## Abstract

To obtain accurate radar-measured wind measurements in tornadoes, differences between air and Doppler velocities must be corrected. These differences can cause large errors in radar estimates of maximum tangential wind speeds, and large errors in single-Doppler retrievals of radial and vertical velocities. Since larger scatterers (e.g., debris) exhibit larger differences from air velocities compared to small scatterers (e.g., raindrops), the dominant scatterer type affecting radar measurements is examined. In this study, radar variables are simulated for common weather radar frequencies using debris and raindrop trajectories computed with a large-eddy simulation model and two electromagnetic scattering models. These simulations include a large range of raindrop and wood board sizes and concentrations, and reveal the significant frequency dependence of the equivalent reflectivity factor and Doppler velocity. At S band, dominant scatterers are wood boards, except when wood board concentrations are very low. In contrast, raindrops are the dominant scatterers at Ka and W bands even when large concentrations of wood boards are present, except for low raindrop concentrations. Dual-wavelength velocity differences exhibit high correlation with air and Doppler velocity differences for most cases, which may enable direct measurements of scatterer-induced Doppler velocity bias in tornadoes. Moreover, dual-wavelength ratios are shown to exhibit strong correlations with dominant scatterer size, except when Rayleigh scatterers are dominant. Finally, vertical velocity retrievals are shown to exhibit lower errors at high frequencies, and large errors remain at centimeter wavelengths even after debris centrifuging corrections are applied in cases with high debris concentration.

## 1. Introduction

Radars measure scatterers’ velocities rather than air velocities, resulting in measurement errors when Doppler radial velocities (herein called Doppler velocities) are used to estimate tornado wind speeds. In tornadoes, scatterers include both debris and hydrometeors. Snow (1984) and Dowell et al. (2005) show that debris is centrifuged outward relative to the air, and Dowell et al. (2005) also showed that tangential and vertical velocities of debris are reduced relative to the airspeed. Simulations by Dowell et al. (2005) showed that differences between air and scatterer velocities can reach tens of meters per second for larger scatterers, and velocity differences increase for smaller tornado diameters and higher tangential velocities. Finally, Doppler velocity measurements for a single scatterer type represent a mass-weighted average, and thus portions of the resolution volume with higher debris concentrations have a greater contribution to Doppler velocity measurements (Lewellen et al. 2008).

Because large differences between air and scatterer velocities occur in tornadoes, these differences must be corrected in Doppler velocity measurements to obtain accurate wind measurements. Given strong scientific interest in understanding near-surface wind speeds (e.g., to assess societal impacts or understand corner flow structure), mitigating debris centrifuging errors close to the ground remains a critical yet elusive goal. Large debris concentrations are highest near the surface (e.g., Wurman et al. 1996; Wurman and Gill 2000; Dowell et al. 2005), leading to the largest differences between air and scatterer velocities. As a result, near-surface radial divergence is often observed in Doppler radar observations (Wurman and Gill 2000; Dowell et al. 2005; Wakimoto et al. 2012), whereas laboratory experiments and numerical simulations exhibit converging flow. Near-surface radial divergence leads to further errors when vertical velocity retrievals are performed, such as the ground-based velocity tracking display (GBVTD; Lee et al. 1999). In Nolan (2013) it was shown that anomalously strong retrieved downdrafts result from increased radial divergence caused by debris centrifuging.

To address the debris centrifuging bias problem, Wakimoto et al. (2012) propose a technique to correct debris centrifuging bias by assuming the scatterers in the tornado are raindrops, and then calculating a median diameter based on radar reflectivity factor. For a tornado case without large visible debris, their analysis showed that divergent near-surface radial flow became convergent after the debris centrifuging correction was applied to the GBVTD, resulting in a more realistic wind field. A limitation to this technique is that the dominant scatterers must be raindrops or small objects with similar electromagnetic and aerodynamic characteristics, otherwise the correction is underestimated. Polarimetric radar observations of tornado debris signatures (TDSs) frequently reveal areas of low copolar cross-correlation coefficient () at S band (Ryzhkov et al. 2005; Kumjian and Ryzhkov 2008; Bodine et al. 2013), C band (Palmer et al. 2011; Schultz et al. 2012a,b), and X band (Bluestein et al. 2007; Snyder and Bluestein 2014), suggesting that the Rayleigh assumption does not apply in such cases. Thus, the raindrop scattering assumption requires further testing.

Two important questions must be addressed to estimate and correct debris centrifuging errors. First, physical scatterer attributes must be determined (sizes, concentrations). Second, if the scatterer(s) characteristics are known, then its impact on the backscattered radar signal must be quantified. Because scatterer types may vary significantly among different tornado cases or throughout a tornado’s lifetime, significant uncertainty exists in determining scatterer type. In some cases, visual observations may corroborate the presence or absence of certain debris types, but debris may be obscured (e.g., by debris clouds or precipitation). Radar observations have the potential to provide information about debris characteristics; however, such inferences to date have been primarily qualitative.

In the present study, experiments are performed to simulate radar observations of tornadoes at multiple frequencies using a large-eddy simulation (LES) model, and to examine how Rayleigh and Mie scatterers affect radar measurements. Several simplified assumptions are made for debris types considered to focus this large parameter space. Tornadoes loft different types of debris from natural surfaces (e.g., grass, leaves, soil, rocks) and man-made structures, including wood boards, insulation, and roof tiles (e.g., Snow et al. 1995; McDonald et al. 2004). In this study we focus on two types of scatterers, wood boards and raindrops, and assume a steady debris source. The electromagnetic models assume simplified shapes of wood boards (spheres and square plates). Since actual wood debris has a wide range of aspect ratios or shapes (e.g., 2 in. × 4 in. wood lumber or 4 ft. × 8 ft. plywood sheets), comparisons of two different shapes help examine the importance of debris shape to radar observations. Debris is also assumed to be randomly oriented based on differential reflectivity () near 0 dB in TDSs (Ryzhkov et al. 2005; Bodine et al. 2013), which indicate that debris has random orientations (in a mean sense). Finally, the analysis examines one tornado vortex flow, so future studies should consider different tornado flows.

Simulations reveal the significant frequency dependence of radar variables. Based on these simulations, recommendations for estimating and mitigating velocity errors associated with debris centrifuging and extracting information about debris characteristics are presented. The primary goals of this study are as follows:

Examine the frequency dependence of the equivalent reflectivity factor and Doppler velocity.

Explore the potential of dual-frequency variables for assessing scatterer size.

Develop dual-frequency methods to estimate differences in air and radar-measured velocities.

Section 2 describes the LES model and trajectory calculations. Then, radar variable simulations are discussed, including the electromagnetic models used for debris and raindrops. In section 3, experiments with varying rain and debris concentrations are performed to examine the frequency dependence of the equivalent reflectivity factor () and Doppler velocity. Conclusions are presented in section 4.

## 2. Radar variable simulations using the LES model

In this section, the LES model used to generate the tornadolike flow and calculate scatterer trajectories is discussed. Then, methods used to simulate and Doppler velocity at different radar frequencies are presented.

### a. LES model and trajectory calculations

The LES model used in this study (Uchida and Ohya 2003; Maruyama 2011) simulates the flow of a vortex chamber (Davies-Jones 1973; Church et al. 1979), producing a wide range of tornado-like flows. The reader is referred to Maruyama (2011) and Bodine (2014) for more details about the numerical calculation scheme of the LES model, but a brief discussion is presented here. The LES model configuration parameters are provided in Table 1. The model domain is divided into two primary regions: convergence and convection regions (Fig. 1). Convergence and convection region heights, and , respectively, exhibit a ratio consistent with tornado vortex chambers and expected values in nature (Church et al. 1979). The convergence and convection regions are separated by an updraft hole radius of . Inflow into the vortex chamber is provided through four inlet regions of length, , with an inflow velocity of . The top outlet hole has a radius of and updraft velocities of . The model is nondimensional, so a characteristic velocity, , is used to provide dimensional scaling. In this study, is 225 m s^{−1}.

In the LES model, 155, 155, and 79 grid points are used in the *x*, *y*, and *z* dimensions, respectively. A constant horizontal grid spacing is used within the updraft radius of 15 m, and a stretched horizontal grid is used at greater radii with horizontal grid spacing ranging from 15 to 155 m. The vertical grid is also stretched from 15 to 93 m with values between 15 and 30 m in the convergence region. Lower boundary conditions are semislip with a surface roughness length of 0.03 m. On other simulator surfaces (e.g., sidewalls and the plate between convergence and convection regions), no-slip boundary conditions are employed. Using these boundary conditions and the configuration specified in Table 1, a vortex breakdown simulation is produced. Mean axisymmetric radial, tangential, and vertical velocities are shown in Figs. 2a–c, respectively. Maximum mean tangential wind speed of 72 m s^{−1} occurs at a radius of 54 m and a height of 32 m.

Using wind data from the vortex breakdown simulation, raindrop and debris trajectories are computed using second-order Runge–Kutta integration to solve the following scatterer trajectory equation:

where is the air velocity and is scatterer velocity. The term is the Tachikawa number (Holmes et al. 2006), defined as

where *ρ* is air density, *A* is scatterer area, is a characteristic velocity, *m* is scatterer mass, and *g* is gravitational acceleration. The term represents a ratio of aerodynamic to gravitational forces and is a nondimensional parameter that determines trajectories of different scatterer types. For nearly spherical scatterers (i.e., raindrops herein), varies as a function of the particle Reynolds number () based on an empirical formula from White (1991):

For wood boards, a constant drag force coefficient of 2 is employed for a square plate (Simiu and Scanlan 1996), such as a plywood sheet. Trajectory calculations were tested in Bodine (2014), and results similar to Dowell et al. (2005) were obtained for radial, tangential, and vertical velocities for idealized vortices.

Two scatterer types are simulated in the present study: raindrops and square wood plates. Raindrop trajectories are computed for the following diameters: 0.5, 1, 1.5, 2, and 4 mm. For each drop size, 1 million trajectories are calculated to provide a sufficient number of trajectories for stable scatterer concentration and velocity statistics. However, because of computational constraints, it is not feasible to explicitly simulate all trajectories required for a drop size distribution (DSD) with a high number concentration (e.g., hundreds or thousands of drops per cubic meter). Thus, we employ a scaling factor so that each drop represents drops. Accordingly, a much larger number of drops can be simulated in the domain as long as accurate statistics for debris concentration and velocity are obtained. The scaling factors used for each drop size are presented in Table 2. The scaling factors are weighted by a Marshall–Palmer distribution (Marshall and Palmer 1948) with a rain rate of 20 mm h^{−1}. Finally, raindrops are initialized randomly within the domain.

For square wood boards, 1000 trajectories per size are computed for 10 different sizes with uniformally distributed lengths (*l*) between 41.6 and 415.7 mm and thicknesses of . Wood boards are initialized randomly within the lowest two grid cells where *w * 10 m s^{−1} to provide more realistic debris lofting.^{1} After a raindrop or wood board hits the surface or exits the simulation domain, a new trajectory is reinitialized. LES mean scatterer concentrations and velocities used in forthcoming radar variable simulations are computed over a 5740-s period with the aforementioned fixed number of trajectories.

### b. Radar variable simulations

To compute radar variables using electromagnetic scattering calculations, raindrop and wood board concentrations and velocities from the LES model are axisymmetrically averaged to a radial grid spacing of 30 m, while the model vertical grid spacing is retained. Radar resolution volume dimensions are often larger than 30 m, particularly at lower frequencies (e.g., S band). However, the primary purpose of this manuscript is to examine the frequency dependence of radar observations, and thus a common grid is selected to facilitate comparisons among different frequencies. For dual-frequency radar observations without spatiotemporally matched radar resolution volumes (e.g., noncollocated radars or collocated radars with different antenna patterns), the frequency dependence of radar observations may differ because scatterer characteristics and positions within the two radars’ resolution volumes differ. Thus, dual-frequency simulations herein most closely represent radar observations from a matched beam radar system, such as the National Center for Atmospheric Research (NCAR) S-PolKa radar (Vivekanandan et al. 2004).

Two types of electromagnetic scattering calculations are presented for comparison, transmission (*T*; Waterman 1969, 1971) matrix calculations and physical optics approximations (POA). T-matrix calculations are commonly used to compute radar variables for hydrometeors and provide accurate values for spheroids. However, T-matrix calculations often fail to converge for scatterers with large eccentricities, high refractive indices, or very large scatterer sizes. Thus, T-matrix calculations are applicable only to a subset of idealized debris types (spheroids) and cannot account for irregularities in debris shapes (e.g., sharp edges). In spite of these limitations, Bodine et al. (2014b) found that T-matrix-derived for debris exhibits similar characteristics to observed dual-wavelength horizontal radar reflectivity factor () differences from the 10 May 2010 Moore–Oklahoma City, Oklahoma, tornado.

The POA model enables calculations of backscatter cross sections () for more realistic debris shapes compared to T-matrix calculations. The term is computed using the POA model for wood boards with *l* ranging from 41.6 to 415.7 mm and thicknesses of , which have the same volume as wood spheres used in the T-matrix calculations.

Using backscatter amplitudes from T-matrix calculations and mean from the POA model (discussed shortly), is computed. Backscatter amplitudes are computed using Fortran T-matrix codes described in Mishchenko et al. (1996) and Mishchenko (2000). From T-matrix calculations, is computed for rain and debris as follows:

where is the complex relative permittivity, *λ* is the radar wavelength, *s* is the backscatter amplitude, and is a function of the refractive index of water (Doviak and Zrnić 1993). The term is the scatterer size distribution, and *D* is the scatterer diameter. Debris is simulated over a large range of sizes and frequencies, so spherical shapes are used in the T-matrix calculations to enable convergence. The term for raindrops is computed based on Eqs. (5) and (6) of Ray (1972), and of dry wood boards are obtained from Daian et al. (2006) and Jebbor et al. (2011) with a value of 2–0.2*j*. Additionally, calculations were performed for wet boards (20% water content) with an of 4.65–1.46j based on Ulaby and El-Rayes (1986) and Senior et al. (1987). Unfortunately, for wood boards have only been measured at S and X bands, and thus some uncertainty exists in their applicability at other wavelengths. A comprehensive investigation of for common debris types is needed but that is beyond the scope of this study. For mean data, is computed using

Mean is computed using the POA model for each square wood board size by averaging over a discrete set of incidence angles that are uniformly distributed over a far-field sphere, with sufficient angular resolution to ensure convergence of the resulting mean value. This requires a larger number of points for larger objects at higher frequencies. Each radar cross section (RCS) is derived from a POA, where the scattered fields are calculated based on truncated equivalent currents on the near side of the board. These currents are derived from oblique incidence plane wave reflection coefficients derived either from the approximate formulation in Senior et al. (1987) for thin wood boards (less than a few tenths of a *λ* in thickness) or on a full two-way plane wave expansion inside of electrically thicker boards [see sections 5.5.2.D and 11.3.2 in Balanis (2012)]. In both techniques, conductive and dielectric losses are carefully taken into account, but the weaker edge diffraction effects are not.

The value of for wood boards computed from T-matrix calculations and the POA model are shown in Fig. 3 for a concentration of 1 m^{−3}. POA is shown for both dry (asterisks) and wet wood boards (circles). For the dry wood board calculations, mean averaged over all diameters is greater for the POA model compared to T-matrix calculations by 0.8–7.4 dB. Atlas (1953) found that mean for prolate or oblate spheroidal ice particles was greater than mean for an equivalent volume sphere, and a similar effect is observed here for wood boards. For a flat plate, specular reflection occurs when the plate is aligned normally to the incident electromagnetic wave and enhances mean particularly at higher frequencies (Knott et al. 1993). Finally, since tornadoes can entrain large concentrations of raindrops and wood boards may contain some moisture content, the impact of changing the wood boards’ water content on is considered. For the wet boards, mean increases by 6.2–7.8 dB for S–W bands, which is consistent with an approximate doubling of .

Large dual-wavelength differences, often called dual-wavelength ratios (DWRs) or dual-frequency ratios (DFRs), are evident for all frequency pairs (Fig. 3). S–C-band and S–X-band DWRs are on the order of 10 and 20 dB, respectively, which are similar to DWRs observed for hail (e.g., Atlas and Ludlam 1961; Snyder et al. 2010; Picca and Ryzhkov 2012) and debris (Bodine et al. 2014b). Between S and W bands, wood plates can produce DWRs exceeding 60 dB. DWRs are also similar for both dry and wet wood boards, since all frequencies exhibit a similar increase in after wetting.

The value for for raindrops is shown in Fig. 4 for common weather radar frequencies. DWRs in rain among centimeter wavelengths are much smaller compared to debris, and thus DWRs among S, C, or X bands could help ascertain whether rain or debris is present. Small differences at centimeter wavelengths occur as a consequence of different , and Mie scattering effects at C and X bands for larger diameters. In contrast, in rain at Ka and W bands is much lower compared to centimeter wavelengths as a consequence of Mie scattering. Between S and W bands, large drops can produce DWRs of 40–50 dB.

Electromagnetic scattering differences among radar frequencies impact Doppler velocity measurements because mean Doppler velocity is a function of individual scatterers’ velocities weighted by their reflectivities and illumination functions (Doviak and Zrnić 1993), as follows:

where the resolution volume center range is and the scatterers’ range is . The illumination function weights the scatterers’ (e.g., due to the antenna or range weighting function). Based on Eq. (6), it is evident that scatterers with large reflectivities or high number concentrations will have a greater impact on Doppler velocity measurements.

Mean reflectivity-weighted velocity is used to approximate Doppler velocity [Eq. (6)], and this approximation generates good results if scatterers are uniformally distributed throughout the resolution volume and are present in relatively high concentrations. Herein mean reflectivity-weighted radial and tangential velocities are referred to as and , respectively, and radial and tangential air velocities are referred to as *U* and *V*, respectively. The radial and tangential velocity measurement errors, associated with assuming the radar measures air velocity, are − *U* and − *V*, respectively. Future studies could employ a realistic radar simulator (e.g., Cheong et al. 2008, 2015) to examine how radar resolution volume size, attenuation, sidelobes, nonuniform debris distributions, etc., affect radar measurements in tornadoes at different frequencies.

Dual-frequency velocity differences will also be computed to assess their potential for estimating Doppler velocity errors. Radial () and tangential () dual-frequency velocity differences will be computed and will represent the velocity difference measured by an idealized dual-frequency radar (i.e., matched beam radar system). *Radial* dual-frequency velocity differences () correspond to simulated dual-frequency velocity differences where the radar beam aligns with radial scatterer motion (e.g., a range–height indicator scan through the vortex center). This is represented as follows:

where and are the two radar frequencies. Likewise, is computed using

and represents the simulated dual-frequency velocity difference where the radar beam aligns with tangential scatterer motion. In a different application, dual-frequency velocity differences have been applied to sizing of ice crystals for vertically pointing radars at Ka and W bands (Matrosov 2011), and greater dual-frequency velocity differences were associated with ice particle size distributions with smaller slope parameters (i.e., larger median volume particle sizes). Differential velocity has also been computed in tornadoes using (single frequency) polarimetric radars by taking a velocity difference between the horizontal and vertical polarizations (Snyder and Bluestein 2014). In this study, they found that higher velocity differences between polarizations occurred in resolution volumes with high spectrum width and low .

## 3. Simulations of frequency dependence of radar measurements in tornadoes

In this section, radar variable simulations of and Doppler velocity are presented at multiple frequencies. Relationships between physical properties of the scatterers (e.g., size) and dual-frequency radar variables are also examined.

### a. Idealized case study

To illustrate the impact of transmit frequency on Doppler velocity measurements in tornadoes, consider the following simplified example with a 10 000-m^{3} resolution volume. In this resolution volume, 100 dry wood plates are simulated with lengths uniformally distributed between 41.6 and 415.7 mm and a thickness of , producing a of 64.5 dB*Z* at S band using the POA model (Table 3). The maximum S-band equivalent reflectivity factor observed in TDSs is approximately 70 dB*Z* (e.g., Ryzhkov et al. 2005; Bunkers and Baxter 2011; Bodine et al. 2013), thus suggesting number concentrations are lower than 1 m^{−3} if the debris is assumed to be wood boards (Fig. 3). The resolution volume also contains 3360 0.5-mm- and 85.1 2-mm-diameter drops per cubic meter (assuming the scaling factors used in Table 2), producing an S-band of 37.6 dB*Z*. In this example, a simple geometry is assumed such that scatterer motion, wind direction, and the radar beam are aligned. To simulate differences between air and radar-measured wind speeds, the wind speed is 50 m s^{−1}, the wood plate velocity is 40 m s^{−1}, and the 0.5- and 2-mm-diameter drops’ velocities are 49 and 47 m s^{−1}, respectively. Slower velocities of the larger scatterers replicate slower tangential velocities of larger scatterers compared to the air velocity. For comparison, Dowell et al. (2005) found that plywood sheet tangential velocities are reduced by about 11 m s^{−1} for their one-dimensional simulation with a vortex radius of 100 m and a maximum tangential velocity of 50 m s^{−1} (see their Table 2, wind profile 3). In contrast, the raindrops exhibit a smaller tangential velocity reduction of less than 5 m s^{−1}.

Simulated and Doppler velocity for the example resolution volume are shown in Table 3. The term exhibits large variations across common weather radar frequencies as a consequence of Mie scattering, consistent with Fig. 3. In contrast, raindrops are predominately in the Rayleigh scattering region for centimeter wavelengths, and thus dual-wavelength differences are small. Raindrops are in the Mie scattering region at W band, resulting in lower .

The large range of dry wood plate for different frequencies produces an important effect on dominant scatterer type and thus Doppler velocity. At S and C bands, dry wood plate exceeds raindrop by at least 10 dB and, consequently, simulated S- and C-band Doppler velocity are within 0.5 m s^{−1} of the wood plates’ velocities (Table 3). However, at Ka and W bands, raindrops produce that exceeds the dry wood plates by 17–20 dB. As a result, the measured Doppler velocities are very close to the velocity of the large drops (which contribute more to than the small drops in this case). At X band, contributions from dry wood plates and raindrops are closer; however, the wood plates still have slightly higher and thus a greater effect on Doppler velocity.

The strong dependence of Doppler velocity on transmit frequency results from the frequency dependence of . For Rayleigh scatterers,

where is a function of the scatterer’s refractive index and *D* is the scatterer’s diameter (Doviak and Zrnić 1993). From Eq. (9), it is apparent that could vary by several orders of magnitude over common weather radar frequencies (e.g., from 3 mm to 10 cm) because of the dependence. For example, for a 1-mm-diameter raindrop at S band is 2.84 10^{−6} mm^{2} compared to 1.1 mm^{2} at W band. In contrast, for wood plates exhibits smaller differences among frequencies. For example, for a 290-mm-length wood board is 2500 mm^{2} at S band and 3400 mm^{2} at W band. For Rayleigh scatterers, it is apparent that the dependence in Eq. (9) is removed when is calculated [Eq. (5)]. However, for scatterers with similar at different frequencies (e.g., dry or wet wood plates), the dependence in Eq. (5) causes much higher at lower frequencies (longer wavelengths), as illustrated in Fig. 3.

Because raindrop concentrations are several orders of magnitude larger than debris concentration (e.g., 10^{3} m^{−3} compared to 0.01 m^{−3}), frequency differences in have a significant impact. At W band, of a wood plate exceeds the 1-mm raindrop by a factor *O*(10^{3}) whereas the raindrop concentration exceeds the wood plate concentration by a factor of 10^{5}. As a result, raindrops are the dominant scatterers when computing . However, at S band, of the raindrop is very small, and thus the wood plates dominate the backscattered radar signal. For a similar example with T-matrix calculations, readers are referred to section 3.1 of Bodine et al. (2014a).

### b. Multiple frequency simulations of radar variables using LES trajectories

In this section, concentrations of dry wood boards and raindrops are varied in different experiments to simulate the impact on , reflectivity-weighted velocity, and dual-frequency radar variables. Simulations are conducted for common weather radar frequencies from S to W bands, and for POA and T-matrix calculations. In the first subsection, a high debris, high raindrop (HDHR) concentration experiment is highlighted for the dry wood board POA calculation. Then, a summary of different experiments is presented for T-matrix experiments and POA calculations for dry and wet wood boards.

Debris and raindrop concentrations vary among tornadoes and even within a single tornado occurrence. Ryzhkov et al. (2005) and Bodine et al. (2013) document the temporal variability of TDSs, indicating variations in debris concentrations or sizes. Spatial distributions and intensities of precipitation vary among supercell thunderstorms, which are often classified as low precipitation (LP), classic, and high precipitation (HP) supercells (e.g., Doswell and Burgess 1993). For supercells with heavier precipitation surrounding the tornado (e.g., classic or HP), greater precipitation entrainment into the tornado may occur. To encompass the expected range of debris and raindrop concentrations in nature, five experiments are performed for a combination of a low, moderate, or high debris concentration scenario (LD, MD, and HD, respectively), and a low, moderate, or high raindrop concentration scenario (LR, MR, and HR, respectively; see Table 2).

For each debris experiment, mean scatterer concentrations are scaled to provide more realistic based on TDS observations and to maintain stable three-dimensional statistics of scatterer velocities and concentrations. For the HD experiment for dry wood boards, S-band maximum T-matrix is 62.3 dB*Z* compared to 67.0 dB*Z* for the POA model. For the LD and MD experiments, debris concentrations are scaled by a factor of and , respectively, resulting in maximum wood board of 10 and 20 dB less than the HD experiment, respectively. Finally, it is worth noting that some combinations of debris and rain concentrations produce identical Doppler velocities, DWRs, and dual-frequency velocity differences because the relative contributions of rain and wood boards to are the same. For example, HDHR and low debris, low raindrop (LDLR) experiments have the same Doppler velocities, DWRs, and dual-frequency velocity differences because the LDLR experiment contains debris and raindrop concentrations that are both scaled by 1/100 from the HDHR experiment. Thus, the experiments producing unique values are presented.

#### 1) HDHR experiment

The first experiment simulates a tornado with HDHR concentrations. Simulated for dry wood boards, raindrops, and all scatterers are shown in Fig. 5 for S, X, and W bands for the POA model. Similar to the idealized example, S-band is dominated by debris, except where debris concentrations are very small (Figs. 5a–c). At X band, rain and debris are dominant scatterers in different regions of the simulated tornado, resulting in a more complex spatial pattern of (Figs. 5d–f). Within the volume enclosed by *r * 200 m and *z * 100 m, exhibits larger contributions from debris. In contrast at S and X bands, even large concentrations of debris have little effect on W-band (Figs. 5g–i), which is evident by very small differences between simulated for rain and all scatterers.

Since exhibits greater contributions from debris at S band, S-band reflectivity-weighted velocities deviate significantly from air velocities. For the POA model, radial and tangential reflectivity-weighted scatterer velocities are shown in Figs. 6a,b, and the difference between radial and tangential reflectivity-weighted scatterer velocities and air velocities ( and , respectively) are shown in Figs. 6c,d. Comparing and to LES model velocities (Fig. 2), a significant reduction in S-band inflow layer depth and maximum inflow velocities occurs, and tangential velocities are reduced within the radius of maximum wind. The exceeds 30 m s^{−1} and occurs where dry wood boards are the dominant scatterers (Fig. 6h). A region of positive occurs in the near-surface inflow region (Fig. 6d; *r * 300 m and *z * 100 m) where debris with higher tangential velocities falls into a region of relatively low tangential velocity.

At X band, significant differences between air and reflectivity-weighted velocities also occur, but the magnitudes are smaller compared to S band (Fig. 7). Dry wood boards remain the dominant scatterers in the lowest 100–200 m, causing a reduction in maximum inflow velocities. The maximum magnitude of radial velocity errors are 31.7 m s^{−1}, and maximum tangential velocity error is 37.4 m s^{−1} (Figs. 7c,d).

In contrast to the centimeter wavelengths, W-band air and reflectivity-weighted velocities are much smaller (Fig. 8). W-band and exhibit close agreement to model wind fields, and the depth and peak inflow velocities are well resolved. The dominant scatterers are raindrops, resulting in smaller magnitudes of and , generally less than 10 m s^{−1}. Some large positive tangential velocity errors occur near the tornado’s central axis near the surface. Since raindrops acquire substantial negative radial velocities in the inflow region, they continue moving toward the central axis of the tornado. These drops acquire higher tangential velocities near the radius of maximum wind, and they continue moving radially inward where tangential velocities are lower, producing positive . Overall, W-band velocity measurements are generally shown to be more robust to debris centrifuging errors even in a case with the highest expected debris concentrations in TDSs (e.g., S-band approaching 70 dB*Z*).

Dual-frequency measurements may provide information about the spatial structure and magnitudes of velocity bias, and dominant scatterer size. Since the S- and W-band radars are sensitive to different scatterer sizes, dual-frequency velocity differences between radial and tangential reflectivity-weighted velocities (S–W DDU and S–W DDV, respectively) exhibit large differences (Figs. 6e,f). The spatial structure of S–W DDU exhibits good correlation S-band with a correlation coefficient of 0.98 (Table 6), and DDU has an RMSE of 2.4 m s^{−1} as an estimator of . DDV also exhibits good correlation with S-band with a correlation coefficient of 0.88 and an RMSE of 2.9 m s^{−1}. DWRs may also provide useful information about debris size. S–W-band DWRs and dominant scatterer radius are shown in Figs. 6g,h. S–W-band DWRs exhibit good correlation (0.87) with dominant scatterer radius (Table 6). Using a DWR between S and X bands, the correlation with debris size is 0.88. Using reflectivity measurements at X band would avoid significant attenuation issues for estimating DWRs at W band.

X–W-band DDU exhibits high correlation with X-band , with a correlation coefficient of 0.95, and X–W-band DDV exhibits a weaker correlation coefficient of 0.75 with X-band (Table 6). Dual-frequency velocity differences poorly estimate where raindrops are dominant scatterers (i.e., DWRs are smaller), and thus the impact of excluding such regions is examined. The correlation coefficient between X–W-band DDV and X-band where DWRs exceed 15 dB (i.e., debris are dominant scatterers) is 0.86.

#### 2) Summary of experiments

The mean within the lowest 300 m for the five experiments is computed for the POA model for dry and wet wood boards (Table 4), and T-matrix electromagnetic models for dry wood boards (Table 5). From left to right, the relative contributions of rain increase; and in the HDLR (LDHR) experiment, debris (rain) has its maximum relative contributions to . As the contribution by rain increases, the mean decreases. At Ka and W bands, the mean errors are less than 4 m s^{−1} for all experiments except HDLR. Thus, for the experiment with the highest debris concentration and lowest raindrop concentration experiment, large debris may become the dominant scatterer at millimeter wavelengths. However, millimeter-wavelength T-matrix mean is lower than the POA model because is lower. In contrast to millimeter wavelengths, the mean errors at S and C bands frequently exceed 10 m s^{−1}. The X band exhibits the greatest sensitivity to the relative contributions of debris and rain, with mean errors of 11.2 m s^{−1} for the HDLR experiment compared to 4.2 m s^{−1} for the LDHR experiment.

DWRs exhibit strong correlations with debris size when debris is the dominant scatterers. For the dry wood board experiments, correlation coefficients between DWR and scatterer size are shown for the two electromagnetic models and different frequency pairs in Tables 6 and 7. Correlation coefficients exceed 0.87 for the high debris experiments for the POA model. Because W-band attenuation can be significant, the S–X DWR would be the most practical and it exhibits the highest overall correlation with scatterer size. For the X–W DWR, correlation coefficients are low for the heavy rain experiments because Rayleigh scatterers become dominant and DWRs decrease.

At present, no method exists to measure differences in air and Doppler velocities in tornadoes using radar observations. Dual-frequency velocity differences [Eqs. (7) and (8)] exhibit strong correlation in some cases, which could enable estimates of differences between air and radar-measured velocity. Correlation coefficients between DDU and , and DDV and , are shown in Tables 6 and 7. S–W and X–W DDU and S- and X-band exhibit high correlation for the three middle POA model experiments (HDMR, HDHR, and MDHR). The highest correlation occurs when debris (raindrops) is the dominant scatterer at the lower (higher) frequency. Accordingly, correlation coefficients are generally lower for the HDLR (LDHR) experiments because debris (raindrops) exhibits a large contribution to velocity at both frequencies.

For wet wood boards in the POA model compared to dry wood boards, the primary difference is that increases for debris and results in higher values of mean (Table 4). The primary trends, however, remain similar to the dry case with mean still decreasing as the contribution of rain increases (i.e., from left to right) and X band exhibiting the largest range of . It is stressed that the raindrop concentration did not change from the dry to wet debris case. So, this comparison reflects a scenario in which the moisture content of the wet board changes, but the background precipitation does not change.

### c. Frequency dependence of vertical wind retrievals

Vertical velocities in tornadoes are often obtained using the GBVTD from retrieved radial winds (e.g., Lee and Wurman 2005; Kosiba and Wurman 2010; Wakimoto et al. 2012). To examine how debris centrifuging errors impact vertical velocity retrievals at different frequencies, vertical velocities are obtained by integrating the Boussinesq form of the continuity equation using simulated for different experiments. Retrieved S-, X-, and W-band vertical velocities are shown in Figs. 9a–c for the HDHR experiment with dry wood boards and the POA model. As illustrated by Nolan (2013), debris centrifuging effects increase downdraft velocities; however, these vertical velocity errors depend on the dominant scatterer type and frequency. For the HDHR experiment, S-band retrievals produce an anomalously strong downdraft, whereas W-band vertical velocity errors are smaller. At S and X bands, a two-cell vortex results (i.e., downdraft reaches the surface), whereas the W-band retrieval produces a vortex breakdown flow consistent with model vertical velocities. Moreover, updraft intensity is increased at S and X bands as a consequence of near-surface anomalous retrieved convergence generated by radial gradients of large debris and at a radius of 200 m (Figs. 5–7).

To assess the effectiveness of the debris centrifuging correction proposed by Wakimoto et al. (2012), the radial velocity bias of 0.5-, 1.5-, and 4-mm-diameter drops is subtracted from simulated , and then vertical velocities are recomputed. The RMSEs for S, X, and W bands for uncorrected and corrected vertical velocity retrievals are shown in Table 8, and corrected vertical velocities are shown for the HDHR experiment for 4-mm-diameter drops (Fig. 9d,e). For S- and X-band corrected vertical velocities, a vortex breakdown flow is obtained after correction, and downdraft velocities are reduced. For all S-band experiments, the correction reduces RMSEs for all drop sizes, although RMSEs still exceed 10 m s^{−1}. At X band, the correction generally reduces RMSEs. For the LDHR experiment, RMSEs increase for the 4-mm-diameter correction and thus estimating drop size may be important. For the HDHR and HDLR experiments, RMSEs exceed 10 m s^{−1} even with a correction for a 4-mm-diameter drop. At W band, corrections for the LDHR and HDHR experiments adversely affect retrievals unless the smallest drop size is used (Table 8). W-band RMSEs exceed 10 m s^{−1} for the HDLR experiment even though the correction is applied.

## 4. Conclusions

Radar variable simulations are conducted with an LES model, scatterer trajectories, and electromagnetic scattering calculations to examine the frequency dependence of and Doppler velocity in tornadoes. Simulations are conducted using a range of raindrop and wood board sizes and concentrations to represent typical small and large scatterers in tornadoes. The simulations reveal the significant frequency dependence of and Doppler velocity. A strong frequency dependence of for small scatterers results in significant changes in dominant scatterer types from S to W band. At S (W) band, dominant scatterers are wood boards (rain), except when their concentrations are very low. As a result, air and simulated Doppler velocities exhibit decreasing mean differences as the radar frequency increases. At intermediate frequencies (e.g., X band), the dominant scatterer type and air and simulated Doppler velocity differences exhibit large variability depending on the relative concentrations of raindrops and wood boards. Finally, higher is observed at each frequency for wet wood boards compared to dry wood boards, resulting in greater mean velocity differences.

Dual-frequency variables for estimating air and Doppler velocity differences and scatterer size are explored. For the middle three POA experiments, high correlations are found between the radial dual-frequency velocity differences (DDU) and the differences between radial air and simulated Doppler velocities. For the HDLR and LDHR experiments, correlations are reduced because the dominant scatterer type is the same at both frequencies (i.e., debris for HDLR and rain for LDHR). When high concentrations of debris are present, DWRs exhibit high correlation with debris size, particularly for S–X-band DWRs. Finally, DWRs were examined for both dry and wet wood boards, and DWR changes ( 2 dB) were smaller than the increase in .

These simulations reveal that diagnosing the dominant scatterer type in tornadoes is a complex process and that the frequency dependence of electromagnetic scattering must be considered. The highest sensitivity to relative contributions of rain and debris occurs at X band, which is a commonly used frequency for mobile radar observations of tornadoes. Thus, dual-frequency or polarimetric radar observations are needed at X band to justify assumptions about dominant scatterer type (e.g., using DWRs or ). Moreover, significant residual errors often remain at S and X bands even after debris centrifuging corrections for raindrops are applied. Thus, for cases where raindrops or Rayleigh scatterers are not the dominant scatterers, additional methods are needed to correct air and Doppler velocity differences. Such corrections are important, as tornadoes with higher enhanced Fujita scale ratings typically have high (e.g., Bunkers and Baxter 2011; Schultz et al. 2012b; Bodine et al. 2013) and likely greater concentrations of lofted debris.

As demonstrated in section 3, dual-frequency velocity differences and DWRs have the potential to estimate scatterer-induced velocity bias and scatterer sizes. Dual-frequency spectral analysis could help distinguish Rayleigh and non-Rayleigh scatterers, and perhaps aid in correcting scatterer-induced velocity bias by identifying velocities of smaller scatterers. To develop corrections for single-frequency polarimetric radars, comparisons between dual-frequency velocity data and polarimetric radar variables may enable robust corrections for velocity errors at single frequencies. Dual-polarization variables such as and differential velocity (Snyder and Bluestein 2014) may exhibit relationships to debris characteristics, in addition to polarimetric spectral densities. Photogrammetry and video observations could also aid in determining how well polarimetric and dual-frequency observations characterize debris size and differences between air and Doppler velocities.

It is emphasized that the results presented in this study are based on simulations, and thus observational validation of the frequency dependence of radar variables is needed. An optimal platform for examining the frequency dependence would be a mobile dual-frequency radar with a matched antenna pattern. Until such systems are developed, collocated mobile radars operating at different frequencies could be used to investigate the frequency dependence if temporal and spatial sampling differences are minimized (e.g., coordinated scans, matched pulse lengths). Dual-frequency capability for fixed radars could also provide opportunities to examine the frequency dependence of radar observations because multiple high-impact tornado events have occurred in proximity to fixed radars (e.g., Ryzhkov et al. 2005; Palmer et al. 2011; Schultz et al. 2012a,b; Atkins et al. 2014; Kurdzo et al. 2015).

Because only one debris type is examined herein, future studies are planned to obtain RCSs of a large set of tornado debris types using laboratory measurements and advanced electromagnetic simulations. These studies will encompass natural debris (e.g., leaves, tree branches, soils) and anthropogenic debris (e.g., wood boards, insulation), and examine more realistic scatterer shapes (2 in. × 4 in. lumber, or rectangular wood sheathing). These efforts include RCSs computed using Ansys high-frequency simulation software (HFSS) and RCS measurements in anechoic chambers at the Advanced Radar Research Center at the University of Oklahoma. Using the RCS data, polarimetric TDSs are simulated using a polarimetric radar time series simulator (Cheong et al. 2015) that ingests wind data from high-resolution models (e.g., LES model herein). Using the radar simulator, polarimetric and dual-frequency radar signatures of different scatterer types can be explored, and new methods to characterize debris size and correct debris centrifuging errors can be developed and tested. Such efforts are critical to advancing scientific understanding of tornado dynamics and improving radar estimates of wind speeds used to assess enhanced Fujita scale ratings.

## Acknowledgments

This study is based upon work supported by the National Science Foundation under Grants. AGS-1303685 and OISE-1209444. The first author is supported by the National Center for Atmospheric Research (NCAR) Advanced Study Program. This study emerged from a substantial collaborative effort between the University of Oklahoma and the Disaster Prevention Research Institute (DPRI) at Kyoto University, and the first author is grateful to Prof. Maruyama and Kyoto University for hosting his stay. The authors would also like to acknowledge the Collaborative Research program of the Joint Usage/Collaborative Research Center for Multidisciplinary Disaster Prevention Study: DPRI Kyoto University. The authors thank Scott Ellis, Wen-Chau Lee, and Tammy Weckwerth for reviewing an early version of the manuscript. Four anonymous reviewers also provided excellent feedback, which improved the clarity of the discussions.

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## Footnotes

^{a}

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

^{1}

To our knowledge, debris lofting characteristics in tornadoes have not been studied or quantified (e.g., initial vertical velocities); thus, debris is lofted only where vertical velocities are sufficiently intense to loft wood boards.