INTRODUCTION

In 1988, the first network of weather radars with a coherent operation mode was deployed. In addition to the data on the power of the received signal common for this technique, they allowed estimating the mean frequency and width of its spectrum. It was believed that this would significantly extend the list of the identified weather hazards. However, no expected breakthrough occurred. The radars started producing the maps of the vertical profiles of wind speed and wind direction, as well as the maps of average radial velocity and the width of the spectrum of radial velocities of particles. However, the information content of these data is still unsatisfactory. This is primarily associated with an insufficient spatial resolution of radars, namely, with a large transversal size of their resolvable volumes. The second factor is the use of data only on the mean spectrum width or, in other words, the average radial velocity of particles in the data processing procedures. The second spectral parameter is the spectrum width, which remains almost unused or is applied only as an auxiliary parameter. At the same time, as the present research shows, it contains enough information for solving the formulated problems, even with the most common values of a width of the beam of the antennas used.

RESEARCH PROBLEM STATEMENT

The above is well illustrated by the situation with tornado detection. It is possible only if the transversal size of the radar resolution cell is at least twice smaller than the phenomenon diameter. Then, during the horizontal scanning, the antenna beam consecutively observes the areas of the vortex with air motions toward the radar and away from it. This leads to the occurrence of the azimuthally neighboring areas with the estimated average radial velocity of particles significantly differing in magnitude on the map of radial velocities.

If the aforementioned condition is not met, the tornado completely fits into the antenna pattern, and no jump occurs in the estimated average radial velocity of particles. The only advantage of such resolution cell over the others is the higher value of the spectrum width of the reflected signal. However, such effect can be caused by many weather phenomena, including not severe ones. In addition, it can originate from a simple decrease in the power of the signal from the observed target area.

Now it should be considered that the mean value of the diameter of the lower part of the tornado funnel is 300–400 m. For the most common antenna beam in modern weather radars [5], the transversal dimension of the resolution cell reaches these values at the distances of 18–24 km. This is the maximum distance at which the confident identification of tornado is possible. With the maximum radar range of 250 km, the area of the confident detection of tornadoes is less than 1% of the entire controlled area.

The situation with the identification of another phenomenon is even worse: this is the formation of the short-term downdraft (microburst). There is no intensive rotation of air in it, which leads to a significantly smaller (as compared to the tornado) spectrum width of the reflected signals from it. Consequently, none of the aforementioned methods is applicable. Therefore, there were attempts to detect this phenomenon based on the indirect features: the diverging flows formed when the microburst meets the ground or, on the contrary, based on the oncoming movement of air masses at the top of the cloud during its formation. However, the registration of the induced jump in the mean frequency of the signal received from the adjacent (along the range) resolution cells is not a simple task, as it can be masked by the increased turbulence in the visible zones, by the presence of airflows induced by other reasons in them, etc.

The issue of the wind shear detection is at the same level. This is caused by very strict requirements for the vertical resolution: the vertical size of the resolution cell should not exceed 30 m, because the measurement unit used in this case is meters per second per 30 m of height. Modern weather radars either do not have the cells with such resolution or have one–two cells. The ratios of the used values of wavelength and antenna size cause the location of the boundaries of the Fresnel far zone, where the antenna pattern is formed, at a distance more than 1.5 km. For the aforementioned beam width of 1°, this is the distance at which the linear size of the antenna pattern becomes equal to the required 30 m. Hence, the radar cannot estimate the wind profile during one azimuthal revolution of the antenna by setting it at a certain elevation and synchronously getting information from all viewed heights. For this purpose, it has to place consecutively the mentioned one–two resolution cells at the heights of interest and to make a scan at each of them in a horizontal plane. Such scanning algorithm requires so much time that other radar functions are forgotten. Therefore, radio profilers are installed at aerodromes in addition to weather radars: these are more effective instruments for solving this problem.

Perhaps, the situation with the identification of squalls is the best among all wind-related severe weather events. In this case, the particle velocity vector is located horizontally, which provides a great Doppler frequency shift. If a squall line is perpendicular to the antenna beam at any distance, the condition is met of exceeding the scale of the event over the size of the resolution cell along the antenna beam (for most of modern weather radars, it is equal to 150 m). However, with changing mutual position of the squall line and antenna beam from the perpendicular to parallel position, the situation essentially changes. Due to the turning of the resolution cell, its size in the direction toward the squall propagation increases and exceeds the scale of the observed phenomenon at some mentioned angle, depending on the distance to the radar. This means that reflections from the surrounding space (background) which mask useful information, are added to the signal reflected from the squall zone. In addition, an increase in this angle means a decrease in the radial component of wind speed in the squall zone with a decrease to zero when reaching a value of 90°, which makes identification still more difficult. The effect of both analyzed factors can be roughly assessed as follows: the squall line is confidently observed only when it is perpendicular to the direction toward the radar \(\pm 45^\circ\). Since the probability density function of the angle characterizing the spatial orientation of the squall is homogeneous, the probability of the correct detection of the phenomenon cannot be higher than 0.5, which is clearly insufficient.

The above means that there are problems with the radar identification of all wind-related weather hazards. The main reason for difficulties is their small scales as compared to the size of the volume resolvable by the radar. There is the situation similar to the smoothing of the function by a moving window. In this case, the role of the window is played by the radar resolution cell. As a result, the clearly pronounced inhomogeneity of the wind field whose size is smaller than the linear size of the antenna beam, may be considered as an insignificant variation or may remain unnoticed at all. This leads to the loss of important information. However, it is lost only due to the fact that the detection algorithms are based on the estimated average frequency of the spectrum of the reflected signal, which is the averaged parameter that neutralizes changes in the spectrum itself when observing such inhomogeneity.

Actually, the information does not disappear but "flows" into the signal spectrum, whose form can be a basis for constructing new algorithms for solving the analyzed problem. However, the evaluation of the spectrum form under conditions of the limited sample size typical of the radar observation of meteorological targets is difficult and poorly formalized. Therefore, it would be better to select any other parameter sensitive to changes in the spectrum form for practical application.

The first natural pretender for such role is the spectrum width defined as the root-mean-square deviation of the frequencies from the mean (or, in other words, as a root of the spectrum variance). Actually, when analyzing the cases listed above in which there are identification problems, all of them can be reduced to one situation in which reflections from two air masses are mixed into a single signal. Each of them is characterized by its own parameters: the power of the reflection, the mean frequency, and the spectrum width. The form of the spectrum of reflections from each mass is assumed to be Gaussian (as it is common for weather radars) [3].

In these conditions, the resulting spectrum is the sum of two Gaussians, its width is determined from the following formula [1]:

$$ \sigma_\omega=\Big(\frac{P_1}{P_1+P_2}\,\sigma^2_{\omega 1} +\frac{P_2}{P_1+P_2}\,\sigma^2_{\omega 2} +\frac{P_1P_2}{(P_1+P_2)^2}\,(\omega_{01}-\omega_{02})^2\Big)^{0.5}$$
(1)

where \(P_i\) is the power of the \(i\)th component of the total signal; \(\omega _{0i}\) is its mean frequency; \(\sigma _{\omega i}\) is its spectrum width.

It is obvious that if there is a rather large (more than \(\sigma _{\omega 1}+\sigma _{\omega 2}\)) difference in the mean frequencies of the components, then under conditions of the rough equality of powers, the width of the total spectrum becomes abnormally large. It is abnormal in a sense that real weather events cannot independently (i.e., with the full filling of the resolving volume with a zone with this phenomenon) form reflections with such parameter.

This situation takes place when observing the wind shear. A method was developed basing on the estimated width of the spectrum of reflections that allows both identifying this phenomenon and determining its parameters using the radar (the magnitude, direction, and altitude of the shear) [2]. The method is based on the fact that the shear has quite a big size in the horizontal plane to be observed during the complete azimuthal revolution of the antenna. At the distances where the powers of reflections from two air masses fitting into the resolving volume are roughly equal, the zones of abnormal wide spectra of a specific form take place. Their structure and estimates of the analyzed parameter are used for obtaining the required information.

Unfortunately, the method is not applicable to the other aforementioned phenomena, because they, firstly, are extensive only along one coordinate and, secondly, do not form signals with a sufficiently large difference in the mean frequencies, so that the width of the spectrum of the reflected signal becomes abnormally large, as follows from (1). The normal value of this parameter cannot be a reliable predictor of the severe weather event, as mentioned above.

As a result, it becomes relevant to search for such parameter that can unambiguously demonstrate that there are two air masses with different values of the particle radial velocity and/or the radial velocity spectrum width in the resolution cell that created the analyzed reflection. It should be noted in the analyzed case that the signal spectrum becomes more complex similarly to radio engineering, in particular, to radar sounding, when inducing a complex pulse. The base equal to the product of the duration of the signal by its spectrum width is considered as an indicator of complexity that determines how many times the pulse can be compressed. It is evident that the analog of this parameter is what we need, we just have to decide what will be the analog of the spectrum width and duration. The first of them is quite clear: this is the width of the energy spectrum of the received random process. The situation with the second one is more complicated, since the duration of the pulse train reflected from the target is not associated with the properties of the target. Here, the fact comes to the rescue that the parameters used to calculate the base are connected with each other by the Fourier transform. Proceeding from this, the analog of the pulse duration is the width of the autocorrelation function of the signal, which is hereinafter called the correlation interval. Among all possible variants of the definition of this parameter, the most convenient for the present study is the root of the variance of the correlation function \(R(\tau )\) calculated in accordance with

$$ \sigma_\tau=\sqrt{\frac{\int^{\infty }_0 \tau^2R(\tau)\,d\tau} {\int^{\infty}_0 R(\tau)\,d\tau}}.$$
(2)

Actually, when viewing the "calm" area of space in the absence of any inhomogeneities, the spectrum of the reflected signal is Gaussian, and the correlation interval is connected to the spectrum width by the relationship

$$ \sigma_\tau\sigma_\omega=1.$$
(3)

When the second component appears in the spectrum, this dependence is disturbed and may be an indicator of the presence of a wind field inhomogeneity in the viewed volume of space. Thus, the problem arises of studying how informative in this respect the proposed characteristic (the base of the analyzed random process) is:

$$ b=\sigma_\tau\sigma_\omega.$$
(4)

METHODS AND RESULTS

The details of the present study have already been published in [4]. Here, we briefly say that the expressions for the energy spectrum width and the correlation interval for a random process with the bimodal Gaussian spectrum consisting of two Gaussians, which passed through the amplitude or phase detector, were derived analytically. The dependences on the ratio of the spectra width, the difference in the mean frequencies, and the ratio of the powers of the components were constructed for different variants of the base calculation. As a result, the advantage of using the data obtained after the phase detection was clearly manifested. It was shown that the value of the base is poorly dependent on the difference in the mean frequencies of the summed signals if the value of this difference is within the resolvable interval \(\Delta \omega =\sigma _{\omega 1} + \sigma _{\omega 2}\). Out of this interval, it demonstrates a fast growth with the increasing mentioned parameter. Figure 1 presents the most interesting dependence of the base value on the ratio of the spectra width of the components with identical mean frequencies. This corresponds to the case of observing the vertical flow at a large distance or the squall line if it is parallel to the antenna beam. The estimates of the mean frequency of reflected signals are useless here, and the spectrum width cannot take abnormal high values. In other words, it is impossible to identify the phenomenon using the traditional method or the new previously described one. The base turns out to be a very sensitive parameter, which is effective both in this case and in the case that was previously considered hopeless.

Fig. 1
figure 1

The dependence of the signal base \(b\) on the ratio of the spectra width of the components \(\sigma _{1}/\sigma _{2}\) for their identical mean frequencies and different ratios of powers \(P_{1}/P_{2}\)

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The arising prospects are well illustrated by the tornado episode. For this purpose, it is sufficient to evaluate a new zone of its confident detection. It was calculated for the most inappropriate conditions:

—the reflectivity of the tornado zone and the background is identical (actually, it is usually higher in the first case);

—the radius of the phenomenon is minimal and equal to 300 m.

Figure 1 shows that the base response remains approximately the same within the interval of the signal components power ratio of 1–0.25. Consequently, the maximum horizontal size of the resolution cell is equal to five tornado radii, or 1500 m. With the antenna beam width of 1°, it is reached at a distance of 90 km. If comparing this result with the one obtained at the beginning of the paper for the traditional method, the growth of the zone of the confident tornado detection at least by 25 times becomes evident.

The next step was to verify theoretical calculations by their comparison with the data obtained by modeling the processes used for the data processing. For this purpose, a random process with the specified parameters of the energy spectrum was formed using the following algorithm:

—the amplitude spectrum of the signal was formed as a sum of two Gaussians with required parameters (power, mean frequency, energy spectrum width);

—each \(i\)th spectral component was multiplied by \(\exp\,(j\varphi _i)\), where \(\varphi _i\) is the random variable with the homogenous distribution;

—the inverse Fourier transform was implemented.

The value of the base was calculated for the obtained signal. The result of averaging over ten different realizations of the series of \(\varphi _i\) was taken as a final value. As a result, a good consistency of theoretical calculations with simulations illustrated in Fig. 2 was achieved.

Fig. 2
figure 2

(1) The simulated and (2) calculated dependences of the base on the relative difference in the mean frequencies of the components

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Then the issues of the practical application of the proposed method were considered. Too much resource is spent for direct calculations in each resolution cell of the spectrum width of the signal and its correlation interval. In modern weather radars, even one of the listed parameters is estimated by a simplified method. Therefore, it is necessary to find economic procedures that allow obtaining the information interesting for us. At the moment, two types of such procedures are known: the procedures based on a priori knowledge of the signal spectrum form and those based on the theory of outliers of a random process.

In our case, the first ones include the paired-pulse method that consists in evaluating the correlation function of the process at two points: at the zero value of the variable (in other words, the signal power) and in case of the shift equal to the period of sounding pulse repetition. This is sufficient to find the desired variable assuming the Gaussian form of the spectrum and, hence, of the correlation function.

An example of the procedure of the second type is the intersection method, that counts the number of intersections of the random process with a certain level (as a rule, zero one). In addition, there are other procedures of the same type (for example, the difference method), but now our aim is to solve a fundamental problem: which of the parameters interesting for the authors is measured in case of the complex bimodal spectrum of the process, rather than the intercomparison of methods. Earlier, when the spectrum was assumed to be Gaussian, such problem was not even considered. The desired parameters were connected by the inverse proportional dependence, and finding one of them automatically meant finding the second one. If the spectrum form becomes more complicated, the idyll is broken, and the issue of what we really measure becomes very relevant.

It is impossible to solve the problem analytically, therefore, the modeling was used. The random process was formed by the above method, to which the investigated procedures were applied. It turned out that both of them estimate the spectrum width of the random process, which is illustrated in Fig. 3.

Fig. 3
figure 3

The comparison of the dependences of the signal spectrum width on the ratio of the spectra width of the components for \(\Delta \omega/\sigma _{\omega 1} = 0.3\), which were obtained by (1) theoretical calculations and (2) the intersection method modeling, as well as (3) using the simulated parameter inverse to the correlation interval

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The result is quite surprising, at least for the paired-pulse method, since it actually measures the parameters of the correlation function.

CONCLUSIONS

The methods for the detection of wind-related weather hazards are insufficiently effective in modern weather radars, which forces to search for new ways of solving this problem. One of such ways is the use of the estimated energy spectrum width of reflected signals. However, based only on these estimates, it is possible to construct an algorithm for detecting only the wind shear characterized by the large horizontal size. Other similar events (tornado, squall, microburst) are proposed to be identified based on the new parameter, namely, the base of the random process at the radar receiver output. The studies have showed that this parameter can be a predictor for solving the stated problems. The fact of its sensitive response to the appearance of two spectra with an identical mean frequency but with different widths in one signal is interesting: this is the case for which the former methods were useless. This affects the solution to the analyzed problem in the following way:

(1) the zone of the confident tornado detection enlarges at least by 25 times;

(2) due to the joining of the cases with the mutual position of the antenna axis and the squall line at the angles of \(0^\circ\pm 45^\circ\), the probability of detection of this severe weather event increases by two times;

(3) the microburst is detected directly, not based on indirect features, which increases the probability of its correct detection.

The investigation of the well-known procedures for estimating the spectrum width of reflected signals demonstrated that they are also applicable to the analyzed case of the bimodal spectrum of the signal, which will considerably reduce the number of mathematical operations during the practical implementation of the method.

FUNDING

The research was supported by the Russian Science Foundation (grant 21-19-00378), https://rscf.ru/project/21-19-00378.