## 1. Introduction

Since its initial implementation in the 1990s, the Weather Surveillance Radar-1988 Doppler (WSR-88D) network has been a critical tool for weather research, forecasting, and severe weather warnings. Government agencies, the public, and private companies utilize data from the WSR-88D network for many purposes in countless radar-based applications. These applications range from the simplest use of a single radar data field to more sophisticated algorithms that combine multiple radar data fields to generate automatic warnings and other derived products, such as quantitative precipitation estimates.

The earliest and most basic radar applications use a single radar field. More sophisticated radar-based applications oftentimes translate radar data to a Cartesian coordinate system. This translation contains its own set of challenges, mostly due to the spatial resolution of radar data. Several remapping schemes to interpolate and smooth the data field exist and have been studied (Trapp and Doswell 2000; Askelson et al. 2000; Zhang et al. 2005). The appropriateness of an interpolation scheme depends on the nature of the radar application. In general, logic is simplified when applied to a Cartesian grid. For example, it is much easier to use satellite and rain gauge data along with radar data to product precipitation estimates if the radar data are on a Cartesian grid.

The inclusion of multiple radars is a natural improvement upon single radar applications. A single radar covers an area with a set of restrictions and individual characteristics. Such restrictions include a finite coverage area, spatial and temporal gaps in radar data coverage, and sensitivity to vertical temperature profiles in the atmosphere. Individual radar characteristics range from beam blockage effects to calibration accuracy. Mapping multiple radar reflectivity fields onto a common 3D Cartesian grid can relieve some of these problems and allows applications to run over larger domains by looking at more than one radar. If one radar is blocked by terrain, a second may be able to fill in gaps. Additionally, it is possible to have more than one radar covering the same grid space. This provides valuable additional information, but requires the efficient and accurate interpolation of multiple observations.

Advances in network communications such as the Collaborative Radar Acquisition Field Test (CRAFT; Droegemeier et al. 2002) and increases in computing power have allowed radar applications to become more complex, using multiple radar data fields over larger domains. The National Severe Storms Laboratory has developed a Four-Dimensional Dynamic Grid (4DDG) algorithm to represent radar data in space and time. The objective of the 4DDG is to accurately and rapidly represent discontinuous radar reflectivity data over a given continuous 4D domain with a minimum of smoothing applied to the original data.

The 4DDG is an improvement upon current operational radar applications in that it represents data in four dimensions, rather than two or three dimensions. Thus, it addresses both spatial remapping and temporal sampling issues. Temporal sampling is an issue because radars sample the atmosphere in discrete intervals that are often not synchronized with surrounding radars. The 4DDG provides a method for handling the varying temporal resolution and discontinuous nature of radar data by utilizing a fourth dimension for temporal smoothing.

The 4DDG algorithm confronts numerous mathematical and computational challenges. These challenges range from addressing the operational nuances of WSR-88D radar sampling characteristics to efficient mathematical and computational methods required for the 4DDG to run in a real-time operational environment.

The organization of this paper is as follows. Section 2 describes the mathematical and computational methods employed in the 4DDG. The 4DDG general algorithm is given in section 3. Case study results from the 4DDG’s different spatial, temporal, and oversampled interpolation schemes are shown in section 4. Finally, section 5 summarizes the 4DDG and future plans for development.

## 2. Mathematical and computational methods

The development of the 4DDG faced several mathematical and computational challenges. This section identifies these challenges and defines the methods used to address them. Mathematical challenges include the translation between spherical and Cartesian coordinate systems, schemes for smoothing radar observations spatially and temporally, handling observations from multiple radars, and complications from oversampling near the radar. The computational challenges are associated with the efficiency needs of the algorithm to be considered operationally viable.

### 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*).

*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]:

*a*is the earth’s radius and

*k*(=4/3) is a ratio that, when multiplied with

_{e}*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.

*λ*

_{1}and

*λ*

_{2}are longitudes and

*ζ*

_{1}and

*ζ*

_{2}are latitudes. Note that the latitudes

*ζ*

_{1}= (

*π*/2) −

*ψ*

_{1}and

*ζ*

_{2}= (

*π*/2) −

*ψ*

_{2}, where

*ψ*

_{1}and

*ψ*

_{2}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:

nearest neighbor (NN),

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

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

*w*

_{remap}and contribute to the overall 4DDG analysis scheme, which is defined in section 3. The calculation of

*w*

_{remap}is based on

*w*is a radar bin’s weight with respect to range,

_{r}*w*is the weight with respect to azimuth, and

_{ϕ}*w*is the weight with respect to elevation angle. The computational detail for each of the weights can be found in Zhang et al. (2005).

_{θ}### 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

*w*,

_{ϕ}*w*, and

_{r}*w*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

_{θ}*z*is the weighted mean reflectivity value at a given grid cell,

_{i}*z*is the reflectivity of the

_{k}*k*th radar bin associated with the grid cell, and

*n*is the number of reflectivity bins affecting the grid cell.

#### 2) Distance-weighted mean

*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 km

^{3}(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.

*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.

*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.

#### 1) Shared memory storage

The 4DDG requires a fast update cycle to perform in a real-time environment. Past algorithms such as the 3D Mosaic (Zhang et al. 2005) run discretely at 5–10-min increments. The 4DDG runs continuously and maintains a shared memory space where outside processes may access it and use the 4DDG 3D reflectivity field for various purposes. This allows the 4DDG to avoid some file I/O overhead.

The shared memory is managed by a software module called Linear Buffer (LB) (Jing and Jain 2000) originally developed by the Radar Operations Center (ROC) for use in the WSR-88D Open System Radar Product Generator. Linear Buffer is designed for message communication between processes, but it can also be used to store data in a random access format. The 4DDG makes use of this feature and is able to update its “3D grid” several thousands of times per second.

#### 2) Precalculated lookup tables

Several of the 4DDG weighting scheme calculations can be performed in advance to save processing time. These calculations are performed in a separate program, and their computed values are stored in lookup tables (LUTs), which are read in as needed. The LUT calculations include all remap and oversampling weights and partial calculations for the mosaic weight. The final mosaic weight is calculated in real time so users can change the adaptable length scale of the mosaic formula [see (10)] without generating new LUTs. Temporal weights [see (11)] cannot be precomputed since the age of the radar data changes with time.

The format of the LUTs is dictated by the way the 4DDG processes radar data (one tilt at a time). There exists one LUT per VCP per tilt per radar for each combination of remap and oversampling schemes. The 4DDG determines which LUT is appropriate based on the current spatial weight settings and the radar’s current tilt and VCP.

Using the LUTs instead of calculating the same values repeatedly saves sufficient time to offset the overhead cost of opening and reading the LUT files. Without the LUTs, the performance of the 4DDG would not be adequate for real-time implementation, given current available hardware. Additional LUT details are discussed in section 3.

## 3. 4DDG algorithm

The 4DDG algorithm is separated into three different programs, the 4DDG LUT, analysis scheme, and reader. The following subsections define these components. Figure 5 is a top-level flowchart of the 4DDG algorithm.

### 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

*Z*is the mosaic reflectivity at a given grid cell,

_{m}*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*,

*Z*is the observed reflectivity from a radar in the domain at time

^{i}_{g,t}*t*, and

*w*is the weight of each reflectivity value with respect to a given grid cell at time

^{i}_{g,t}*t*.

*t*, and

*w*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

^{i}_{g,t}*t*and superscript

*i*, the weight from (12) is

*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*is the temporal weighting function. Again, note that oversample weights may be included in

_{t}*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.

## 4. Case studies

The 4DDG has several spatial and temporal weighting schemes. The use of these schemes is often based on the weather event occurring or the application of the 4DDG output. The effects of the 4DDG spatial weighting schemes for different weather regimes are described and reviewed in detail in Zhang et al. (2005). Therefore, the focus of the following case studies will be the performance of the 4DDG temporal weighting scheme, which is defined by (11) and the oversampling schemes.

In each temporal case study we examine the 4DDG reflectivity fields with and without temporal weighting. To illustrate how different time scales are better suited for different weather events, reflectivity fields using a time scale of 2 and 15 min were generated for each case. The weighting functions for these time scales are shown in Fig. 6.

The stratiform and convective cases have unique 3D domains defined such that two radars populate the grid and each radar has influence over the entire domain. The northeast and southwest corners of the domain are determined by the location of these radars (Fig. 7). The domain’s size and resolution vary for each event and are summarized in Table 2.

After investigating the effects of temporal weighting on convective and stratiform cases, the oversampled weight schemes are examined.

### 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 dB*Z* to less than 20 dB*Z*. 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 dB*Z*) 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 dB*Z* 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 dB*Z* 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 dB*Z* are ignored and values of less than 25 dB*Z* 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+ dB*Z* outline the storm structure nicely, while critical reflectivity values such as the 55-dB*Z* (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 dB*Z*). 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).

## 5. Summary

The 4DDG is an experimental radar-based application developed at the National Severe Storms Laboratory. This paper discussed the many challenges faced by the 4DDG and the strategic methodologies used to address them. In addition, brief case study results were presented to illustrate the utility of using a temporal weighting scheme to mosaic radar data for different weather regimes. Oversampled weighting schemes were also explored by comparing four different weighting techniques.

To be considered operationally viable, the 4DDG had to address several mathematical and computational challenges. The majority of these consisted of translating between spherical and Cartesian coordinates, interpolating and extrapolating reflectivity fields (remapping/oversampling), combining observations from multiple radars (mosaicking), and aging the reflectivity data while allowing it to affect the domain (temporal).

To accommodate different weather regimes, the 4DDG has three remapping schemes available:

nearest neighbor,

1D vertical linear interpolation, and

2D horizontal linear interpolation.

The nearest-neighbor scheme is best left for nonoperational research purposes as it leaves many artifacts and does a poor job of retaining storm structure. Convective events consist of vertically oriented storms and are best represented by 1D vertical linear interpolation. Stratiform events are more horizontally uniform and are best remapped using 2D horizontal linear interpolation.

Oversampled weight schemes are used by the 4DDG to handle the dense set of observations that exist near the radar. Without these schemes, some observations would be lost. The following oversampled weight schemes are provided:

arithmetic mean,

distance-weighted mean, and

maximum reflectivity.

In addition to remapping and oversampling, the 4DDG must also mosaic reflectivity data to optimally use observations from multiple radars. The 4DDG uses an exponentially decaying distance-weighting function to mosaic reflectivity data from multiple radars. A radar’s sampling resolution worsens with distance. The mosaic weight scheme intuitively allows for reflectivity values closer to a radar to have more influence than those observed by a far away radar.

The 4DDG advances radar-based applications by including temporal smoothing. This smoothing is achieved by using an exponentially decaying weight function based on the age of the observation. The time scale of the weighting scheme can be adjusted to different weather regimes. It is expected that convective events will require a short time scale, while stratiform events will benefit from a longer time scale.

The various spatial and temporal weighting schemes employed by the 4DDG are meant to most accurately represent 3D reflectivity structures while keeping the data as raw as possible. The output grid is not designed for producing higher-order derivatives. Therefore, it should be noted that users of the 4DDG output should consider applying filters to the field as is appropriate for their purposes.

The computational challenges of the 4DDG are related to its development as a real-time application. To help increase processing speed, the 4DDG uses lookup tables to reduce the number of calculations the algorithm must perform to apply new data to the 3D domain. The 4DDG also reduces processing time by using shared memory to output its final 3D reflectivity field. Outside programs can then read the shared memory for use in other algorithms or to be written out to file.

Case studies presented in this paper showed the effects of the 4DDG temporal and oversampled weight schemes. Convective and stratiform events were processed by the 4DDG using no temporal weight, a temporal weight with a short time scale, and a temporal weight with a long time scale. As expected, both events were better represented using some form of temporal weighting. The shorter temporal scale (e.g., 2 min), which allows for an observation’s influence to decrease quickly with time, proved best for the fast-developing convective event. Conversely, the stratiform case was better represented using the longer time scale (e.g., 15 min), allowing temporal smoothing to fill spatial gaps in radar coverage.

The convective case was also examined to see the effects of different oversampled weight schemes. It was found that any oversampled scheme was preferable over none. The three schemes offered by the 4DDG had different advantages. Each scheme is best depending on how the 4DDG final 3D reflectivity field is used (e.g., modeling, severe weather applications).

## Acknowledgments

The authors are thankful to Dr. Zhongqi Jing of the National Weather Service Radar Operations Center for his help in developing the shared memory management for the 4DDG. The authors would also like to thank Dr. S. Lakshmivarahan of the University of Oklahoma School of Computer Science for his review that helped in the preparation of this manuscript. Major funding for this research was provided under the Federal Aviation Administration (FAA) Aviation Weather Research Program Advanced Weather Radar Technologies Product Development Team MOU, and partial funding was provided under NOAA–University of Oklahoma Cooperative Agreement NA17RJ1227, U.S. Department of Commerce. This research is in response to requirements and funding by the FAA. The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA.

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