a. Translation of coordinate systems: Spherical and Cartesian

A fundamental process of mapping and mosaicking radar data to a Cartesian grid is the translation between a radar bin’s location in spherical and Cartesian coordinates. In spherical coordinates, a radar bin location is described using the azimuth, range, and elevation angle or (ϕ, r, θ) of the bin’s center with respect to the radar location. Relating this to Cartesian coordinates (x, y, z) requires radar beam geometry, great circle principles, and latitude–longitude conversion. Radar beam geometry and great circle principles are briefly explained in the following subsections. Figure 1 shows the general flow of calculations required for transitioning between spherical and Cartesian coordinates.

1) Radar beam propagation geometry

Radar beam propagation equations are the basis for the calculation of different aspects of a radar bin’s location. Figure 2 shows a number of spatial variables that are important for our purposes, as they relate to a bin’s location. At times, it is necessary to calculate a bin’s height (h), elevation angle (θ), distance with respect to the earth’s surface (s), or range along an elevation angle (r).

Given two of the four variables, r, h, s, and θ can be derived from the following equations based on the equivalent earth model [Eq. (2.28) of Doviak and Zrnić 1993]:

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where a is the earth’s radius and k_e (=4/3) is a ratio that, when multiplied with a, approximates the earth’s equivalent radius a_e.

One disadvantage of the equivalent earth model is its assumption that n is linearly dependent on h. This can lead to inaccuracies when the atmosphere’s vertical temperature profile does not decrease with height as expected (e.g., in a temperature inversion). A strong temperature inversion can effectively trap a radar beam, refracting it toward the earth’s surface and diverting it away from the path predicted by (1), (2), and (3). Despite this disadvantage, these equations are expected to generally provide the 4DDG weighting schemes with accurate information regarding radar beam propagation.

2) Great circle principles

Calculations based on great circle principles are equally essential for translating between spherical and Cartesian coordinates. A basic application of great circle principles is calculating the arc distance s and heading B between two known points on the earth’s surface. Figure 3 illustrates an example.

For mapping and mosaicking purposes, consider a radar located at point 1 and a radar bin (or a Cartesian grid cell) located at point 2. The following equations show how the azimuth and arc distance can be calculated:

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where λ₁ and λ₂ are longitudes and ζ₁ and ζ₂ are latitudes. Note that the latitudes ζ₁ = (π/2) − ψ₁ and ζ₂ = (π/2) − ψ₂, where ψ₁ and ψ₂ are colatitudes as shown in Fig. 3. Equation (5) arises from the law of cosines for spherical triangles. Solving (5) for B will give the heading from point 1 to point 2.

The inverse of this basic application is valuable as well. Given a radar bin’s azimuth, its arc distance from the radar, and the radar’s location, it is possible to compute the bin’s latitude and longitude. From these coordinates, it is possible to use grid conversion calculations to find the Cartesian coordinates of the radar bin.

b. The remapping challenge

To accurately represent radar reflectivity fields on a 3D Cartesian grid, the 4DDG must address the sampling characteristics of radar, incorporating the spatial and temporal variability of radar observations and their discontinuous nature. The following subsections outline the various schemes employed by the 4DDG to overcome them.

Consider the task of mapping one radar reflectivity field onto a 3D Cartesian grid. Radar data coordinates are described using azimuth, range, and elevation angle or (ϕ, r, θ). Due to the nature of spherical coordinates, radar data do not have constant spacing in all directions. Assuming the radar is a WSR-88D, the range direction has a fixed spacing of 1 km for reflectivity. However, as one’s distance from the radar increases, the data spacing with respect to the azimuthal and elevation angle directions degrades. The spacing with respect to elevation angle varies depending on range from the radar, height, and radar volume coverage patterns (VCPs) since angular distances between tilts can differ. For example, given a WSR-88D radar in VCP 21 mode, the data spacing is worse than 1 km for tilt five at a range of 30 km and for tilt six past a range of 15 km (Zhang et al. 2005). Note the focus here is on 1-km data spacing since the 4DDG Cartesian grid spacing is 1 km. Figure 4 shows the distribution of radar data for VCP 21. Notice how the elevation angle data spacing decreases significantly after tilt five (4.3°). The elevation angles associated with the WSR-88D VCPs are shown in Table 1.

The spatial variability of radar data causes portions of the Cartesian domain to be oversampled near the radar, undersampled far from the radar, and not sampled at all between far spaced tilts (e.g., VCP 21 tilts five and six), below the lowest tilt, and above the highest tilt. Several weighting schemes are employed to handle these scenarios given different weather events.

Depending on a user-defined option the 4DDG will use one of the following methods that were presented in Zhang et al. (2005) for remapping:

1. nearest neighbor (NN),

2. one-dimensional vertical linear interpolation (VI) plus NN, and

3. two-dimensional horizontal linear interpolation (HI) plus VI.

All three methods calculate the components of w_remap and contribute to the overall 4DDG analysis scheme, which is defined in section 3. The calculation of w_remap is based on

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where w_r is a radar bin’s weight with respect to range, w_ϕ is the weight with respect to azimuth, and w_θ is the weight with respect to elevation angle. The computational detail for each of the weights can be found in Zhang et al. (2005).

c. The oversampling problem

The previous section discussed the various remap weight formulas and methods used in the 4DDG. It was assumed at the time that the size of a grid cell was less than the size of a radar bin (undersampled). However, grid cells that are located close to a radar may be oversampled, which results from the nature of radar data in spherical coordinates.

For a 1° azimuthal beamwidth, the azimuthal size of a radar bin is smaller than a 4DDG grid cell within 60 km of the radar. Thus some grid cells are sampled by more than one radar bin. The previous weighting schemes do not take into account this possibility and may miss some reflectivity values.

To avoid data loss, the 4DDG may employ one of three methods for use in calculating a final reflectivity value for grid cells oversampled by radar bins. These three methods are arithmetic mean, distance weighted mean, and maximum reflectivity. The maximum reflectivity scheme is sufficiently self-explanatory. Of the radar bins affecting an oversampled grid cell, the maximum reflectivity value is retained and all others are ignored. The following subsections discuss the arithmetic mean and distance-weighted mean schemes. Note that the weights computed by these schemes are included as part of the 4DDG remapping weights.

1) Arithmetic mean

The arithmetic mean method assigns a weight of one to w_ϕ, w_r, and w_θ for each radar bin whose center falls within the grid cell. These weights, when applied in (7), result in the 4DDG calculating a grid cell’s final reflectivity using a mean reflectivity as seen in

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where z_i is the weighted mean reflectivity value at a given grid cell, z_k is the reflectivity of the kth radar bin associated with the grid cell, and n is the number of reflectivity bins affecting the grid cell.

2) Distance-weighted mean

Oversampling in the 4DDG can also be handled by a distance-weighted mean method. Radar bins whose centers are located in an oversampled grid cell are weighted based on their position with respect to the grid cell’s center. The influence a radar bin has on the grid cell is determined by

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where w is the resultant weight (w_remap), d is the distance between the radar bin center and grid cell center, and R is the radius of influence, which is equal to the cross-diagonal distance of 1 km³ (approximately the size of one grid cell).

d. Spatial mosaic weight

In addition to remapping single radar reflectivity fields, one must consider a multiple radar scenario where more than one radar collects reflectivity data for the same point in space. The 4DDG provides a methodology that takes advantage of multiple observations while handling those observations that do not agree with each other.

The requirements of the 4DDG mosaic weighting scheme are formed by the general assumption that, due to changes in resolution, radar observations closer to the radar are more accurate than those farther away. A desirable mosaic weighting scheme should therefore employ a rapidly decreasing weight, which retains high-resolution features in raw radar data at close ranges and information pertaining to storm severity (e.g., maximum reflectivity values). A mosaic weighting scheme should also employ positive weight values for long ranges, which ensures the radar has influence over the entire region it covers. Weighting functions such as linear interpolation and Cressman (1959) fail to have both of these key features.

The shape of an exponential weighting function is easily adjustable to achieve a rapid decrease with range while retaining a positive weight value. Thus, an exponentially decaying function is especially useful for radar analysis and is employed in the 4DDG algorithm. Equation (10) is the exponential decaying weight function used by the 4DDG for mosaicking:

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Here r is the distance of the observation from its respective radar and R is an adaptable length scale (i.e., 25 km).

e. Temporal weight

The previous discussion addresses only three of the four dimensions that the 4DDG represents. The fourth dimension is time, which is used to address the temporal variability of radar data. Given two radars, the likelihood of their data being synchronized with respect to each other is low. The issue becomes how to accurately represent reflectivity fields at a given time and compensate for the varying age of the radar data. The 4DDG records a history of reflectivity values (i.e., 10 minute’s worth) and combines them to produce the radar reflectivity fields at the present time.

No research exists on the effects of different temporal weighting schemes on mosaicked reflectivity fields. Therefore, in an initial effort, the 4DDG uses an exponentially decaying weighting function

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where t is the age of the reflectivity data and T is an adaptable time scale (i.e., 2 min). The weighting function in (11) is of the same type as the mosaic weight scheme used by the 4DDG. Its characteristic of a rapidly decreasing and positive weight make it well suited for temporal smoothing reflectivity fields. Newer reflectivity values can be given a large influence while older observations are still allowed to affect the final reflectivity field.

f. 4DDG computational efficiency

The 4DDG must process data efficiently to run within a real-time operational environment. Given the initial target minimum requirements, the 4DDG must run on multiple large domains while ingesting numerous radars (an average of 35 per domain) such that all incoming data are reflected within 5 min of their arrival in the system. For example, a domain ingesting 35 radars’ worth of data running in VCP 11 will generate 490 tilts of data in 5 min. Thus, the 4DDG may require an update rate as high as one tilt every 0.62 s to avoid time lags between data ingest and a final 3D reflectivity grid.

To reduce the run time of the 4DDG, several time-saving techniques were employed. These include shared memory data storage and precalculation of various weighting scheme values.

a. 4DDG LUT

The 4DDG LUT is an offline algorithm that generates the LUTs used by the 4DDG analysis scheme. Separate LUTs are required for each radar, VCP, tilt, and remap/oversample weighting scheme combination.

LUTs are generated in a set based on a given VCP. To generate one set, the algorithm iterates through each grid cell in a 3D domain and converts its position to spherical coordinates with respect to a specific radar. If a grid cell is slightly above the highest tilt in the VCP or slightly below the lowest tilt, its position is extrapolated to the closest tilt. Weights for all other grid cells covered by the VCP are calculated using one of the three available remap schemes (see section 2). In addition, the range of the grid cell from the radar is computed and stored for calculating the mosaic weight later.

It is possible that some bins are missed due to oversampling (see section 2). It is assumed that including these additional observations is desirable and oversampling weights are computed. The algorithm iterates through all radar bins not assigned to at least one grid cell, and the bin’s Cartesian coordinates are calculated. The bin’s weight is determined by one of the three available oversampling schemes.

Upon computing all necessary remap/oversampling weights and mosaic weight distances, 4DDG LUT writes out one LUT for each tilt in the VCP. Each LUT contains information pertaining to the number of grid cells each radar bin influences, grid cell indices, remap weights, oversampling weights, and ranges of grid cells from the radar.

b. 4DDG analysis scheme

The 4DDG analysis scheme is the main component of the 4DDG algorithm. Running continuously, the 4DDG analysis scheme ingests radar data on a tilt-by-tilt basis. The appropriate LUT is chosen based on the radar, tilt, VCP, and weight scheme options. Values from the LUTs are used to apply new reflectivity data to the system. Several “history” arrays store reflectivity values, remap weights, mosaic weights, and ages of the observations for each grid cell in the domain. Once the reflectivity data reach a certain age (e.g., 10 min) it and all accompanying information is pruned from the history arrays. After the history arrays are updated and pruned, the 4DDG uses a combination of its spatial and temporal weighting schemes to update a 3D domain of mosaicked reflectivity data stored in shared memory. The combination of schemes is defined by

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Here, Z_m is the mosaic reflectivity at a given grid cell, N is the number of observed reflectivity values affecting a given cell within the 3D domain, T is the number of times for which there is an observation available at bin i, Zⁱ_g,t is the observed reflectivity from a radar in the domain at time t, and wⁱ_g,t is the weight of each reflectivity value with respect to a given grid cell at time t.

The weight t, and wⁱ_g,t is a combination of remap, mosaic, and temporal weights. Note that remap weights may include weights from one of the oversampling weight schemes. Dropping the subscript t and superscript i, the weight from (12) is

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where w_remap is the spatial weighting function for remapping a single radar reflectivity field from spherical to Cartesian coordinates, w_mosaic is the spatial weighting function for mosaicking multiple radar reflectivity values, and w_t is the temporal weighting function. Again, note that oversample weights may be included in w_remap. Detailed descriptions of how the weights of (13) are computed can be found in Zhang et al. (2005).

c. 4DDG reader

The 4DDG reader plays a small but essential role in the 4DDG algorithm. The 4DDG final 3D reflectivity field is stored in shared memory (a linear buffer). The 4DDG reader is responsible for accessing the linear buffer and writing its data to various file formats. Based on output options, the 4DDG reader writes out to file individual products ranging from the entire 3D reflectivity field to horizontal and vertical cross sections. This occurs at regularly scheduled intervals (e.g., every 5 min) as set by the user.

a. Convective case

Convective events are characterized by vertically oriented storms and develop at a rapid pace with an average lifespan of 20–30 min (Glickman 2000, p. 121). The vertical structure of convective storms is best represented by the 4DDG 1D vertical interpolation (VI) remapping scheme, which is used for this case. The arithmetic mean oversampling scheme is used to include all data near the radars. The time scale of the 4DDG temporal weighting scheme is varied to illustrate how different time scales are more appropriate for convective cases. The outcome of using of no temporal weighting is also explored to show the utility of the temporal weighting itself. The expected result is that shorter time scales will work best, considering the fast development life cycle of convective storms.

Three runs of the 4DDG were performed for this case. First, temporal weighting is turned off, allowing new reflectivity values to overwrite preexisting data. The second and third runs used time weighting with time scales of 2 and 15 min, respectively. The domain used for this Oklahoma convective event on 14 August 2002 is shown in Fig. 8. The radars used for this case are WSR-88Ds located in Frederick (KFDR) and Oklahoma City (KTLX). The vertical cross section, also defined in Fig. 8, is seen in Fig. 9 for all three runs.

Examining Fig. 9, several important differences are immediately apparent. The sections labeled “A” highlight an area where the advantages of temporal smoothing are most evident. The region of higher reflectivity in area A shows how temporal smoothing can help retain storm structure. The storm structure in Fig. 9a is discontinuous and difficult to identify. The temporal smoothing seen in Figs. 9b and 9c enhances the reflectivity data’s spatial representation and allows for a more physically realistic storm structure.

Area “C” identifies another scenario where temporal weighting is an advantage. The domain for the convective case is such that both KTLX and KFDR have influence over the entire domain. As the range from the radar increases the resolution of the data sampling becomes worse. The large blocks of reflectivity values seen above KTLX are actually observed by KFDR. In Fig. 9a, this is most evident due to the lack of temporal smoothing. Figures 9b and 9c use temporal weighting to preserve some of the data from KTLX.

Notice the sudden change in reflectivity values in area C, where values drop from 30 dBZ to less than 20 dBZ. The areas of lower reflectivity are from KFDR. These values have a strong effect on the data points directly above KTLX due to the region directly above the radar where the radar does not sample (e.g., cone of silence). Therefore, there are no higher-resolution reflectivity values from KTLX to help sharpen the low-resolution observations from KFDR. Still, using temporal weighting helps to retain some storm structure.

Given that convective storms develop rapidly, it is hypothesized that using the temporal weighting function defined in (11) with a smaller time scale T would be more appropriate. This proves correct when comparing area “B” of Figs. 9b and 9c, where T is set to 2 and 15 min, respectively. Area B in Fig. 9b shows a core of high reflectivity; the same area in Fig. 9c shows three areas of high reflectivity. This is an example of the temporal smoothing allowing past reflectivity values to have influence for too long (a large value for T). This results in the erroneous multiple storm cores. Thus, temporal smoothing in convective cases is desirable but only for short time scales (e.g., 2 min).

b. Stratiform case

Stratiform weather events are horizontally oriented systems. Their structure is distorted when using 1D VI alone (Zhang et al. 2005). The 4DDG HI scheme (see section 2) can more accurately represent stratiform events while filling gaps in radar coverage and will be used with the arithmetic mean oversampling scheme to remap reflectivity fields.

As with the convective case, the 4DDG temporal weighting is the focus. A stratiform case in Oklahoma was run with no temporal weighting and temporal weighting with 2- and 15-min time scales. Figure 10 defines a domain and cross section for the stratiform event, which occurred on 18 March 2002 and was detected by radars in Tulsa (KINX) and Oklahoma City (KTLX).

The results from three temporal smoothing methods are shown in Fig. 11. As with the convective case, areas A, B, and C highlight regions where the effects of temporal smoothing (or lack there of) are most evident. The region labeled A is an example of how temporal weighting can help with some of the problems associated with radar sampling. In Figs. 11a and 11b a small area of higher reflectivity (approximately 40 dBZ) is present. The temporal smoothing using a smaller time scale smoothes this area and reduces its size. Figure 11c is missing this area due to its larger time scale (stronger temporal smoothing). Stratiform events are very slow to develop. The extent of the area of 35 dBZ is likely in error due to the poor spatial resolution of KINX. The Tulsa radar observations have a strong influence directly over KTLX due to the “cone of silence” discussed in the previous section. The stronger temporal smoothing in Fig. 11c reduces this feature.

The radar “cone of silence” problem is corrected in other areas as well. Area B identifies another incident where temporal smoothing for both large and small time scales can help retain storm structure. The high-resolution data near KTLX (apparent at B in Figs. 11b,c), although slightly older, is still better than the newer low-resolution data observed by KINX (apparent at B in Fig. 11a). The 4DDG temporal weighting scheme allows the higher-resolution data to correct the reflectivity field.

Finally, area C is a region where the effects of the 4DDG vertical interpolation scheme still appear despite the use of horizontal interpolation. These artifacts are most visible in Fig. 11b where areas of 25 and 30 dBZ are stretched vertically. The smaller time scale in Fig. 11b is not capable of smoothing these artifacts, which are better handled by the larger time scale of Fig. 11c.

The effects of short and long time scales on a stratiform case are muted compared to convective events. The results in Fig. 11 show how temporal smoothing can fill spatial gaps and smooth over random errors in the observed fields. In a convective case, as seen in Fig. 11c, a long time scale leads to false areas of high reflectivity. For rapidly developing events, the temporal smoothing can no longer fill spatial gaps accurately. However, the slow development of stratiform cases allows us to take advantage of a long time scale.

c. Oversampling scheme performance

The spatial resolution of radar data is such that some areas of the 4DDG domain are undersampled while others are oversampled. Those grid cells near the radar are oversampled and, as a result, radar observations may be lost unless some scheme is used to explicitly include them. The 4DDG uses oversampled weighting schemes to avoid this data loss.

The 4DDG has three oversampled schemes available for use. Previous sections defined these schemes, which are arithmetic mean, distance-weighted mean, and maximum reflectivity. In addition to these schemes, it is possible to turn off the oversampling scheme completely, although it is not advised.

Figure 12 shows the results of using different oversampling weights near a radar. The data used for this comparison are borrowed from the convective case, at 0230 UTC 14 August 2002, which was examined in a previous subsection. The reflectivity values of interest are those located in the southwest corner of the domain at a constant height of 2 km, within approximately 30 km of KFDR. To better isolate the effects of different oversampling schemes, a minimal amount of other smoothing is applied. Therefore, no temporal weight is used and the remap scheme is nearest neighbor.

Comparing the four reflectivity fields of Fig. 12, the advantages of applying an oversampling weight scheme are immediately apparent. By ignoring the oversampled problem and applying no weight scheme, the resulting reflectivity field fails to accurately represent the shape of the storm in the area and misses important values of high reflectivity (see Fig. 12a). This is most noticeable within 10 km of the radar where reflectivity values of 30–35 dBZ are ignored and values of less than 25 dBZ are shown instead.

The maximum reflectivity oversample scheme does a better job of representing the storm structure close to the radar and retains important reflectivity values. Areas of 45+ dBZ outline the storm structure nicely, while critical reflectivity values such as the 55-dBZ (bright red) pixels in central regions of the plot are kept. The characteristics of this weighting scheme make it well suited for severe weather applications, which often rely on well-defined storm structures and large reflectivity gradients to trigger identifications.

The arithmetic mean and distance-weighted mean oversampled schemes yield similar results as seen in Figs. 12c and 12d. They retain storm structure better than the maximum reflectivity method, but tend to smooth areas of higher reflectivity (see areas with greater than 55 dBZ). These oversampled weighting schemes are best suited for nonsevere weather applications such as quantitative precipitation estimates and numerical models.

Most importantly, Fig. 12 shows that any oversampled weight scheme is preferable over none at all. The schemes available in the 4DDG each have their own unique advantages, but they are equally suitable for use. Multiple oversampled weight schemes are provided by the 4DDG, allowing the user to manipulate the final 3D reflectivity field such that is best suited for their needs (e.g., modeling, severe storm applications, hydrometeorological applications).