a. Overview of simulated atmospheric fields

The model-simulated dataset used to test our radar simulator was produced by the ARPS model (Xue et al. 2000, 2001), which was also used by May et al. (2007). The simulation used here has twice as much horizontal resolution, however. 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 (see Xue et al. 2000, 2001, 2003 for details). For the current simulation, only the Kessler-type warm-rain microphysics is used. The simulation had a horizontal grid spacing of 25 m over a 48 km × 48 km domain during the period of time that data are used here, and the vertical grid is stretched and has a vertical spacing of 20 m at the surface. The simulation was for a tornadic supercell thunderstorm, and the storm was initiated by a thermal bubble in a horizontally homogeneous environment defined by the 20 May 1977 Del City, Oklahoma, sounding reported in Ray et al. (1981).

For the ARPS simulation, a horizontal resolution of 50 m was first used and the entire simulation length was over 4 h. An intense tornado developed after 3.5 h into the simulation on the 50-m grid. Over a half-hour period centering on the time of the most intense tornado, a simulation using a 25-m resolution is performed, starting from an initial condition interpolated from the 50-m grid. On this grid, an F4–F5 intensity tornado with a maximum ground-relative wind speed of over 120 m s⁻¹ was obtained, with a pressure drop of over 80 hPa at the center of the tornado vortex. The simulated tornado vortex is about 200 m in diameter near the ground. Because of the small size and great intensity of the tornado vortex, spatial and temporal variabilities associated with the tornado are extreme. Gridded outputs with all model fields are available at 1-s intervals from this simulation. Detailed analysis of the simulated tornado is not very important here and will be reported elsewhere.

In Xue et al. (2007), a single time output from the same 25-m simulation near the time of most intense tornado was sampled by a simple Gaussian beam-pattern-based radar simulator to create simulated radial velocity data for testing a variational velocity analysis scheme combined with azimuthal oversampling. Liu et al. (2007) further used the dataset to test a wavelet-based tornado detection algorithm.

In the proposed time series simulator, at any given instance in time, we will use two time level outputs from this 25-m model simulation as input to our radar emulator to create simulated observations.

b. Spatial/temporal evolution of scatterers

The radar echo from a transmitted pulse is a complex function of continuous fields of reflectivity and velocity, which depend on the true atmospheric conditions and are thus impractical for analytical solution. However, it is possible to approximate this function by using a large number of discrete scatterers that are randomly distributed over the domain of interest. This is essentially a spatial sampling problem and the composite signal is obtained by coherently integrating the reflected signals from each discrete scatterer. Compared to the volume-scattering case, the returned signal from a single discrete scatterer is much simpler and analytical solutions exist. Volume scattering is approximated by summing the returned echoes from tens of thousands of discrete scatterers that exist within the field of view of the radar. This process is termed Monte Carlo integration (Metropolis and Ulam 1949).

The coherently integrated signal from of all the discrete scatterers is referred to as the composite signal. Each sample of the composite signal is a complex sum of the baseband signal from all the individual scatterers, which can be expressed mathematically as follows:

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where superscript (·)^(k) indicates the index of the discrete scatterers, N is the total number of scatterers, A^(k) represents the amplitude of the signal from the kth scatterer, ψ^(k) represents the phase of the kth scatterer, and N represents additive white Gaussian noise signal. Equation (1) is the general form for producing the composite signal and can be readily extended to simulate systems that use multiple receivers, multiple frequencies, and/or multiple pulse repetition times (PRTs), for example.

Figure 1 is an artist’s depiction of the general structure of the proposed weather radar simulator algorithm. To initialize the simulator, an enclosing volume is defined that includes the radar’s field of view plus a small margin to mitigate undesirable effects caused at the fringes of the volume. A set of scatterers (typically thousands) is randomly positioned within this enclosing volume, with a uniform distribution. At each sample time (PRT), the composite returned signal is derived using Eq. (1). As time progresses, the position of each discrete scatterer is updated based on its velocity and PRT. There are two components in updating and replacing the scatterers. First, scatterers that exit the enclosing volume (due to the position update) are replaced with randomly positioned new scatterers. It should be emphasized here that in order to properly emulate the composite signal using Monte Carlo integration, the spatial uniformity of the scatterer distribution must be maintained. Second, in order to avoid the scenario in which convergent (or divergent) flows cluster together (or spread apart) the scatterers, it was necessary to implement a random replacement routine. Clearly, this strategy depends on the wind field structure. For example, for radar simulations from a vertically pointing radar, with dominant transverse wind (horizontal), such a routine would not be necessary (Holdsworth and Reid 1995; Yu et al. 2000). In our case, however, random replacement was needed given the flows present in the storm simulation fields. Randomly replacing scatterers at each sampling time, such that all targets are replaced every 5 s, provides satisfactory results. This process is equivalent to considering that each discrete scatterer has a limited lifetime.

The positions of each scatterer are updated with the instantaneous velocity field and a random component that relates to subgrid turbulent kinetic energy (TKE). The velocity and TKE fields are extracted and interpolated from a set of pregenerated fields, which will be discussed later. The position update can be mathematically described as follows:

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where X^(k)(n) = [x y z] represents the position vector of the kth scatterer at time n, and V^(k) = [ũ υ̃ w̃] represents the velocity vector of the kth scatterer and T_s represents the sampling interval, that is, the PRT of the radar system.

As mentioned previously, each velocity component is obtained from the wind velocity and subgrid TKE fields. They are described as follows:

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where the second term of Eqs. (3), (4), and (5) represents an instantaneous perturbation of velocity vectors due to the subgrid TKE. It is obtained by scaling the output of a random number generator ε that has a normal distribution and unity variance.

In Eq. (1), the phase of the backscattered signal from each discrete scatterer depends on the number of cycles the signal has gone through during its travel from the radar to the target and back to the radar. To be more precise, the phase is also a function of backscattered complex amplitude of each particle, but this complication is not necessary to realistically simulate the radial velocity of the scatterer. In other words, we want to produce time series data that carry the signatures of a Doppler spectrum that is representative of the wind field distributions from the ARPS model. Thus, only the phase change from pulse to pulse is needed and not the phase due to the scatterer. The phase is given by the following equation:

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where D^(k) represents the two-way distance of the kth scatterer and λ represents the wavelength of the radar system. It is easily justified to assume that the initial phase of the transmit signal is zero. The Doppler shift of each target is created by the change of phase with time, which is controlled by the position update Eq. (2).

In the following sections, the important amplitude term in Eq. (1) will be discussed. The spatial weighting functions (range and angle), used in the Monte Carlo integration process, will first be presented. Then, the reflectivity parameterization scheme will be provided.

c. Weighting functions

Ignoring system losses, the amplitude of the kth scatterer A^(k) depends on the transmit power, antenna pattern, range weighting function, reflectivity, and range. The weighting functions (angle and range) account for the varied contribution from each scatterer at a specific angle and range. The reflectivity of each scatterer is a function of atmospheric conditions, which is derived from realistic physical parameters, and will be described in the next section. Radar parameters that are shared among all discrete scatterers, such as transmit power and antenna gain, are set constant. Given the perfect calibration possible with simulations, these common parameters are considered relatively unimportant in the process of computing the amplitude. The amplitude for a scatterer at an arbitrary location (x, y, z) can be described as

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where r represents the range of the scatterer from the radar, w_a is the angular weighting function of the two-way beam pattern, w_r is the range weighting function, and Z_e represents the parameterized equivalent reflectivity factor. For notational convenience, the dependence of w_a, w_r, and Z on position (x, y, z) is not explicitly stated but is assumed throughout this paper. It should be emphasized that the backscattered power is inversely proportional to r⁴ since point targets are used in the simulator. Through the Monte Carlo integration process, however, the range dependence will be reduced to r ² given the volume integration performed.

The range weighting function, shown in Fig. 1, simulates the effect of pulse shape and receiver filtering on each scatterer. Scatterers near the center of the range gate and near the boresight of the radar will have the maximum weighting. For a narrowband receiver with a time–bandwidth product equal to unity, the (power) range weighting function can be sufficiently approximated with a Gaussian function centered around the desired range r₀ and is given by Doviak and Zrnić (1993):

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where σ_r = 0.35Δr in which Δr represents the range resolution given by

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The speed of light is denoted by c (3 × 10⁸ m s⁻¹) and τ is the pulse duration.

For a typical parabolic dish antenna, the normalized one-way transmit beam pattern has a sine functionality with the largest gain concentrated on the main lobe of the pattern. This function is well approximated by the following equation (Doviak and Zrnić 1993):

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The angular distance from the beam axis (boresight) is given by ϕ and J₂ is the Bessel function of the first kind (second order). For a monostatic radar system that uses the same antenna for transmit and receive, the power-normalized two-way beam pattern is simply the square of Eq. (10), given by

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where w_tx and w_rx would be equal, in this case. For more advanced applications, such as phased array antennas and imaging radars, the simulator has been designed to allow for the case of different transmit and receive beam patterns. Examples from each case will be presented in section 4.

d. Parameterization of reflectivity

The fundamental control of each discrete scatterer is governed by the ARPS-generated atmospheric fields. However, reflectivity factor in Eq. (7) is not a standard output parameter of the ARPS model and must therefore be calculated from known model microphysical parameters. Based on the work of Smith et al. (1975) and for the purpose of radar data assimilation, Tong and Xue (2005) developed the scheme that is used here to characterize the equivalent reflectivity factor of the individual scatterers under the Rayleigh assumption. In that work, the total equivalent reflectivity factor was given by

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where Z_r, Z_s, and Z_h represent the reflectivity factors from the three precipitating hydrometeors, that is, the rainwater, snow, and hail, respectively. Of course, it is more standard to present reflectivity factor in units of dBZ, which can be calculated using the following equation:

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In the work of Tong and Xue (2005), the rainwater component of the equivalent reflectivity factor was determined to have the following form:

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where ρ is the air density (kg m⁻³), q_r is the rainwater mixing ratio (kg kg⁻¹), and N_r = 8.0 × 10⁶ m⁻¹ is the intercept parameter in the assumed Marshall–Palmer exponential raindrop size distribution. Regions with snow have possible contributions to Z_e from both dry and wet snow. In such cases, a 0°C threshold in air temperature is used to distinguish the respective contributions. This component of the total equivalent reflectivity factor due to snow has been shown to have the following form (Tong and Xue 2005):

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where ρ_s = 100 kg m⁻³ is the density of snow, ρ_i = 917 kg m⁻³ is the density of ice, N_s = 3.0 × 10⁶ m⁻⁴ is the intercept parameter for snow, |K_i|² = 0.176 is the dielectric factor for ice, and |K_r|² = 0.93 is the dielectric factor for water. Finally, for the case of hail, the wet hail formation is used and given by Smith et al. (1975):

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where ρ_h = 913 kg m⁻³ is the density of hail, and N_h = 4.0 × 10⁴ m⁻⁴ is the intercept parameter of hail.

Given the atmospheric fields generated by the ARPS model and the scheme for calculating the equivalent reflectivity factor, it is now possible to produce the needed fields for the radar simulator. An example of the ARPS fields from the 25-m tornadic thunderstorm simulation is provided in Fig. 2. The upper three panels (from left to right) provide the wind fields (horizontal wind vectors plus vertical velocity in shades), air density, and rainwater mixing ratio, respectively. Assuming a simulated radar position to the right of the displayed fields, the radial velocity (Fig. 2d) is derived from the three-dimensional wind by simple projection. The potential temperature is shown in Fig 2e. Equivalent reflectivity factor derived from the ARPS fields is shown in Fig. 2f. Note that even though our reflectivity simulation can handle the effect of ice species, our current dataset does not contain ice. As expected, regions of high Z_e follow regions with high rainwater mixing ratio. These fields are used in Eq. (1) to produce time series data. Therefore, we will consider the data in Fig. 2 as ground truth, with the resolution provided by the inherent grid spacing of the ARPS output.

a. Single range gate canonical example

A simple one range gate simulation is presented in this section. To illustrate simulating time series data that are solely modulated by the motions of the atmosphere, we have overridden the reflectivity values from the ARPS model and set them to a constant value for all scatterers. A radar is configured to have a 2° antenna beamwidth and a 1.5-μs pulse width (Δr ≈ 225 m), directed to a volume located 20 km away from the radar. With this configuration, the resultant domain is approximately 700 m × 300 m × 450 m, that is, a domain size that can contain 27 resolution volume cells (one additional cell on each side) in order to effectively represent the contribution from neighboring gates. This simulation domain encompasses more than 7000 grid points from the ARPS model. For this example, 540 discrete scatterers (20 per resolution volume) were used.

Figure 4 shows the histogram of radial velocities derived from the true wind vectors and the Doppler spectrum of the time series data from the simulator. One can see that the simulator correctly produced a time series that has a velocity spectrum that agrees with the wind field being injected into the simulator through the table lookup and interpolation process.

b. Parabolic-dish scanning radar

As a first test of the proposed radar simulator, a mechanically scanned, parabolic-dish radar, which closely mimics the parameters of the Weather Surveillance Radar-1988 Doppler (WSR-88D), was simulated. The operating frequency was set to 2.7 GHz with a symmetric, two-way, 3-dB beamwidth of 0.95°. A pulse of 1.57 μs in length was used, providing 235-m range resolution. An aliasing velocity of 27.8 m s⁻¹ resulted from the 1-ms PRT. An elevation angle of 0.5° was used with the first of 22 range gates set at 8 km. The radar was scanned over a 24° swath of the ARPS simulated atmosphere, which contained a tornado producing a hook echo in the reflectivity plots. The subvolume of the model data used for the radar simulator (see Fig. 1) had a size of 6 km × 6 km × 0.24 km. Given that the resolution volume at 10 km (middle of the domain) is approximately 6.46 × 10⁶ m³, at least 2.67 × 10⁴ scatterers are needed in order to meet the aforementioned condition of 20 scatterers per resolution volume. Therefore, 30 000 discrete scatterers were used for this example. The radar was scanned over the 24° angular region with a 1° azimuthal sampling interval, although finer sampling could easily be achieved. The antenna rotation rate was set to produce 50 samples for each azimuth angle with a 50-ms dwell time. Gaussian-distributed, complex noise was added to the time series data in order to simulate electronic receiver noise. Since the noise floor should be approximately constant for a particular radar system, the additive noise power was chosen to produce an average signal-to-noise ratio (SNR) of 70 dB over the entire domain (azimuth angles and range) scanned by the radar.

Using the 50 time series samples, the autocorrelation function at lags 0, 1, and 2 were estimated for each range gate and azimuth angle. These results were used with covariance processing to produce estimates of reflectivity, radial velocity, and spectrum width, which are shown in the top three panels of Fig. 5. The simulated radar is located to the right of the figure, which determined the orientation of the radial velocity. The limits of plot axes indicate the size of the input data grid or enclosing volume. As can be seen, the enclosing volume is larger than the actual simulated region, which mitigates artifacts caused by boundary effects as scatterers enter/exit the volume (Holdsworth and Reid 1995).

Markers on the moment fields indicate the locations, for which time series and corresponding Doppler spectral estimates are shown at the bottom of the figure. Five examples were chosen to sample a variety of atmospheric conditions. The first example (denoted by a circle) corresponds to an extremely low-SNR case with the expected flat spectral shape. The time series show low-amplitude fluctuations typical of noise-dominated signals. Note that the axis scales for the time series plots are not constant. The next example (denoted by a triangle) has the strongest backscattered power from the set and shows an inbound velocity of approximately −11.78 m s⁻¹. The reflectivity is over 60 dBZ near the core of the mesocyclone as indicated by large dashed circle in upper-left panel. The last three examples provide illustrations of partial aliasing, due to the 27.8 m s⁻¹ unambiguous velocity and various reflectivity levels. In general, the estimated Doppler spectra have a shape (Gaussian) that is expected from volume-filled atmospheric scatter, lending credibility to the Monte Carlo integration scheme used.

c. Phased array imaging radar

As another example of the utility of the radar simulator, a more advanced radar design is now used. Here, a phased array radar system has been simulated with a transmit frequency of 3.2 GHz, a pulse width of 1.57 μs (Δr = 235 m), and a PRT of 1 ms. The radar is scanning at an elevation angle of 1.5° at a range of 10 km for the first gate. Twenty-two (22) range gates are used with an 18° azimuthal coverage (0.75° sampling) to observe the cyclonic circulation present in the ARPS data. To test aircraft clutter mitigation using adaptive beam forming and to illustrate the flexibility of the radar simulator, a strong point target (aircraft) has been inserted into the atmospheric fields of the ARPS model, flying toward the northeast within the field of view, at a speed of 28 m s⁻¹. For simplicity, the reflectivity of the aircraft has been chosen to be an arbitrarily high value of 80 dBZ in order to have a target that obscures the weather signals. The enclosing volume for this simulation is approximately 6 km × 5.2 km × 0.5 km.

The simulated radar uses a spoiled transmit beam that has a uniform 18° azimuth coverage. In practice, this can be accomplished by either using a subset of transmit elements that is closer to the center of the array, or an independent transmitter that has a wide beam pattern. In elevation, the transmit beam has a beamwidth (Δθ) of 1.5° with a Gaussian weighting applied so that scatterers on the scanning elevation plane contribute most significantly. Such a radar design will allow the scanned region to be simultaneously observed from all directions. The 93 receive elements of the array are shown in Fig. 6 and are assumed to be omnidirectional. In this case of an imaging radar, the simulator will produce time series data for each of the receive elements. The azimuthal resolution can be estimated using (Stoica and Moses 1997)

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where L = (m − 1)d/λ is the array length measured in wavelengths, m is the number of elements across the longest aperture, that is, m = 31, and d is the element spacing, which is 10 cm for this particular configuration. The azimuthal angular resolution (Δϕ) is therefore approximately 1.79°. Using a similar procedure to the previous example to calculate the average number of scatterers for the closest range gate and assuming that the elevation weighting limits the resolution volume as Δϕ = 1.5°, ΔV ≈ r ²(Δθ)(Δϕ)(Δr) = 1.92 × 10⁷ m³. Therefore, at least 1.62 × 10⁴ discrete scatterers are needed for proper sampling. The number of scatterers was set to 20 000 to provide some margin of error.

Using the 93 sets of time series data—one for each receive element—imaging or beam forming is performed using array processing schemes well established in the atmospheric profiling literature (e.g., Palmer et al. 1998; Cheong et al. 2004b). By adjusting the phase and amplitude for each of the signals, it is possible to produce an image of reflectivity, radial velocity, and spectrum width simultaneously for each of the desired azimuth angles (Palmer et al. 1998). As such, no beam-smearing results, and temporal evolution in the atmosphere does not distort the estimated fields.

Here, two imaging methods (Fourier and Capon) are compared. The first results were obtained using the standard technique based on Fourier array processing. A Hann window weighting was applied across the array in order to suppress sidelobes in the resulting beam pattern. Estimated fields of reflectivity, radial velocity, and spectrum width are shown in Fig. 7 for a time sequence of 10 scans. Note that the simulated radar is situated to the left of the figure. Each scan was produced from a 120-ms average (120 pulses with a temporal spacing of 5 s). Over this time period, the spatial autocorrelation function was estimated, which is an essential signal-processing step for beam forming (Palmer et al. 1998; Cheong et al. 2004b). The simulated aircraft is seen to progress from the southeast to the northwest of the reflectivity image through the 10-scan sequence. It should be noted that the aircraft echo causes an extreme distortion of the radial velocity and spectrum width estimates. Although the aircraft is essentially a point target, its effect is seen over a large azimuthal swath. Obviously, this is an effect of the sidelobes of the array pattern, even with tapering the array weights. Fourier imaging is plagued by such effects given that the element weights are designed to produce the highest gain in the desired direction but does not taken into account any undesirable signals.

Capon imaging was first used for atmospheric radar applications by Palmer et al. (1998). Over the years, it has proven to be an excellent algorithm for high-resolution studies where clutter mitigation is a concern (e.g., Cheong et al. 2004b). The algorithm is adaptive with the observed data since the array weights are adjusted (scan to scan) in order to minimize interference to the formed beam. In other words, as the beam direction is scanned across the field of view, the element weights (amplitude and phase) are adapted to form nulls in the direction of interference, while keeping a constant gain in the desired direction. The results are evident in Fig. 8, in comparison to Fig. 7, where the aircraft echo is still observed but its adverse effects are limited to only a few closely spaced cells. By adapting the element weights to the observed data, the Capon method is capable of observing nearby weather echoes while nulling point sources, such as the simulated aircraft echo. In addition, it has been shown that the Capon algorithm can increase resolution at only minimal computational expense.

In this section, examples showing the effectiveness of studies of radar signal-processing techniques using the proposed radar simulator have been presented. It should be emphasized that without time series capabilities, which was not the goal of many previous simulators, a thorough investigation of the technical aspects of radar algorithm development would be significantly hindered.