1. Introduction

The design of a weather radar system and its scanning strategy involves trade-offs based upon features to be observed and the cost of building and deploying the radar system. Design trade-offs are often difficult to quantify in terms of their impacts on detecting and tracking features of interest. Moreover, the development of optimal scanning and the refinement of radar-based algorithms require large datasets to test the full range of environmental conditions and radar operating parameters to yield robust results. Recent advances in numerical modeling have made it possible to simulate convective storms at very fine scales over a broad range of environmental conditions (e.g., Wicker and Wilhelmson 1995; Lewellen et al. 1997). Coupling a software radar emulator with high-resolution numerical simulations, one can generate large sets of simulated radar data that span a wide range of radar operating characteristics. These simulated datasets can be used to quantify the impact of radar design and operational mode on the diagnosis of storm features by automated algorithms.

Many approaches have been taken previously in simulating radar data, varying in sophistication from simple time series simulation (Zrnic 1975) to reflectivity calculation (Chandrasekar and Bringi 1987; Krajewski et al. 1993) to full simulation of radar returns from each pulse (Capsoni and D’Amico 1998; Capsoni et al. 2001). Zrnic (1975) generated simulated time series radar data and Doppler spectra using an assumed Gaussian distribution of velocities within the resolution volume. Chandrasekar and Bringi (1987) looked at the variation of simulated reflectivity values as a function of raindrop size distribution parameters. Similarly, Krajewski et al. (1993) calculated values of reflectivity factor and differential reflectivity using rainfall rates from a numerical model, with an assumed drop size distribution. Neither of these studies was concerned with Doppler velocity or the impacts of scanning strategies. Wood and Brown (1997) evaluated the effects of Weather Surveillance Radar-1988 Doppler (WSR-88D; Crum and Alberty 1993) scanning strategies on the sampling of mesocyclones and tornadoes. The effects of the scanning strategy were accounted for by using an effective beamwidth for the radar, which was used to scan an analytic vortex with a uniform reflectivity field. Capsoni and D’Amico (1998) simulated the pulse-to-pulse time series of radar data by combining the simulated returns from individual hydrometeors within a radar volume. This work was extended to generate polarimetric signatures by Capsoni et al. (2001). Because of the computational requirements of this approach, the radar data were generated for only a single range gate only, and thus many aspects of the scanning radar were not simulated.

This work describes a radar emulator designed to simulate the expected average returns from a scanning Doppler radar. Starting with output from a high-resolution numerical simulation, the emulator generates fields of power, equivalent reflectivity factor, Doppler velocity, and Doppler spectrum width based on the radar configuration and scanning strategy used. Here we show that the emulator is capable of simulating several radar data characteristics, including range resolution, azimuthal over- and undersampling, nonstandard (anomalous) propagation, Rayleigh attenuation, antenna sidelobes, velocity aliasing, and range aliasing.

As an example of its use for research, the emulator is applied to output from a numerical simulation of an F3 intensity (Fujita 1971) supercell tornado simulated by the Advanced Regional Prediction System (ARPS; Xue et al. 2000, 2001) to evaluate the ability of 2° beamwidth radars to directly detect the circulation associated with the tornado. This application is motivated by the first integrated project of the Center for Collaborative Adaptive Sensing of the Atmosphere (CASA; Brotzge et al. 2005), which recently deployed four such radars in the Oklahoma test bed.

2. Radar emulator design

b. Scattering

For computational efficiency, backscattering and extinction cross 65 per unit volume of air are precalculated at each model grid point. Moreover, only Rayleigh scattering by liquid hydrometeors is currently included. According to Battan (1973) and Doviak and Zrnic (1993), the Rayleigh approximation implies the following relationships for the backscattering cross section σ_b and the extinction cross section σ_e of a sphere of liquid water with diameter D:

View Expanded

View Expanded

View Expanded

where λ is radar wavelength, and m is the complex index of refraction for liquid water. The emulator currently assumes a monodisperse distribution of cloud droplets and a Marshall and Palmer (1948) distribution of raindrops, which is consistent with the microphysics scheme used in many numerical models, including the ARPS. The use of a more general gamma distribution (Ulbrich 1983) would be possible. These assumed distributions permit the calculation of the total backscattering and extinction cross sections per unit volume from the cloud water ρ_c, and rainwater ρ_r concentrations (in kg m⁻³) at each grid point:

View Expanded

View Expanded

where ρ_l is the density of liquid water; R_m is the median radius for the cloud droplets; and N₀ is the Marshall–Palmer distribution intercept parameter; R_m and N₀ are assumed to have values of 50 μm and 8000 drops m⁻³ mm⁻¹, respectively. The Debye formula, shown by Saxton (1946, 278–325) to be applicable for the microwave region, is used to calculate the complex index of refraction explicitly. The Debye formula is

View Expanded

where ɛ₁ is the static dielectric constant, ɛ₂ is the optical dielectric constant, λ₀ is the transition wavelength, and i equals−1. Values of λ₀, ɛ₁, and ɛ₂ as a function of temperature were taken from Kerr (1951) and are based on Ryde and Ryde (1945). This formulation of m² allows the emulator to capture the temperature and wavelength dependencies of K_w.

It is important to note that the Rayleigh approximation has been assumed for both scattering and attenuation. The range where the Rayleigh approximation is accurate for attenuation is much more limited than that for backscatter (Battan 1973). Therefore, at wavelengths shorter than approximately 10 cm, the attenuation simulated here will grossly underestimate the actual attenuation. This limitation will be addressed in future work by using scattering parameters calculated using Mie theory for spherical scatterers (Mie 1908) and/or the T-matrix method for nonspherical scatterers (Waterman 1965).

c. Sampling of input fields

To sample the virtual model atmosphere, the emulator calculates radar variables along the path of individual pulses at the interval specified by the pulse repetition time (PRT). This allows the input model fields, as well as the state of the radar (such as antenna pointing angle), to change for individual pulses. While the emulator is currently configured for a mechanically scanning antenna, pulse-by-pulse calculation can be used to emulate measurements for phased-array radar as well. The pulse generated within the emulator defines the volume of space that contributes to a sample taken along the radar beam. It is bound in elevation and azimuth by a fixed multiple of the half-power beamwidth. This multiple is chosen based on the number of sidelobes that are desired for simulation in the antenna pattern. The pulse is bound in range by the specified pulse length. This volume of space is subdivided into individual pulse elements defined in angular coordinates such that, at the maximum range from the radar, the dimensions of each pulse element are 10% smaller than the model grid spacing at that range. While there is flexibility in how many subdivisions are made in the pulse element, having too many increases the computation requirements without changing the results. Subdividing the pulse volume into subelements that become larger than the grid spacing at the maximum range of the radar will result in undersampling of the model input fields and will change the emulator output.

Each pulse element is assigned values of extinction cross section, backscattering cross section, and radial velocity that correspond to the grid point nearest to the element’s location in space. Nearest-neighbor sampling is chosen over interpolation to improve the computational efficiency of the emulator. Because the pulse elements are generally much smaller than the grid cells, this sampling method provides sufficient accuracy.

The radial velocity is calculated by the projection of the total wind velocity vector onto the radar beam:

View Expanded

where V_r is the radial velocity, u is the x component of the wind, υ is the y component of the wind, w is the z component of the wind, w_t is the average terminal fall speed for the hydrometeors, θ is the azimuth angle measured clockwise from north, and ϕ is the elevation angle. The average hydrometeor terminal fall speed for the grid box is calculated as a backscatter cross-section weighted average given by

View Expanded

where η is the total grid reflectivity, ρ is the air density of the grid box, ρ₀ is the reference density, N(D) is the drop size distribution, and V_t(D) is the terminal fall speed as a function of diameter, which is calculated using the fitted relationship of Brandes et al. (2002):

View Expanded

where V_t is in meters per second and D is in millimeters. The weighting by backscatter cross section makes the terminal fall speed more representative of the velocity seen by the radar than a simple mass-weighted average.

The pulse itself is propagated through the numerical grid using a ray-tracing technique. For each range gate, the height of each pulse element is determined separately by taking into account the atmospheric index of refraction experienced by that particular ray element. This allows for differential propagation across the radar beam. The change in the height above ground Δh and change in range from the radar (along the surface of the earth) Δr can be calculated from the incremental change in range along the path Δs as

View Expanded

View Expanded

View Expanded

where a is the radius of the earth, h is the previous height of the element above ground, n is the index of refraction at height h, n₀ is the index of refraction at the radar, and ϕ is the initial elevation angle of the element (Doviak and Zrnic 1993). The index of refraction is calculated from the model temperature T, water vapor pressure e, and air pressure p, using the relation provided by Bean and Dutton (1966):

View Expanded

where C_d, C_w1, and C_w2, have values of 0.776 K Pa⁻¹, 0.716 K Pa⁻¹, and 3.7 × 10³ K² Pa⁻¹, respectively. The element’s range from the radar along the surface of the earth is then converted to standard two-dimensional Cartesian coordinates, which are used to determine the location of the element on the model grid.

The pulse volume is allowed to propagate through the environment as far as twice the unambiguous range R_a,

View Expanded

where T_s is the PRT, and c is the speed of light. Allowing the pulse to propagate 2R_a from the radar means that after one PRT from the time the radar is started there are two pulses propagating through the model field at any given instant. Thus, when a sample is taken, the returns from both pulses are assigned to the gate, producing the effects of range aliasing. Range aliasing can be disabled if desired.

d. The calculation of returned power

The entire pulse volume is stepped forward in range while keeping track of the total extinction cross section along the path. This running total is kept for each pulse element, which allows for the calculation of differential attenuation across the pulse. As the pulse is propagated through the model grid of the simulated atmosphere, it is periodically sampled at an interval in range dictated by the specified gate spacing. This allows for the gate and pulse lengths to be independent. When a pulse sample is taken, three values are calculated: power, power-weighted average radial velocity, and power-weighted radial velocity variance. The power weightings are performed over all of the pulse elements as follows:

View Expanded

View Expanded

where V_r is the power-weighted average radial velocity, σ_vr is the power-weighted variance of radial velocities, P_i is the power for a particular pulse element, and V_i is the radial velocity for a particular pulse element. The estimate of variance here, σ_vr, is not unbiased but is chosen to simplify the computations. Since values from thousands of pulse elements are used in the calculation, the difference between biased and unbiased estimates is negligible.

As given in Doviak and Zrnic (1993), the power P for a sample taken at range r₀ is given by

View Expanded

where

View Expanded

View Expanded

and P_t is the transmitted power, g is the system gain, λ is the wavelength, r is range from the radar, l is the attenuation factor, f ² is the normalized antenna pattern, η is the reflectivity (backscattering cross section per unit volume), θ is the azimuth angle relative to the beam center, ϕ is the elevation angle relative to beam center, and W is the range-weighting function. The emulator approximates this integral with a sum over the finite elements within the pulse volume,

View Expanded

where ΔV is the volume of a pulse element, and all quantities subscripted with i are values for a particular pulse element. The emulator assumes a Gaussian range-weighting function and a normalized antenna pattern with the following form (Doviak and Zrnic 1993):

View Expanded

where J₂ is the second-order Bessel function of the first kind, θ is the angular offset from boresight, and D_a is the diameter of the antenna, which for (21) above can be calculated from the half-power beamwidth θ₁ as

View Expanded

where λ is the wavelength. Doviak and Zrnic (1993) state that (21) describes the antenna pattern for the first few sidelobes quite well for a parabolic antenna. However, (21) is limited in that it gives sidelobes of a fixed level and location (e.g., Fig. 1), prohibiting configuration of sidelobes with arbitrary magnitude.

e. Moment calculation

The sampling of model data is repeated for the number of pulses that are to be averaged for a radial of data, as specified by the scanning strategy. Moment data (power, Doppler velocity, and Doppler spectrum width) are then generated at each range gate along the radial. Power is calculated as the average of all power samples for the specified number of pulses at that range gate. Note that this is the expected mean power that an actual weather radar would produce if random power fluctuations were successfully removed by the pulse averaging and the radar system had no noise.

Radial velocity is calculated as the power-weighted average of all velocity samples (one per pulse) at that range gate. To emulate velocity aliasing, this average is restricted to a value within the Nyquist interval and is given by

View Expanded

where

View Expanded

View Expanded

View Expanded

where V_a is the aliased velocity value; V_r is the original (unaliased) radial velocity; V_NYQ is the Nyquist (or aliasing) velocity; and n, an integer, is the number of Nyquist intervals by which the V_a differs from V_r. One advantage of emulated data is that the unaliased Doppler velocity is known. Spectrum width is calculated as the power-weighted average of the variance for each sample, which is the variance of all velocity values within the pulse. Initial attempts at emulating spectrum width used only the variance of the individual velocity samples that were themselves an average over the entire pulse. That approach produced unreasonably low spectrum width. By taking into account the variance of all velocity values within all pulses, the spectrum width takes into account the effect of antenna rotation and wind shear across the radar beam. However, we have neglected subgrid-scale atmospheric turbulence. Moreover, since the emulator does not generate a true power spectrum at each range gate, the emulated spectrum width does not take into account a limited Nyquist interval or the pulse-to-pulse variability associated with random phase changes from scatterers moving relative to the transmitted wavelength. In addition to the three moments above, equivalent reflectivity factor (Z_e) is calculated from the average power P_r, using

View Expanded

where τ is the pulse duration.

3. Demonstration of emulator capabilities

Emulated data were generated for different radar characteristics to illustrate the emulator’s capabilities and to demonstrate the impact of radar design on data quality. The input is from a numerical simulation of a supercell thunderstorm produced using the Advanced Regional Prediction System (ARPS; Xue et al. 2000, 2001). The ARPS is a fully compressible and nonhydrostatic prediction model, and its prognostic state variables include wind components u, υ, w; potential temperature θ; pressure p; the mixing ratios for water vapor q_υ; cloud water q_c; rainwater q_r; cloud ice q_i; snow q_s; and hail q_h; plus the turbulent kinetic energy used by the 1.5-order subgrid-scale turbulent closure scheme.

For the current simulation, only liquid-phase Kessler (1969) microphysics is used. The simulation had a horizontal grid spacing of 50 m over a 48 km by 48 km domain and a vertically stretched grid that goes from the surface to 16 km. The stretching is specified by a hyperbolic tangent function, having a minimum spacing of 20 m at the surface, 380-m spacing at the top of the model, and a mean spacing of 200 m (Xue et al. 1995). The model thunderstorm was initiated by a thermal bubble in a horizontally homogeneous environment defined by the 20 May 1977 Del City, Oklahoma, supercell sounding reported in Ray et al. (1981). Detailed analysis of the simulated storm is unimportant here as the simulation serves only as input to the emulator. Furthermore, only a single time step taken during the most intense portion of the tornadic stage of the simulated storm is used. The impact of storm evolution on radar-derived storm structure will be the subject of future studies.

Figure 2 shows the rainwater mixing ratio and storm-relative velocity at the surface at 13 500 s into the simulation, the time used to produce the emulated data. An intense supercell thunderstorm with characteristic v-notch and hook (Lemon and Doswell 1979) in the rainwater field is evident. The simulated tornado vortex is about 200 m in diameter, with maximum winds of about 75 m s⁻¹.

Table 2 lists the parameters used to define the radar and scanning strategy in each of the experiments discussed below.

4. Application to tornado detection

To illustrate the research and operational value of the radar emulator, the detectability of tornadic signatures is examined as a function of the radar range from the tornado and the azimuthal sampling interval. The emulated radar characteristics (Table 3) follow those of the Integrated Project 1 (IP1) radars deployed by the CASA Engineering Research Center (Brotzge et al. 2005). To keep cost low, these radars have a broad beam (2° half-power beamwidth), use relatively low power (25 kW), and operate at X band. One of the goals of CASA is to improve the detection of low-level hazardous weather, such as tornadoes, by placing the radars close to each other and by performing collaborative adaptive sampling of the lowest 3 km of the atmosphere. The average radar spacing of the IP1 network is about 30 km.

Using the known location and intensity of the tornado in the model as a baseline, this study examines the values of several tornado intensity metrics, including maximum velocity V_max, maximum radial velocity difference ΔV, diameter D, and axisymmetric vorticity ζ_a, as determined directly from the emulated radial velocity data. They are examined as functions of range (3, 10, 30, and 50 km) from the tornado, using both azimuthally matched sampling (2° intervals) and oversampling (at 1° intervals). The axisymmetric vorticity, defined as the vorticity for an axisymmetric vortex having the same ΔV and diameter as the tornado, is given by

View Expanded

The intensity parameters are used to quantify the range dependency of tornado detection by 2° beam X-band radars. To eliminate the impact of dealiasing algorithms, this quantitative analysis assumes perfectly dealiased Doppler velocities and, hence, represents the best-case scenario. Furthermore, the known location of the tornado is used to choose the gates for the calculation of the parameters, as opposed to choosing a location based on the position of the velocity maxima in the data. The values of these parameters for all cases, with different range and azimuthal sampling combinations, are listed in Table 4. It should also be noted that the viewing angle chosen here (from northeast) minimizes the impact of attenuation at X band.

At 3-km range, the tornado can be clearly identified in both the matched and oversampled moment data (Figs. 14 and 15). The tornado resides in the tip of a well-defined hook echo in a region of enhanced spectrum width. At this range, the strong flow within the tornado can be diagnosed even in the aliased Doppler radial velocity field, especially when the storm is azimuthally oversampled by the radar beam. Indeed, the oversampled unaliased velocity field (Fig. 15d) shows separated inbound and outbound velocity maxima in the tornado vortex. Separation between maximum radial velocities is a useful criterion for resolving a tornadic circulation. Even at this close range, however, the peak Doppler velocity of the tornado measured by the radar (Table 4) is greatly decreased from the true value of 78 m s⁻¹. The distance between peak Doppler velocities, 216 m, agrees well with the ∼200 m distance between velocity maxima in the model field. Axisymmetric vorticities of 0.864 and 1.024 s⁻¹ for the matched and oversampled cases, respectively, further indicate that the tornadic circulation is well resolved by the 2° half-power beamwidth radar at 3-km range.

Moving the radar to 10 km away from the tornado resulted in degradation of the radar-derived structure as the geometric width of the beam increased (Figs. 16 and 17). While the hook echo and maxima in spectrum width were still fairly well resolved in the matched sampling case, the peak Doppler velocities associated with the tornado were located much farther apart (705 m), with the peak velocity down to 35.2 m s⁻¹. Consequently, the estimated vorticity decreased to 0.163 s⁻¹, or 20% of the value obtained at 3-km range. It should be noted that the peak inbound velocity measured with the tornado was only 22.4 m s⁻¹, less than the 24.1 m s⁻¹ inbound velocity associated with the mesocyclone. It is only when the storm was azimuthally oversampled that the tornadic circulation was resolved at 10-km range with the current radar system. With 1° azimuthal sampling (Fig. 17), the distance between the velocity maxima decreased to 529 m, which was the main factor for the increase in the axisymmetric vorticity to 0.237 s⁻¹.

At 30-km range with matched sampling, the structure of the hook echo and spectrum width field is so degraded that the location of the tornado is no longer well diagnosed by the radar parameters (Fig. 18). Even the nonaliased velocity field no longer shows separate velocity maxima for the tornado and mesocyclone. Instead, a single maximum is located several gates away from the known location of the tornado. The tornado-scale flow is no longer resolved because the half-power beamwidth is over 1 km wide at this range, roughly 5 times the diameter of the tornado. Also at this range, the poor resolution of the data makes distinguishing storm shear from regions of aliased velocity a challenge (Fig. 18c). Using the known location of the tornado, a vorticity estimate of 0.064 s⁻¹ is calculated for this circulation. Such a low vorticity estimate would likely not be associated with a strong tornadic mesocyclone. It should be noted that the inbound velocity measured and used in the calculation is only 1.6 m s⁻¹. Even when oversampling is performed (Fig. 19) the tornadic circulation is still not resolved by a 2° beamwidth radar when it is located 30 km away. While the separation between the maximum inbound and outbound velocities in the mesocyclone and a region of enhanced spectrum width exists, there is no indication of a tornado vortex signature (Brown et al. 1978). Furthermore, dealiasing the X-band radial velocity field becomes challenging, as the beam containing the tornado appears to be embedded in a broad-scale region of aliased inbound velocities. In reality, the minimum in radial velocity associated with the tornado separates aliased inbound velocities from the true receding flow.

At 50-km range, the half-power beamwidth is 1.7 km across, reducing even the mesocyclone-scale circulation to gate-to-gate shear and completely obscuring the tornado-scale flow (Figs. 20, 21). It would be very difficult to detect reliably a tornado with the size and intensity as in the simulation using only the moment data at this range.

5. Conclusions

A Doppler radar emulator based on Rayleigh scattering has been developed that simulates a wide range of radar operating characteristics, including range and azimuthal oversampling, velocity and range aliasing, Rayleigh attenuation, second-trip echoes, antenna sidelobes, and anomalous propagation. The emulator calculates returned power, equivalent radar reflectivity factor, Doppler velocity, and Doppler spectrum width from cloud model output containing fields of wind, temperature, moisture, and hydrometeor species. The capabilities of the emulator are demonstrated using a high-spatial-resolution simulation of a tornado embedded within a supercell thunderstorm. It is shown that the emulator is a useful tool for evaluating the capabilities and trade-offs in the design, deployment, and operation of radar systems. Given that the emulator can produce numerous synthetic datasets for a wide range of storm types and radar characteristics, we believe that such a tool can be a significant aid in the development of radar algorithms. Such realistically emulated data can also be used in observing system simulation experiments (OSSEs) such as those of Xue et al. (2006) for examining the potential impact of radar data on thunderstorm analysis and prediction.

Using the output from a 50-m horizontal-resolution simulation of a supercell storm that explicitly resolves an F3 intensity tornado of about 200 m in diameter, the basic capabilities of the emulator are first tested in a set of experiments that examines the effects of radar wavelength, beamwidth, azimuthal oversampling, gate length, sidelobes, pulse repetition frequency, and the effects of velocity and range aliasing. Results consistent with theory are observed from the simulated data.

As an example of the emulator’s many potential applications, the detection of the simulated tornado described above by a 2° half-power beamwidth X-band radar is examined. The emulated data show that the strength of the diagnosed tornado circulation decreases rapidly with range, with the tornado-scale flow becoming unresolved at and beyond 30 km. Azimuthal oversampling improves the ability to diagnose the tornado vortex, especially from the 10- to 30-km ranges. At shorter ranges, the 2° beam-matched azimuthal sampling is sufficient. It is important to note that the simulated tornado examined here represents the top 10% of tornadoes occurring in nature in terms of intensity. The much more prevalent weaker and/or smaller tornadoes will be even harder to detect.

A significant problem demonstrated by the emulator is the impact of velocity aliasing at X band on the potential diagnosis of the circulations. Correct dealiasing is crucial to tornado detection when the detection algorithm mainly relies on the radial velocity data (e.g., Burgess et al. 1993; Liu et al. 2007). Any method that can increase the effective Nyquist velocity, such as the use of staggered PRT (Gray et al. 1989), would likely be helpful.

Future studies will include more radar operating parameters as well as the use of objective algorithms to evaluate tornado detection for broad-beam X-band radars. This work is motivated by the Integrated Project I (IP1) of the Collaborative Adaptive Sensing of the Atmosphere (CASA; Brotzge et al. 2005) Engineering Research Center that recently deployed four 2° half-power beamwidth X-band radars in central Oklahoma. One of the goals of such inexpensive networks is tornado detection, based on the promise of being able to observe at low altitudes (0–3 km) and at short ranges (less than 30 km), and do so in a collaborative and adaptive manner. In the future, the radar emulator will be further enhanced to include Mie scattering, as well as scattering from ice-phase hydrometeors, and the antenna routine will be modified to allow for emulation (using many model time steps) of electronically steered phased-array radars that can point the beam in arbitrary directions on a pulse-to-pulse basis. This will enable applied research in the phased array radar (Forsyth et al. 2005) program and further enhance the educational utility of the radar emulator.