a. Pulse compression

Pulse compression involves transmitting a wideband signal coded in either frequency or phase and decoding the return signal through filtering, which results in increased average transmit power and enhanced range resolution compared to an identical system that does not include pulse compression (Skolnik 1990). Phase codes partition the transmitted pulse into segments of equal time duration, or subpulses, and then switch the phase of the signal at specified segments. In particular, binary phase codes switch the phase between two values, usually 0 and π, 0 and 1, or + and −. The amount of compression possible is equivalent to the time–bandwidth product (BT) of the code, which is the product of the signal bandwidth and signal total duration. Bandwidth of a phase-coded signal is calculated with

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where τ is taken to be the code subpulse length. The return signal power increase is proportional to the code length while the range resolution is inversely related to bandwidth as given by

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This shows that decreasing subpulse duration results in a corresponding enhancement in range resolution regardless of the total pulse length. See Skolnik (1990) and Nathanson (1991) for more details.

One problem with pulse compression is that it creates range sidelobes, artifacts produced by the compression process whereby returns from other ranges contaminate the signal at the desired range. This corruption alters the in-phase and quadrature (I and Q) time series data so that estimates based upon these data could be in error. Fortunately, sidelobe suppression filters can be included in the pulse compression scheme that reduce the amount of corruption in the data, resulting in more accurate estimations. This addition comes at the cost of a slightly decreased signal-to-noise ratio (SNR), increased minimum range, and increased complexity. The ISL metric indicates the amount of sidelobe corruption by comparing the total power contained within the sidelobes to the main lobe:

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In this equation, x₀ refers to the main lobe magnitude while x_i refers to the ith-range sidelobe, with improvement being indicated by a reduction in its value. This metric is of great importance in meteorological applications since the backscattered signal is spread over several range gates. An attractive property of the Barker codes is that they have uniformly distributed sidelobes about the main lobe when decoded with a matched filter and have a mainlobe peak that is higher than the sidelobes by a factor equal to the code length (Skolnik 1990; Nathanson 1991). For example, a 5-bit Barker code in conjunction with a matched filter will produce a main lobe 5 times higher than the sidelobes. Errors also can be produced by targets with radial velocities that can alter the phase of the backscattered signal, but they are negligible at this transmitting frequency and pulse duration since the phase change is much less than 45° (Nathanson 1991). While other pulse compression schemes involve changing the transmitted pulse frequency, we have chosen to concentrate on binary phase coding.

b. Simulation procedure

The Time Series Weather Radar Simulator (TSWRS; Cheong et al. 2008) was used to create in-phase and quadrature (I and Q) time series data for a case involving a tornadic supercell as modeled by the Advanced Regional Prediction System (ARPS; Xue et al. 2003). The TSWRS is a three-dimensional radar simulator consisting of an ensemble of thousands of scatterers placed within the field of view (i.e., all the radar resolution volumes along a radial) of the virtual radar. The meteorological fields used as input to the simulator correspond to output data from the ARPS numerical simulation model developed at the Center for the Analysis and Prediction of Storms (CAPS) at the University of Oklahoma. The spatial and temporal resolutions of the ARPS output used in this study were 25 m and 1 s, respectively. To begin the simulation process, scatterer characteristics are initialized from a known ARPS dataset where a scatterer can be thought of as a point target, which has been given properties that correspond to the meteorological parameters of a radar resolution volume. At the next time step, the scatterer positions are updated according to the wind field and their corresponding properties are also updated at their new locations. The return signal amplitude and phase from each scatterer are then processed via Monte Carlo integration to calculate time series of the desired meteorological signals. The I and Q data were then processed to calculate the velocity difference (ΔV), spectrum width (σ_υ), spectral flatness (σ_s), phase of radially integrated bispectrum (P; Wang et al. 2008; Yu et al. 2007), and eigenratio (χ; Yeary et al. 2007) for use in a fuzzy logic tornado detection algorithm (FLTDA).

The simulation begins with the input of ARPS data into the TSWRS and the initialization of the scatterer properties. The simulations used 30 000 scatterers for the standard-resolution case while those incorporating pulse compression increased the number of scatterers proportionally to the increase in range resolution so that the same average scatterer density of 20 per resolution volume would be created. This scatterer density was chosen to balance computational speed with enough targets to create a realistic simulation of radar return signals. Next, the pulse is propagated throughout the radar field of view on a gate-by-gate basis, switching the phase code over the appropriate range gates. The radar then receives the returns from the scatterers and composes the signal. Mathematically, this step, taken from Mudukutore and Chandrasekar (1998), can be described by

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After the signal is composed, the data are decoded through the filtering process to produce estimates of the reflectivity, radial velocity, and the parameters needed for input into the tornado detection algorithm using the covariance method (Doviak and Zrnić 1993). A graphical depiction of this process is shown in Fig. 1. An average signal-to-noise ratio of 70 dB was set for cases using the TSWRS to provide for a verification of the concept before evaluating it under more realistic conditions.

While the matched filter gives an impulse response that is a time-reversed replica of the transmitted signal (Cook and Bernfeld 1993), the mismatched filter operates on the output of the matched filter for this application. The mismatched filter was designed using the method from Key et al. (1959) for the 5- and 13-bit codes. The mismatched filter coefficients calculated from this process were then inserted into the simulation after matched filtering to provide for improved sidelobe suppression. The ISLs achieved are −8.0 and −18.8 dB for the 5-bit Barker code and −11.5 and −25.0 dB for the 13-bit Barker code using matched and mismatched filtering, respectively. The additional sidelobe suppression offered by the mismatched filters means that targets outside the range of interest would not have as large of a corrupting effect on the calculated parameters, such as reflectivity, for that particular range bin.

c. Experimental weather radar procedure

To validate the weather radar simulator results as well as examine any additional effects that may be encountered in an operational environment, data obtained from an experimental weather radar have been artificially phase coded to simulate pulse compression. The radar used in this experiment, is the prototype Weather Surveillance Radar-1988 Doppler (WSR-88D) at Norman, Oklahoma (KOUN), which has been modified to include dual-polarization (Doviak et al. 2002). Pulse compression was not used in the field data collection but rather was simulated by artificially switching the phase of the raw time series data over the appropriate number of range gates in accordance with the code. For example, a 13-bit Barker code pulse would extend over 13 range gates with the phase of the data being altered to replicate the code as it propagated throughout the radar field of view. The individual range gates were then summed to produce a composite signal that was then processed as in the radar simulator.

d. Fuzzy logic tornado detection

The tornado detection algorithm employed here is a modification of the NFTDA system proposed by Wang et al. (2008), derived from Yeary et al. (2007). The algorithm uses fuzzy logic that incorporates the five parameters previously mentioned, (gate-to-gate velocity change ΔV; spectrum width σ_υ; spectral flatness σ_s; eigenratio χ; and phase of radially integrated bispectrum, PRIB or P), simultaneously in order to make a decision as to the existence of a tornado. To perform this function, the algorithm uses membership functions to denote the degree of membership in the “yes” or “no” set given a particular parameter value. The neural network has been removed from the process in order to stabilize comparisons between data obtained from nonpulse compression and pulse compression simulations. The neural network optimizes the membership functions to maximize the performance of the algorithm, which is counter to the intent here of simply comparing noncompressed and compressed results given a set of membership functions.

FLTDA utilizes the S- and Z-shaped membership functions in the fuzzification step, which can be described completely by a lower and an upper breaking point. Use of these one-sided membership functions implies that the input parameters can be described linguistically as being either “high” or “low.” The S-shaped membership function is given by

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where x₁ and x₂ represent the lower and upper breaking points, respectively; i is the input parameter; and j represents the tornadic or nontornadic case (Wang et al. 2008).

The rule inference step evaluates the strength of each rule, via the membership function results, in order to provide input into the defuzzification step. As in the NFTDA, the Mamdani system (Ross 2005) of rule inference was chosen, where the outputs of the fuzzification step are multiplied together to determine the rule strength. Defuzzification completes the process by simply using the maximum value obtained from the rule inference process to make the final decision of whether or not a tornado is present.

e. Performance evaluation

The metrics used to describe the performance of the FLTDA are the same as those used in forecast verification (Murphy 1996). The process begins with the definition of a set of scenarios that describes the possible outcomes of the detection process. This is analogous to the construction of a 2 × 2 contingency table denoting hits, misses, false alarms, and nulls. A hit is defined as detecting a tornado within the area defined as containing a tornado. A miss is defined as not detecting a tornado when there was one present within the time step. A false alarm occurs when a tornado is detected outside of the area containing the tornado or if a detection is made when no tornado is present. The null state simply means that no detection was made when the tornado was not present. For each time step, or image, the location of the detections was compared to the area defined as containing the tornado when it was present. This area was defined as a circle having a radius of 200 m, which matched the estimated size of the vortex using ARPS data. The location of the tornado was determined by the location of the minimum pressure in the domain corresponding to the image. Formation of the tornado was determined to occur when a fully developed vortex formed in the density information at the lowest height of the ARPS simulation.

The performance of the algorithm was calculated over the entire simulation on a per image basis by calculating skill scores described below. The intent of the method was to analyze both the spatial and temporal levels of accuracy of the algorithm so that multiple hits and false detections for a single image were considered as one event. If a hit and a false detection were made for an image, then both events would be scored. This type of scoring led to sample sizes that could exceed the number of images for a simulation. The overall performance of the system could not be totally described by one metric, so four metrics were used consistent with that found in Mitchell et al. (1998). Shown below are the probability of detection (POD), false alarm ratio (FAR), critical success index (CSI), and the Heidke skill score (HSS) for each simulation (Wilks 2005):

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In all of these equations, a is the number of hits, b is the number of false alarms, c is the number of misses, and d is the number of nulls.

The POD describes the likelihood of detecting an event given that the event occurs, with a score of 1 corresponding to perfect detection. The FAR is the ratio of false detections to total detections and is typically used in conjunction with POD. A score of 0 means that no false detections were made. CSI indicates the proportion of events correctly detected given that null cases are removed, where a score of 1 is perfect. Finally, the HSS measures the accuracy of the algorithm relative to that which would be detected by random chance. In other words, it calculates a score after removing hits that would have occurred solely by random chance.

The process of tornado detection and its performance hinges upon there being actual differences between tornadic and nontornadic signatures as well as the data collection method being able to accurately estimate those signatures. Brooks (2004) described a model for tornado-warning performance that is similar to the process defined here. The analogous point made was that greater separation between distributions generally leads to better performance. For the FLTDA, the distributions are those created for the five input parameters through sampling of the simulation. These distributions served as the foundation for defining the membership functions and adjusting them accordingly to improve detection performance. In terms of distributions, tornadic signatures typically have large gate-to-gate velocity differences, large spectrum widths, small spectral flatness, large eigenratios, and large PRIB.

The tornadic and nontornadic distributions of the parameters were created by setting a search area around the tornado and then dividing the signatures inside and outside of the search area between tornadic and nontornadic sets, respectively. This process was completed for all images, which were then summed to produce histograms of the signatures. The histograms were then used for comparison against the membership functions to examine the applicability of those breaking points to that simulation. If significant deviations between the membership functions and the histograms were found and believed to be a significant source of error, then the breaking points for those membership functions were adjusted. This process was repeated until the histograms and membership functions appeared to match in a qualitative sense.

a. Weather radar simulator data

The test case for all simulations consisted of 99 images representing a small time segment of a tornadic supercell thunderstorm as modeled by the ARPS. Data were gathered using the TSWRS operating at 3.2 GHz. The pulse width was fixed at 1.57 μs with a pulse repetition interval of 1 ms, giving an unambiguous range and velocity of 150 km and 23.5 m s⁻¹ respectively.

b. KOUN data

A tornado was tracked by KOUN on 10 May 2003 that provided I and Q data for volume scans at 0337, 0343, and 0349 UTC. The range resolution was 250 m with an unambiguous velocity of 35 m s⁻¹ with data being gathered every 1° with a 1° beam resolution. During these times, the tornado was located approximately 40 km away from the radar and reached a maximum intensity of EF4 on the enhanced Fujita scale. Phase coding was accomplished by switching the phase of I and Q data over the appropriate number of range gates, which were then added together to simulate a long, coded pulse. Processing of the data was completed using the same method as that used for the TSWRS scenarios.

a. Weather radar simulator results

The TSWRS integrated a high-resolution dataset that contained a tornado within the simulated supercell thunderstorm. The radar range resolution was set to 235 m and served as the standard resolution definition. This resolution was chosen in order to maintain consistency with other informal studies that utilized the weather radar simulator. Due to computational limitations in terms of time, memory, and the extracted ARPS dataset, it was not possible to simulate Barker codes that would recover range resolution to 235 m. This forced the analysis to take the approach of comparing results between uncoded and coded data with smaller range gate spacing (i.e., finer resolution).

The pulse compression process correlates the return signals in range since the pulse extends over a number of range gates. The amount of interference impressed upon the desired signal from all the other range gates is expressed through the ISL. Therefore, a reduction in ISL should produce more accurate estimates of meteorological parameters such as reflectivity. Indeed, this is the case as is shown in Table 1, which presents the mean and standard deviation of Z, υ_r, and all five of the fuzzy logic parameters. However, it was found that many of the parameters displayed varying degrees of bias. One notable case was that for spectral flatness, which was observed to have a strong, negative bias. The root cause is that the pulse compression process acts as a weighted averaging process that serves to reduce the variance of the I and Q data. This implies that the results differ at the spectral level, as shown in Fig. 2 where the mismatched filter better approximates the uncoded spectrum. This effect on the input parameters is summarized in Table 2 where the biases are separated into tornadic and nontornadic cases. The statistics were averaged for both matched and mismatched filters with a majority of the bias deriving from the matched filter cases. Of note, it was found that the errors for tornadic cases showed a greater error than nontornadic cases. The PRIB was the only parameter that did not demonstrate any bias in excess of 10%, while the eigenratio exhibited a strong positive bias for both tornadic and nontornadic signatures.

A contour plot of error in estimating Z using both matched and mismatched filters is shown in Fig. 3. The contour plots are normalized using the logarithm of the data to bring out the fine details. The x axis is the uncoded reflectivity gradient calculated along a radial, which has been smoothed by a rectangular window, in other words, a 13-point moving average. The first point to be made is that while decreasing ISL only reduces the mean error slightly, it does significantly reduce the variation of the error considerably over a wide band of gradients. The second point is that the error is asymmetric about both horizontal and vertical axes. While a complete explanation for this behavior is not proposed here, the asymmetry in error (i.e., about the 0-dBZ axis) may be due to the sidelobe structure of the code. The 13-bit as well as the 5-bit Barker codes are the only two code of this type that have all positive sidelobes, implying that the code would be biased toward overestimating the reflectivity since any interference would be additive. The asymmetry in the vertical direction (i.e., about the 0 dBZ m⁻¹ axis) could simply be due to the simulation not containing enough samples that had a strong positive reflectivity gradient to show up in the data. To put the scale into perspective, a value of 0.1 corresponds to roughly 5 out of the 600 000 total points calculated.

Separating the signatures between tornadic and nontornadic cases resulted in Fig. 4, where some clear distinctions can be made between parameters. Most of the nontornadic signatures have a ΔV of less than 10 m s⁻¹ while the tornadic signatures range fairly uniformly in value from 20 to over 40 m s⁻¹. For spectral flatness, the nontornadic signatures tend to increase in frequency starting at around 10 dB, peaking at about 20 dB, and then trailing off by about 25 dB. A less clear example is that for PRIB, where the tornadic and nontornadic signatures exhibit a Gaussian shape with the only difference being that the tornadic signatures have a slightly higher frequency at higher PRIBs.

Detection statistics of the FLTDA process that utilize ΔV, spectrum width, and spectral flatness as the input parameters are shown in Table 3. High POD, CSI, and HSS are coupled with a low FAR as is desired. The membership functions were manually optimized to maximize the performance of the algorithm for the matched filter case. Although the matched filter data have been shown to be less accurate than those derived from the mismatched filter, they can still produce high levels of performance. The reason being that the fuzzy logic approach leverages differences between distributions separately from the accuracy of the data resulting in essentially a decoupled problem. Greater separation between the tornadic and nontornadic signatures generally improves performance. It was also found that not all information inserted into the process improved performance. Some of the distributions, such as PRIB, have significant overlap between the two types of signatures that can result in erroneous classification. Examining Table 3 again shows that the mismatched filter performs worse than the matched filter. The cause is that the mismatched filter has several “misses” in its simulation that can be attributed to small ΔV’s that serve to nullify the other “hit” results.

b. KOUN results

Processing data obtained from KOUN in a similar manner as for the simulator, the algorithm detected the tornado at the appropriate locations in the three volume scans as demonstrated in Figs. 5 and 6. However, the algorithm did not successfully detect the tornado at 0337 UTC using 5- and 13-bit Barker codes with their respective mismatched filters. Suppression of the eigenratio at locations coincident with other tornadic indicators appears to be the cause of these misses. The algorithm did detect the tornado for all other cases regardless of processing method. While locating the tornado was successful, the number of detections resulting from pulse compression increased for most cases compared to that obtained without pulse compression. Reducing the ISL appears to alleviate this issue although this may be an artifact of the small number of samples used. The other item of note is that the PRIB is suppressed from the values obtained without pulse compression. It is unknown at this time as to the root cause of this observation, but the overall structure of the PRIB field is retained for all cases, which still makes it useful as a discriminator in the FLTDA process.