g., Fulton 1998), all radar bins whose centers fall within the Cartesian grid have the same weight regardless of how much of the radar bin area falls within the grid cell. This approach can introduce significant discrepancies in certain situations. For example, it is possible that the bin’s center falls within a particular grid cell with less than or just over 50% of the bin’s area contained within that cell, but it has the same weight as a bin that falls completely inside the grid cell. Conversely, a bin’s contribution will be completely ignored if its center falls outside the grid cell, even if a substantial portion of the bin lies within the grid cell.

#### 3) Distance-weighted averaging

Distance-weighted averaging (e.g., Harrison et al. 2009) is supposed to be an improvement on simple averaging as it tries to take into account the contribution from each radar bin by multiplying the bin estimate by a weight based on its distance from the gridcell center. This method is similar to the inverse distance interpolation method that is used for interpolation of rainfall data based on rain gauge observations. It is worth mentioning that some distance-weighted averaging schemes used more than one elevation. However, it still uses the bins whose centers fall within the grid cell or within a certain distance from it. This approach also has some important limitations. For example, a radar bin whose center does not fall within the Cartesian grid cell (or a certain radius from the grid center) is still ignored no matter how much of its area falls within the grid. Moreover, in reality, the contribution of a radar bin does not depend on its distance from the gridcell center only but also on the orientation of the radar bin.

### b. Precise remapping

In the context of rainfall rates, approximate methods can violate the principle of mass conservation and introduce errors in radar measurement, which is extremely important for hydrological applications of weather radar-rainfall estimates. For example, Sharif et al. (2002) and Vivoni et al. (2007) showed that errors in radar-rainfall estimation can be amplified in hydrologic model predictions. As suggested by Henja and Michelson (1999), we developed a geometrically precise algorithm for remapping by using the beam radar geometry to define the points (two dimensions) or planes (three dimensions) of intersection of the Cartesian grid and collocated conical radar bins. For example, the three-dimensional algorithm is useful for merging radar with cloud model output (e.g., Sharif et al. 2002) and visualizing three-dimensional radar data (e.g., Terblanche et al. 2001). For the sake of simplicity, focus here will be on two-dimensional rainfall fields computed from radar reflectivity for typical meteorological and hydrologic applications.

In our two-dimensional analysis, “radar bin area” refers to the vertical projection of the radar pulse volume onto Earth's surface. Any pair of adjacent points is connected by a curve or a straight line. These lines divide the enclosing grid cell into a number of polygons (one of the sides can be a curve). Each polygon represents the contribution from a different radar bin. Depending on the gridcell resolution, the algorithm also identifies cases where a radar bin falls completely inside a grid cell or vice versa. The algorithm then computes the area of each polygon and multiplies it by the bin’s precipitation estimate and then divides the sum of these products by the gridcell area to compute the average precipitation over the grid cell. For example, in Fig. 1 the average rainfall will be computed from seven bins for the square grid cell marked “1” in Fig. 1a and from four bins for the cell marked “2” in Fig. 1b.

For any radar location with fixed scan geometry, the remapping algorithm result can be saved in a lookup table. The lookup table includes the index of each Cartesian grid (*x*, *y* or *i*, *j*), the radar bins that contribute to the grid rainfall (defined by their range and azimuth value), and the partial contribution of each bin to the total area of the grid. The code can also save information about the intersection points of each grid with the polar bins. If the radar is operated in an indexed beam mode (i.e., the starting azimuth of the scan is the same for consecutive scans or is different by multiples of the azimuthal spacing) the lookup can be used as is; otherwise, a simple algorithm can modify the lookup table as the azimuth is shifted by retrieving and readjusting information about the intersection points between each grid and the radar bins. Furthermore, a very simple and fast algorithm can generate a lookup table for a grid of coarser resolution from a table that has been developed for a grid of finer resolution by just adding the contribution by all the radar bins that contributed to each small grid that falls within the larger grid. So, it will be more computationally efficient to develop the main lookup table for the finest possible grid resolution (e.g., 10 × 10 m^{2}) and use it to build tables for larger grids. Lookup tables have been used in other interpolation methods (e.g., Mittermaier and Terblanche 1997).

## 3. Comparison of approximate methods and precise remapping

We show the importance of the precise remapping method by quantifying the magnitude of differences in gridded rainfall accumulations when using other remapping methods. From Fig. 1, it is clear that the difference in estimated precipitation between approximate and precise remapping methods depends on the relative sizes and orientations of the radar bins and the grid. However, it is also a function of precipitation gradient (i.e., the difference in rainfall estimates among adjacent bins). In the case of convective precipitation, high gradients are frequently observed that create large differences in instantaneous rainfall estimates among adjacent radar bins.

*R*

_{p}over a Cartesian grid cell of area

*A*

_{g}can be expressed as

*a*

_{i}is the partial area of polar bin

*i*that falls within the Cartesian grid cell and

*r*

_{i}is the rain rate for that bin.

*R*

_{a}only represents the nearest bin or the average for the polar bins whose centers fall within the Cartesian grid cell. The difference between approximate methods and precise remapping is illustrated in Fig. 2, which shows the intersection of 1° × 1 km radar bins and 1 × 1 km

^{2}Cartesian grid cells to illustrate the differences between approximate methods and mass-conserving, precise remapping. For the nearest-neighbor method:

The intersection of 1 km × 1° polar radar bins with a 1 × 1 km^{2} square grid at 30 km from the radar.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The intersection of 1 km × 1° polar radar bins with a 1 × 1 km^{2} square grid at 30 km from the radar.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The intersection of 1 km × 1° polar radar bins with a 1 × 1 km^{2} square grid at 30 km from the radar.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

*R*

_{a}will be equal to the rain rate for that bin. In simple averaging:

*N*is the number of radar bins whose centers fall within the grid cell. In Fig. 2, those are bins 2 and 6. In distance-weighted averaging:

*n*is typically taken as 1 or 2. Again, in Fig. 2 the bins considered are bins 2 and 6 unless a certain radius from the center of the grid cell is used.

It is to be noted that in approximate averaging all radar bins with centers that overlie a Cartesian grid are either assigned the same weight or a weight related to their distance from the grid center regardless of how much of each polar bin’s area falls within the grid. For example, in Fig. 2, for the grid immediately to the right of the one with the dotted boundary, in simple averaging the two bins with centers overlying the grid cell will have the same weight although one of the bins falls almost entirely within the cell while only half of the second overlies the cell. For the same cell, distance-weighted averaging will assign an excessively high weight to one bin because its center almost coincides with the gridcell center. Also, depending on the sizes of the radar bin and grid cell, and distance from the radar, there may be only one bin or no bin with its center overlying the grid cell, thereby reducing linear averaging and distance-weighted averaging to the nearest-neighbor method. For all approximate methods, *R*_{a} will be strictly accurate only when the interpolation grid falls entirely within one polar bin.

*R*

_{a}, is the rain rate for bin 2, which occupies about one-fourth of the area of the grid. For linear and distance-weighted averaging,

*R*

_{a}is the average rain rate for the area shaded by the dashed lines. For approximate methods,

*R*

_{a}is almost always estimated for a subarea of the grid cell (the area shaded by the dashed lines). In the following analysis, we call this subarea

*A*

_{a}. The area of the grid cell that is not considered (excluded) in approximate methods [i.e., the area contributed by bins 1, 3, 4, 5, 7, and 8 (the unshaded part)], will be called

*A*

_{x}. Again,

*A*

_{x}is nonzero in the vast majority of cases and it can be much larger than

*A*

_{a}. It is to be noted that

*R*

_{a}is an estimate of the average rain rate for

*A*

_{a}(the shaded area), which itself is not 100% accurate in general. For simplicity, we will assume that

*R*

_{a}is 100% accurate. The average rain rate for the excluded area of the grid (i.e.,

*A*

_{x}) that is computed precisely (e.g., from the six radar bins composing the unshaded part of the grid cell) will be given by

*R*

_{x}. The precise average rain rate (in linear space) for the grid cell can be expressed as

*A*

_{g}=

*A*

_{a}+

*A*

_{x}.

From Figs. 1 and 2 it is clear that the area ratio *y* will depend on the grid orientation relative to the radar bins, distance from the radar, and the difference in area between the polar bin and the grid. As the radar bins grow larger away from the radar, the area ratio *y* is likely to become smaller with fewer bin centers overlying the grid cells. This is illustrated in Fig. 3, which shows the intersection of 1° × 0.15 km radar bins and 1 × 1 km^{2} Cartesian grid cells as the bin widths approach and then slightly exceed 1 km. It can be visualized that in some cases only one radar bin will have its center inside a grid cell with that bin occupying a tiny fraction of the gridcell area (*y* being less than 0.08 for the grid cells near the center of Fig. 3). As the bins become wider, there will be an increasing number of grid cells within which no radar bin center will fall. For these grids, the nearest bin will be used and the ratio *y* can also be smaller still.

The intersection of 0.15 km × 1° polar radar bins with a 1 × 1 km^{2} square grid at 58 km from the radar.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The intersection of 0.15 km × 1° polar radar bins with a 1 × 1 km^{2} square grid at 58 km from the radar.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The intersection of 0.15 km × 1° polar radar bins with a 1 × 1 km^{2} square grid at 58 km from the radar.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

## 4. Quantifying the differences between approximate methods and precise remapping

An algorithm was developed to compute the values of the area ratio (i.e., *y*) at a variable distance from the radar for different radar bin and Cartesian grid resolutions. The results are shown in Fig. 4. In Fig. 4a, the values of *y* are shown for all 1 × 1 km^{2} grid cells at a distance of up to 140 km from the radar with a bin size of 1° × 0.15 km. The histogram of the average value of the area ratio at any distance from the radar is shown in Fig. 4b. The variability of the average value of the area ratio with distance for different grid resolutions for 1° × 0.25 km radar bins is shown in Figs. 4c and 4d. At far ranges, the average value of *y* is well below 0.5 for 1 × 1 km^{2} grids.

(a) A scatterplot of the computed values of *y* [see Eqs. (1)–(3) for definition] for a 1 × 1 km^{2} grid placed as a mesh surrounding a radar of 1° × 0.15 km radar bins as a function of the distance from the radar and (b) a histogram of the average values of *y* for 10-km range bins. The histograms of the average values of *y* for (c) 1 × 1 and (d) 4 × 4 km^{2} grids for 1° × 0.25 km radar bins are shown.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

(a) A scatterplot of the computed values of *y* [see Eqs. (1)–(3) for definition] for a 1 × 1 km^{2} grid placed as a mesh surrounding a radar of 1° × 0.15 km radar bins as a function of the distance from the radar and (b) a histogram of the average values of *y* for 10-km range bins. The histograms of the average values of *y* for (c) 1 × 1 and (d) 4 × 4 km^{2} grids for 1° × 0.25 km radar bins are shown.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

(a) A scatterplot of the computed values of *y* [see Eqs. (1)–(3) for definition] for a 1 × 1 km^{2} grid placed as a mesh surrounding a radar of 1° × 0.15 km radar bins as a function of the distance from the radar and (b) a histogram of the average values of *y* for 10-km range bins. The histograms of the average values of *y* for (c) 1 × 1 and (d) 4 × 4 km^{2} grids for 1° × 0.25 km radar bins are shown.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

Realistically, the value of the rain-rate ratio *x* can take any value between zero (*R*_{a} is zero and *R*_{x} nonzero) and infinity (*R*_{a} is nonzero and *R*_{x} zero). Knowing the realistic ranges of the values of the ratios *x* and *y*, Eq. (7) can be used to estimate the RE, the error in the rate estimate computed by an approximate method compared to the mass-conserving, precise method as a function of *x* and *y*. The RE is shown in Fig. 5 as a percentage for values of *x* between 0.05 and 20 and for *y* between 0.05 and 1. It can be seen that the error can be very large when the difference between *R*_{a} and *R*_{x} is large and *y* is small. This will be the case at the edges of convective cells and for regions of light precipitation where the relative difference in rain rate among adjacent radar bins will be large (e.g., 2 and 0.5 mm h^{−1}). The unique situations when either *R*_{a} or *R*_{x} is zero (*x* = 0 or *x* = ∞) are not shown in Fig. 5. When *x* = 0, the error expressed by Eq. (7) will be equal to 100% for all values of *y* while it will be equal to −100(1 − *y*)/*y*% when *x* is equal to infinity. In the latter case, the error can be extremely high when the width of the bin reaches or exceeds the grid size.

The RE in grid-averaged rainfall rate computed by precise and approximate methods as a function of rain-rate ratio *x* and the area ratio *y* [see Eq. (7) for definitions of RE, *x*, and *y*].

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The RE in grid-averaged rainfall rate computed by precise and approximate methods as a function of rain-rate ratio *x* and the area ratio *y* [see Eq. (7) for definitions of RE, *x*, and *y*].

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The RE in grid-averaged rainfall rate computed by precise and approximate methods as a function of rain-rate ratio *x* and the area ratio *y* [see Eq. (7) for definitions of RE, *x*, and *y*].

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

As stated in the derivation of Eq. (7), we assume that *R*_{a} is 100% accurate for simplicity. Actual errors associated with approximate remapping are likely to be higher than the errors shown in Fig. 5 because *R*_{a} is not exactly the true average rain rate over *A*_{a}, as explained above. More sophisticated remapping methods (e.g., Wurman and Gill 2000), which are still approximations, may produce results closer to those of precise remapping but they would require equal or higher computational effort than precise remapping. For example, an algorithm developed by NCAR requires dividing the Cartesian grid into 50 × 50 m^{2} pixels and applying the nearest-neighbor approach to each pixel before computing the average for the grid (REORDER algorithm; Wurman and Gill 2000).

Neither remapping nor averaging should be performed on radar reflectivity in a two-dimensional linear space as this may lead to a significant overestimation of rainfall if reflectivity is converted to rainfall rate after remapping [see Randeu and Schonhuber (2000) for a discussion and analysis]. Reflectivity should be converted to rainfall at the highest available resolution before temporal or spatial averaging (Morin et al. 2003) or remapping is conducted. This is true for rainfall accumulations. However, for visualization of three-dimensional radar data, reflectivity should be regridded in algorithmic space as done in the Thunderstorm Identification, Tracking, Analysis and Nowcasting (TITAN) tool developed by NCAR (Dixon and Wiener 1993). TITAN is based on a Cartesian grid and can use a method called Digital Signal Processing for Logarithmic, Linear, and Quadratic Responses (DISPLACE) averaging (Terblanche 1996; Mittermaier and Terblanche 1997), which uses a two-stage distance-weighted scheme (using four data points per elevation, and then a second vertical step) that was entirely based on lookup tables (Terblanche et al. 2001).

## 5. Example from observational data

The sensitivities of radar-rainfall estimates to the method used for remapping radar observations onto Cartesian coordinates for different sizes of the radar bin and interpolation grid were quantified using rainfall data estimated by two weather radars collocated during a field experiment. In the spring of 1997, NCAR deployed its S-band dual-polarization Doppler radar (S-Pol) as part of the 1997 Cooperative Atmosphere–Surface Exchange Study (CASES-97). The S-Pol was placed about 10 km west-northwest of the Wichita, Kansas, WSR-88D (KICT). Radar measurement resolution was 1° × 1 km^{2} for WSR-88D and 1° × 0.15 km for S-Pol. The temporal resolution was approximately 5 min for WSR-88D and less than 2 min for S-Pol. Measurements were collected with both radars for 10 rainfall events in May and June 1997. These events occurred in different days. An event starts with the first radar detection of rainfall and is terminated when rainfall stops for 10 min. The no-rain period is then removed. The events used in this study lasted between 2 and 9 h. Rainfall estimates were based on the default WSR-88D *Z–R* relationship (*Z* = 300*R*^{1.4}; Fulton 1999), where *Z* is the reflectivity factor (mm^{6} m^{−3}) and *R* is the rain rate (mm h^{−1}), for both radars. Rainfall rate estimates were made on polar grids using reflectivity values from the 0.5° elevation angle. In many cases using a single elevation angle (the lowest) is often the best, especially in regions with a convective regime with upright reflectivity cores and much greater gradients in the horizontal. However, for regions not dominated by convection, with a greater proportion of stratiform rain there is often a benefit in using multiple elevations to help with the bright band. In this case, the regridding algorithm can also help to mitigate against brightband effects.

Rain-rate and event-total rainfall estimates from the two radars were computed using the four interpolation methods over a grid mesh covering the entire radar domain. Averaged rainfall was computed over Cartesian grids of sizes ranging from 0.5 × 0.5 to 4 × 4 km^{2}. A grid cell was included in the comparison only when all bins overlaying the grid cell reported precipitation. Rain-rate values below 3 mm h^{−1} were discarded. The choice of 3 mm h^{−1} is specific to this. We are aware that other places around the world may be very interested in a regridding scheme that considers even 0.5 mm h^{−1} rain rates.

The relative error in grid rainfall rate interpolated using approximate remapping as expressed Eq. (7) was computed for different grid sizes for estimates of the two radars. Results for the S-Pol are shown in Fig. 6. A scatterplot of the absolute value of the error of linear averaging for a 0.5 × 0.5 km^{2} grid is shown in Fig. 6a as a function of the coefficient of variation of the rain rate within the grid (computed using a rain rate for all bins intersecting the grid). The differences can be well above 100% when the variability of rainfall rate is high. These are regions at the edges of convective cells and regions of light precipitation where the relative difference among adjacent radar bins can be large (e.g., 6 and 3 mm h^{−1}). Again, in many regions, even 0.5 mm h^{−1} rain rates are considered significant, in which case the relative differences between adjacent bins can be very large.

(a) A scatterplot of the absolute value of the relative error grid rain rates computed using linear averaging for 0.5 × 0.5 km^{2} grids as a function of the coefficient of variation of the rainfall rate within the grid. The average relative error for linear averaging (solid lines), distance-weighted averaging (dashed lines), and nearest-neighbor (dotted lines), for grid sizes of (b) 0.5, (c) 1.0, and (d) 4.0 km. S-Pol data were used.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

(a) A scatterplot of the absolute value of the relative error grid rain rates computed using linear averaging for 0.5 × 0.5 km^{2} grids as a function of the coefficient of variation of the rainfall rate within the grid. The average relative error for linear averaging (solid lines), distance-weighted averaging (dashed lines), and nearest-neighbor (dotted lines), for grid sizes of (b) 0.5, (c) 1.0, and (d) 4.0 km. S-Pol data were used.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

(a) A scatterplot of the absolute value of the relative error grid rain rates computed using linear averaging for 0.5 × 0.5 km^{2} grids as a function of the coefficient of variation of the rainfall rate within the grid. The average relative error for linear averaging (solid lines), distance-weighted averaging (dashed lines), and nearest-neighbor (dotted lines), for grid sizes of (b) 0.5, (c) 1.0, and (d) 4.0 km. S-Pol data were used.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The averages of the absolute value of the percent error for different approximate methods and gridcell sizes are shown in Figs. 6b–d. As expected, the errors are much larger for the nearest-neighbor approach. The differences are generally smaller for larger gridcell sizes. This is due to the fact that as the gridcell size increases the number of radar bins falling completely within the interpolation area increases and the relative effect of the errors introduced by computing the contribution of bins falling partially within the interpolation area decreases. Results for linear-average and distance-weighted-average remapping are similar for small gridcell sizes but start to diverge for larger cell sizes. The absolute value of the relative error of approximate remapping is higher for the distance-weighted method than it is for linear averaging. As mentioned earlier, although distance-weighted averaging tries to consider the relative contribution from each bin, it does not consider the size and orientation of the bin.

The best-fit linear regression lines for the absolute value of the percent error in the approximate methods relative to the mass-conserving estimate for the 1 × 1 km^{2} grid for WSR-88D data are shown in Fig. 7. The differences between remapping methods were significantly smaller for WSR-88D data compared to S-Pol data due to the larger bin size and lower temporal resolution of WSR-88D as the larger bins smooth the precipitation gradients. However, the gradients of the average error lines are similar to those of S-Pol data.

The averages of the absolute value of the percent difference between precise and linear-average (solid line), distance-weighted-average (dashed line), and nearest-neighbor (dotted line) remapping for a 1 × 1 km^{2} grid using WSR-88D data.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The averages of the absolute value of the percent difference between precise and linear-average (solid line), distance-weighted-average (dashed line), and nearest-neighbor (dotted line) remapping for a 1 × 1 km^{2} grid using WSR-88D data.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The averages of the absolute value of the percent difference between precise and linear-average (solid line), distance-weighted-average (dashed line), and nearest-neighbor (dotted line) remapping for a 1 × 1 km^{2} grid using WSR-88D data.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The errors of approximate remapping approaches are also significant for total event accumulations. Figure 8 shows the distribution of relative errors of total event accumulations for 0.5 × 0.5 km^{2} grid cells computed using approximate average remapping applied to S-Pol rainfall accumulations. Though one might expect that for a particular grid underestimates and overestimates at particular time intervals within the event duration would tend to balance each other, the differences in total event accumulations can still be high, especially for small accumulations, for which they can be well above 100% (not shown in the figure to improve legibility). The errors in total event accumulations are smaller for larger gridcell sizes and for WSR-88D rainfall accumulations. As expected, the errors of the nearest-neighbor method are much higher (Fig. 9). Errors of the nearest-neighbor method increase with the size of the grid cell.

Distribution of the percent difference in grid-averaged total event accumulation using linear-average remapping compared to precise remapping. Results are segregated into logarithmically spaced bins of accumulated rainfall (3–10, 10–31.6, 31.6–100, and >100 mm). Original radar data gate spacing of 150 m and beamwidth of 1.0° remapped to 0.5 × 0.5 km^{2} grids using S-Pol data.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

Distribution of the percent difference in grid-averaged total event accumulation using linear-average remapping compared to precise remapping. Results are segregated into logarithmically spaced bins of accumulated rainfall (3–10, 10–31.6, 31.6–100, and >100 mm). Original radar data gate spacing of 150 m and beamwidth of 1.0° remapped to 0.5 × 0.5 km^{2} grids using S-Pol data.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

Distribution of the percent difference in grid-averaged total event accumulation using linear-average remapping compared to precise remapping. Results are segregated into logarithmically spaced bins of accumulated rainfall (3–10, 10–31.6, 31.6–100, and >100 mm). Original radar data gate spacing of 150 m and beamwidth of 1.0° remapped to 0.5 × 0.5 km^{2} grids using S-Pol data.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

As in Fig. 8, but for nearest-neighbor relative to precise remapping.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

As in Fig. 8, but for nearest-neighbor relative to precise remapping.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

As in Fig. 8, but for nearest-neighbor relative to precise remapping.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

## 6. Sensitivity of estimated radar rainfall to the resolution of the interpolation grid

Comparisons between radar estimates and gauge observations typically involve choosing the most representative radar bin (typically the nearest) or averaging of the radar observations over a certain area centered on the gauge location (e.g., Fulton 1998). Different practitioners use different averaging and interpolation methods. Equations (5)–(7) can also be used to calculate the difference in the value of the interpolated rainfall as the interpolation grid resolution becomes coarser. In this case *R*_{a} would be the average rainfall rate for the smaller grid, *R*_{x} is the average rainfall rate for the part of the larger grid outside the smaller grid, *A*_{a} is the area of the smaller grid, and *A*_{g} is the area of the larger grid.

S-Pol observations from the 1997 experiment were used to quantify the dependence of the grid-averaged rainfall rate on the grid resolution. Figure 10 shows the difference in the estimated rainfall rates as a function of the coefficient of variation for different grid resolutions compared to rainfall rate estimated for a concentric 0.5 × 0.5 km^{2} grid. As expected, the change is larger for larger grid sizes. Figure 10a shows a scatterplot of changes in grid rainfall over a 1 × 1 km^{2} grid compared to a concentric 0.5 × 0.5 km^{2} grid as a function of the coefficient of variation of rainfall rate within the larger grid when precise remapping is used. The average value of the difference in grid rainfall for different values of the coefficient of variation for concentric grid of sizes of 1 × 1 and 4 × 4 km^{2} for precise remapping are shown in Fig. 10b. The same changes are shown in Figs. 10c and 10d for distance-weighted-average remapping. Similar changes can be observed when linear-average remapping was used. The results show that precise remapping is much less sensitive to changes in the size of the interpolation area than approximate remapping because it accounts for any part of a bin falling within the grid cell.

(a) A scatterplot of the absolute value of the percent difference in grid-averaged rain rate over a 1 × 1 km^{2} grid compared to a concentric 0.5 × 0.5 km^{2} grid as a function of the coefficient of variation of rainfall rate within the larger grid. (b) The linear regression–based average value of the change for concentric grid of 1 × 1 (solid line), 2 × 2 (dotted line), and 4 × 4 km^{2} (dashed line) when precise, mass-conserving remapping was used. (c),(d) As in (a),(b), but for distance-weighted-average remapping.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

(a) A scatterplot of the absolute value of the percent difference in grid-averaged rain rate over a 1 × 1 km^{2} grid compared to a concentric 0.5 × 0.5 km^{2} grid as a function of the coefficient of variation of rainfall rate within the larger grid. (b) The linear regression–based average value of the change for concentric grid of 1 × 1 (solid line), 2 × 2 (dotted line), and 4 × 4 km^{2} (dashed line) when precise, mass-conserving remapping was used. (c),(d) As in (a),(b), but for distance-weighted-average remapping.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

(a) A scatterplot of the absolute value of the percent difference in grid-averaged rain rate over a 1 × 1 km^{2} grid compared to a concentric 0.5 × 0.5 km^{2} grid as a function of the coefficient of variation of rainfall rate within the larger grid. (b) The linear regression–based average value of the change for concentric grid of 1 × 1 (solid line), 2 × 2 (dotted line), and 4 × 4 km^{2} (dashed line) when precise, mass-conserving remapping was used. (c),(d) As in (a),(b), but for distance-weighted-average remapping.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

The dependence of grid-averaged rainfall on the grid resolution was weaker for the WSR-88D data because of its larger bin size compared to S-Pol. Since the rainfall gradients were already smoothed over the larger bin size, the differences in estimated rainfall between adjacent bins tended to be smaller. The changes in WSR-88D grid-averaged rainfall (not shown) with increasing grid size are about 4–6 times smaller than the changes with the S-Pol data, but they can still be significant for the higher values of rainfall rate coefficients of variation. Total event accumulation is also sensitive to the grid resolution as can be seen in Fig. 11, which shows the distribution of the difference in total event accumulations when a different grid resolution is used. The percent differences in grid rainfall estimates can be very high, especially for smaller rainfall accumulations. In general, the change in averaged rainfall was found to increase with increasing gridcell size.

Distribution of the percent difference in total event accumulation averaged over a 1 × 1 km^{2} grid and that averaged over a concentric 0.5 × 0.5 km^{2} grid. Results are segregated into logarithmically spaced bins of accumulated rainfall (3–10, 10–31.6, 31.6–100, and >100 mm). S-Pol data were used.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

Distribution of the percent difference in total event accumulation averaged over a 1 × 1 km^{2} grid and that averaged over a concentric 0.5 × 0.5 km^{2} grid. Results are segregated into logarithmically spaced bins of accumulated rainfall (3–10, 10–31.6, 31.6–100, and >100 mm). S-Pol data were used.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

Distribution of the percent difference in total event accumulation averaged over a 1 × 1 km^{2} grid and that averaged over a concentric 0.5 × 0.5 km^{2} grid. Results are segregated into logarithmically spaced bins of accumulated rainfall (3–10, 10–31.6, 31.6–100, and >100 mm). S-Pol data were used.

Citation: Journal of Hydrometeorology 15, 6; 10.1175/JHM-D-14-0058.1

## 7. Summary and conclusions

This paper presented a mass-conserving, geometrically precise method for remapping radar data onto a Cartesian grid. The method is based on exactly computing the proportion of each individual radar bin’s area (or volume) falling within the Cartesian grid cell as suggested in previous studies. Thus, the weight applied to a given gate data value is based on the extent to which the gate’s sample volume is contained within the interpolation grid cell. The algorithm calculations are repetitive, making coding very flexible. For example, processing of grid cells is not required to follow a certain sequence, not all grid cells need to be processed at the same time, and the code execution can be easily parallelized. The processing time is dependent on the number of Cartesian grid cells. After performing the calculations, lookup tables can be constructed for future use (i.e., computations need to be performed only once even if the radar is not operated in an indexed beam mode). Therefore, computational load is not a major challenge for this method.

The results presented in this paper apply to square grids but very similar results were found for circular interpolation areas of similar resolutions. Moreover, the results discussed in this paper are focused on two-dimensional remapping and radar-rainfall rate rather than reflectivity since averaging of reflectivity will introduce errors into reflectivity–rainfall rate conversion. The proposed method is particularly useful in situations where a grid point is sampled by multiple radars with possibly different sizes of pulse volumes. Since the proposed remapping method computes partial bin areas (volumes) precisely, it will be possible to apply weights based on the distance from the radar or the center of interpolation area more realistically as well as weights that reflect the variability of radar power within each bin, if needed. Also, if deemed necessary, advection effects can be easily incorporated by simply rotating or translating the projection of the radar bins. The proposed remapping method will not affect radar quality control procedures such as corrections for ground clutter, anomalous propagation, and signal jamming that are typically performed on the polar samples.

The proposed precise remapping was compared to popular approximate remapping approaches such as linear averaging, nearest-neighbor, and distance-weighted averaging. Examination of the intersection of polar radar bins with a Cartesian grid mesh and theoretical analysis revealed that the difference in interpolated grid-averaged rainfall rate between precise and approximate remapping will depend on the proportion of the gridcell area covered by bins that are considered in approximate interpolation (bin whose centers fall within the grid cell) and the rainfall rate variability among adjacent grid cells. The proportion of gridcell area considered in approximate remapping depends on the difference between the bin size and gridcell size. The proportion can be very small at far ranges if this difference is large. Simulations based on bin sizes of 1° × 0.15 km and 1° × 0.25 km and grid resolutions of 1 × 1 km^{2} and 4 × 4 km^{2} demonstrate that the average values of the proportion of the gridcell area considered in approximate remapping is not much larger than 50% at far ranges.

Rainfall observations collected by two radars during a field experiment were used to quantify the differences in grid-averaged rainfall rate estimates using precise and approximate remapping approaches. The difference in estimated grid rainfall rate is quite large when the variability of radar estimates is high within the interpolation grid, especially for finer grid resolutions (though one exception is for the nearest-neighbor approach, where the difference increases with the gridcell size). The difference can exceed 100% for the high-resolution radar (1° × 0.15 km bins) for all interpolation areas. This can be the case for convective cells and light rainfall region within a storm. The differences are much smaller for WSR-88D (1° × 1 km bins) because rainfall rate gradients were already smoothed by the larger radar bins.

The same theoretical basis for estimation of the difference in grid-averaged rainfall rate between precise and approximate remapping approaches was also applied to estimate the difference in estimated rainfall rate for concentric Cartesian grids of different resolutions. This latter difference is also highly dependent on the variability of rainfall rate among adjacent radar bins. However, we found that the sensitivity to the gridcell size is much lower in the case of precise remapping in comparison to approximate methods. This information is critical since radar bias is typically estimated by comparing rain gauge observations to radar estimates averaged over a certain area around the gauge location. So, for the case of convective rainfall, the computed bias can be highly sensitive to the remapping method and the size of the interpolation area.

The authors hope that this study will help start serious discussions on setting some standards for radar data remapping and radar bias computations for operational use. The results clearly indicate that higher-resolution rainfall observations are more sensitive to the interpolation method and size of the interpolation area than the legacy 1° × 1 km WSR-88D observations. WSR-88D data are now available at the super resolution of 0.5° × 0.25 km (Kumjian et al. 2010), which makes the effect of the remapping method on the estimated rainfall and radar bias more significant as discussed above. With short-range bins comes the ability to better define reflectivity gradients in range with even better than S-Pol’s azimuthal resolution. The S-Pol’s 1° beam creates discrepancies in the dimensions of the sampling volume with azimuth and range, especially at short ranges. The interpolation grid has to match the underlying resolution of the raw data. Hence, in practice a variable grid with distance from the radar is possibly the most appropriate.

Finally, we caution that the research described in this paper is focused on the geometrical representation of remapped radar product and is not intended to address the accuracy of radar-rainfall estimates, which is controlled by several factors including the radar hardware, range, vertical profile of the atmosphere, precipitation gradients, and precipitation microphysics. Other factors not considered include wind drift, beam shape (for 3D applications), and data synchronization/validity time. We certainly do not want to imply that the vertical dimension can be ignored as it helps improve rainfall estimation in many situations such as under brightband conditions (e.g., Mittermaier et al. 2006).

## Acknowledgments

This study was partially funded by NSF Grant EPS-1135483 to the second author.

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