This guide represents a portion of my advanced knowledge of radar meteorology, especially as it pertains to analyzing data containing severe storms (e.g., tornadic supercells). I was inspired to create this after the tragic yet also meteorologically significant event that occurred in Canadian County, Oklahoma on 31 May 2013 – that being the record widest tornado to ever be documented, the El Reno EF-3 tornado. Note that the EF-3 rating was based on damage surveyed by the National Weather Service. At least two separate research-grade mobile Doppler radars each confirmed EF-5 winds were present within this tornado. Much of this guide explains how the radar reflectivity, Doppler radial velocity, and spectrum width products that could be used to analyze this tornado are created. The bulk of this material was obtained from a weather radar signal processing course I took while a graduate student at the University of Oklahoma in 2011. It was taught by Dr. Robert Palmer, currently the Associate Vice President for Research at the University of Oklahoma. This work is dedicated to him since he was one of, if not the best teacher I ever had in a meteorology course, and the course was one of, if not the most enjoyable meteorology course I have ever taken. Part of the material covered in this guide also came from Doppler Radar and Weather Observations, 2nd Ed., by Doviak and Zrnić, 1984. I produced all illustrations in this guide using various software display packages (e.g., Matplotlib, Matlab, and GRLevel2AE). This guide is not meant to be a polished piece of work. It is also not meant to supplement, compliment, or in any way relate to a separate radar guide I have published online, “How to read and interpret Doppler weather radar.” While it has been proofread (by me), some typographical, grammatical, and literary errors may still remain. Factual errors should be absent, however. Nonetheless, if you discover any error while reading this guide, please let me know by sending me an email at jeffduda319@gmail.com. Original completion date: 11 September 2013 First publication online: 2 December 2014 Most recent revisions and online update: 21 August 2015

Introduction – what is RADAR? The term RADAR stands for Radio Detection And Ranging. Let’s break that acronym down further:

Radio: this refers to the preferred wavelength of radiation used in the technology. Radio waves are waves of length typically larger than 1 millimeter. Common wavelengths used in weather radar range from a few millimeters up to 10 or so centimeters. Specific wavelengths are generally chosen to optimize some feature of the radar system. For example, longer waves are less susceptible to attenuation and give more reliable measurements of large rain drops and small hail stones, whereas shorter waves allow for better spatial resolution (i.e., smaller beamwidth).

Detection: when radar was in its infancy, the developers were more interested in simply detecting the presence of an object within the radar beam. Because of that, they used “continuous wave” (CW) radar, which is a radar beam comprised of a continuous stream of energy rather than of bursts, as in “pulse radar.” When continuous wave radar is used, only detection is reasonably capable. This part of the technology only answers the question of whether or not an object is present.

Ranging: this refers to the determination of distance from the radar. This is only possible for practical use when pulsed radar is used. For CW radar, the beam can only discriminate among distances equal to the wavelength or less. Thus, when dealing with radio waves on the order of millimeters to centimeters, CW waves alias (i.e., repeat themselves) every few millimeters or centimeters. For example, for a wavelength of 1 cm, a returned signal from CW radar will be unable to tell you if an object is between 3.0 and 3.01 meters or between 30000.0 and 30000.01 meters away. The phase of the returned wave would be identical in each case. For this reason, we use pulsed radar, which sends out short bursts of energy instead. Pulsed radars send out pulses then wait for the returned signal. Since all electromagnetic waves travel at the speed of light the precise distance to the target can be determined given the length of time before the signal returns. This gives pulsed radars a huge advantage over CW radars in the sense that the maximum detectable distance from a pulsed radar may be several (as many as 10 or possibly even more) orders of magnitude greater than that from CW radar. Furthermore, the lack of practical range discrimination for CW radar makes it generally not useful for meteorological purposes.

Fundamental mechanical and electronic components of a radar system So we know that we will need to send pulses for our radar to work. How do we do that? The schematic below discusses the fundamental components of a pulsed radar system.

It starts with a signal generator, sometimes referred to as a stable load oscillator. This generates the oscillating signal that will be transmitted.

A pulse modulator acts on the continuously generated (i.e., CW) signal to shut it off periodically. This results in the pulsed signal. The length of the pulse can be controlled and affects the sensitivity of the radar as well as the spatial resolution of the radar products. We use the term pulse length (represented by the Greek letter tau, or τ) to describe how long each pulse lasts. We use the term pulse repetition frequency (PRF) to describe how often pulses are generated. The PRF is very important in something called the “Doppler dilemma,” which we will discuss later.

An amplifier adds power to the signal so that sufficient energy may be returned to the radar to detect either small objects or very distant objects.

A transmit-receive switch allows the signal to travel along the waveguide to the antenna while stopping signal from feeding back into the other parts of the system and damaging them. This switch puts the system in one of two states: transmitting – where the energy is allowed to move along the waveguide to the antenna, and receiving – where the received wave is allowed to move along the same waveguide (the waveguide is like a one-way street in that you can’t have signals going both ways along the waveguide at the same time; it will confuse the system as to which signal is coming and which is going) coming back from the antenna.

The antenna guides the emitted signal towards a specific target in space and receives energy coming back from that same target region. There is actually an entire branch of science called antenna theory which deals with all the aspects of designing an antenna properly. We won’t spend much time on that, as it is not terribly significant for the purposes of this guide.

For Doppler radars, there are additional components that allow for the determination of velocity. The additional components are mixers that add signals of known frequencies to the returned signal, then mix and filter the resulting signal. This produces two channels of data, called the I (in-phase), and Q (quadrature) components of what is essentially a complex-valued echo voltage. “Echo voltage” is the name given to the returned signal. It is a sinusoidal signal much like the transmitted signal, except there will be phase, amplitude, and frequency differences that will be used to determine the returned power (reflectivity) and frequency (velocity) of the meteorological targets the pulse sampled. The I/Q components are called “base band” and constitute level I data. The computation of the typical products from radar (reflectivity, velocity, and spectrum width) are all based on the base band data.

One pulse isn’t good enough... To get Doppler velocity measurements, it is not good enough to send out only one pulse along one azimuth. This is because the returned pulse will be so short in duration that it will only comprise a tiny fraction of the period of the signal. Thus it would be very difficult to get an accurate estimate of the frequency of the returned wave with just one pulse. For this and other reasons, a series of pulses are actually sent out at a single azimuth. The total time the radar spends sending out pulses along a single azimuth (a combination of pulse lengths and PRFs) is called the dwell time. The dwell time is a factor in the accuracy and certainty of the measurements as well as the sensitivity of the radar to small or distant objects and the update frequency of the radar products. So what we’re left with is actually a time series of I and Q components that oscillates in time. A sample graph of Level I data from a single range gate (the data representing signal from a fixed distance from the radar) at a single azimuth is shown below.

Figure 1. Time series of I/Q components from a weather signal. The unit of time is seconds. The dwell time of this signal is thus about 0.0275 s, or

about 27.5 ms.

Range resolution The range resolution of the radar is determined by the pulse length. The longer the pulse is, the more difficult it is to distinguish between returns from pulses spaced closely to each other and the poorer the range resolution. For example, consider a pulse length of 1 μs (a common value for meteorological uses). The spatial width of that pulse is then given using the well-known relationship

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑠𝑝𝑒𝑒𝑑 ∗ 𝑡𝑖𝑚𝑒 where we know speed to be that of light, or about 3x10

8 m/s (which will be denoted as c). The math is then

𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑝𝑢𝑙𝑠𝑒 = 3 ×108𝑚

𝑠∗ 10−6𝑠 = 3 × 102𝑚 = 300 𝑚

The pulse will begin reflecting as soon as it encounters an object. This means that the leading edge of the pulse will begin travelling back towards the radar at the speed of light, so that the relative velocity in question is actually twice the speed of light. Thus we divide this result by 2 to get the range resolution. This means that objects as far apart as 150 m will be sampled by the same radar pulse, which means the returned signal will contain the combined effects of both objects; they will be indistinguishable. The following set of graphics illustrates this issue.

Illustration of range resolution – Part 1

The pulse now interacts with the far scatterer and has completely interacted with the nearer scatterer. More energy

begins to reflect (faint blue shade).

Start with an emitted pulse with length in time of τ moving left to right. Thus the spatial width of the pulse is cτ. Two

scatterers (red and blue dots) separated by more than cτ/2 (the range resolution) are located downrange of the pulse.

As the pulse interacts with the nearer (red) scatterer, energy begins reflecting back towards the radar while the rest of the

pulse continues on.

The pulse has now passed both scatterers. It continues on, although it has lost some energy from what it imprinted on the

scatterers (a feature called attenuation), but unless the objects are very large or highly absorbent, the beam is negligibly

attenuated. Since there was sufficient separation between the scatterers, the information returning to the radar will be

separate for each scatterer, and the radar will be able to resolve their spatial dislocation.

The pulse continues on. It is almost all the way past the nearer scatterer and is almost to the far scatterer.

Illustration of range resolution – part 2

Now the situation is slightly different. The two different scatterers are now separated by a distance less than the range

resolution.

scatterers spaced

less than cτ/2 apart

pulse width (τ)

The pulse begins interacting with the nearer scatterer (red), but the close spacing of the scatterers and the width of the pulse

are about to result in the pulse interacting with both scatterers at the same time.

Since the pulse is interacting with both scatterers, some of the energy in the returning pulse contains energy reflected by

both scatterers.

The pulse continues on. Meanwhile, the reflected pulse contains information from both scatterers, and the radar system will

be unable to distinguish the contribution to the total returned energy from either scatterer. They will effectively be the same

scatterer as far as the receiver is concerned.

The above illustrations were simplified to isolate only one set of scatterers along a radial, or line of constant azimuth, which is the angle between some reference line (in radar meteorology, 0° represents north) and the direction the radar is pointing. All points along a radial share the same azimuth, but span the range. In reality, the signal will return the entire time the radar is in its listening mode. We use the term duty cycle to describe the proportion of the time the radar is listening. The duty cycle is defined as

𝑑𝑢𝑡𝑦 𝑐𝑦𝑐𝑙𝑒 = 𝜏 ∗ 𝑃𝑅𝐹 which is really just the ratio of the amount of time the radar is “on”/sending pulses (i.e., the pulse length) to the total t ime of interest (the pulse repetition time, which is just the reciprocal of the pulse repetition frequency). A Typical duty cycle is <1%. If the duty cycle is too high, the radar spends too much time transmitting pulses, which causes the system to heat up too much, which poses a danger to the entire system. Not only that, but a longer pulse length also reduces the range resolution and increases the minimum distance at which scatterers can be detected. The minimum distance is generally more limited by the time it takes the transmit/receive switch to change from one state to the other, but if the pulse length exceeds this time, it will become the factor that determines this minimum distance. Typical minimum distances are around 1.0 nautical miles. This highlights one of the many “gives-and-takes” of radar meteorology, in which there is no perfect way to configure the radar system for every application. Changing one parameter to improve one aspect of the system will degrade some other aspect. There are many examples of this in radar meteorology, and they will be discussed later. In actual meteorological applications, multiple pulses are sent out along an azimuth. There are two main reasons for this. 1) Each pulse acts as a single measurement of the quantity we are trying to measure (returned power). Because of noise in the system and other random error associated with thermal differences in the radar system, the atmosphere, and the scatters themselves, any single measurement will contain error. As is usually done in other areas of meteorology, taking multiple samples enables a more accurate and certain estimate of the property we seek to measure. 2) As introduced above, each pulse is too short to capture a significant period of oscillation of the returning signal, so velocity estimates generally cannot be obtained from just one pulse. We need at least two pulses to get any idea of the velocity. What results is an extended time series of signal returning to the radar. Fig. 1 showed only a subset of the full returned signal that the processor in the radar system sees. What this means is we can sample the returning signal as often as we want. While it typically doesn’t make sense to sample more often than allowed by the range resolution, nevertheless this is sometimes still done, but with the understanding that there is correlation between samples. This is called oversampling. Otherwise, samples can be obtained far enough apart such that each sample can be considered independent of nearby samples. Once a series of samples are obtained, the Doppler spectrum, or power-weighted frequency spectrum, is derived from the autocorrelation of this series. The autocorrelation function points out any periodicity in the signal. Obviously, since the radar emits an oscillating signal, we would expect an oscillating signal to return. That doesn’t have to be the case, however, especially when sampling ground clutter or clear air. In those extreme cases, the autocorrelation function may be just a peak at lag 0 for noise (since noise is effectively random and completely uncorrelated), or may remain high for a very long time for ground clutter. We want our samples to decorrelate so that there is some independence among samples. Referring back to point 1 in this section, the more independent samples we have, the better our estimate. As mentioned above, the three products generated (reflectivity, velocity, and spectrum width) are actually derived from, or are exactly given by, the zeroth, first, and second moments of the Doppler spectrum. The Doppler spectrum is a frequency-domain plot showing the amplitude of the various frequency components of the returned signal. Fourier transforms are used to convert the autocorrelation function of our signal into a Doppler spectrum. The Doppler spectrum is very similar to a probability distribution function, except it is not normalized so that the total area under the spectrum is not 1.0. However, it still acts the same way. For Doppler spectra, the moments are given by

𝑃𝑡ℎ 𝑚𝑜𝑚𝑒𝑛𝑡 = 𝑀(𝑃) = ∫ 𝑓𝑃𝑆(𝑓)𝑑𝑓

+∞

−∞

where f, representing frequency, is the independent variable, and S(f) is the Doppler spectrum. It is now pertinent to introduce many other concepts relevant to the remainder of the discussion regarding Doppler spectra. We will return to this after discussing aliasing, the Doppler dilemma, and the weather radar equation.

Sidebar: Aliasing Aliasing is something that happens when an analog (i.e., continuous) periodic signal is sampled too infrequently for the result to actually mimic the analog sample itself. This is more easily explained using illustrations.

Figure 2. Example of aliasing of a 1 Hz analog signal (thick black line). A 10 Hz sampling is indicated in green. A 2 Hz sampling at the

Nyquist interval is indicated in cyan. An aliased signal (sampled at 1.375 Hz) indicated in red. The difference between the top and bottom panels is the phase offset of the analog signal, indicated at the top of each panel.

In the above example, an analog signal with a frequency of 1 Hz (one cycle per second) is being sampled at various rates. There is actually a theorem floating around in this. The Nyquist theorem states that to properly sample a signal (i.e., to avoid aliasing), you must sample a signal at least twice per cycle. This cutoff value is called the Nyquist frequency, or, when converted to a Doppler velocity, the Nyquist, folding, or aliasing velocity. In the example signal, the Nyquist frequency for the analog signal is 2 Hz since the analog signal is 1 Hz. This means we must sample this signal at least two times per second in order to capture the frequency of the signal correctly. If we sample at exactly this frequency, we get the correct frequency, but the signal appears attenuated (the cyan curve in the upper panel). However, what is tricky about sampling exactly at the Nyquist frequency is the phase offset. If the phase offset is just right, we may sample a flat line (bottom panel). This is something to be careful of, as this can cause the measured frequency to differ even though we sampled frequently enough. NOTE: the lines in the plot above are artificial! We only get data at the measured points (squares). Our brain naturally fills in the space between with data, however. It does not have to be a line, but that is the simplest order of interpolation between two points, so I added it anyway. This does help illustrate what happens when you do not sample frequently enough. The red trace shows what our data would look like if we sampled too infrequently (in this example, at 1.375 Hz). Since we are below the Nyquist frequency, the returned signal does not have the correct frequency content and appears irregular. In radar meteorology, the situation is a little different than the example above. Instead of varying the sampling frequency (the PRF), we fix the sampling frequency, and the analog signal varies in frequency. This is depicted in the following diagram.

Figure 3. Illustration of adequate signal sampling frequencies the way it is done in radar meteorology. The analog signal, which varies

from panel to panel, is in black, and the sampled signal (at 10 Hz in each panel) is in red.

In this example, our sampling frequency is fixed at 10 Hz. The same theory still applies however, just in the other direction. Now the highest frequency we can sample without aliasing is half that, or 5 Hz, since we need at least two samples per cycle to avoid aliasing. As long as the signal we are trying to measure has a frequency at or below 5 Hz, we adequately sample it, as is seen in the upper two panels. However, if the signal has a frequency higher than 5 Hz, our sampled signal becomes aliased. What saves the science of radar meteorology is the relationship between the frequencies of the analog and aliased signals. Given a fixed sample rate X, we know the Nyquist frequency for the sampling is X/2. If the analog frequency Y exceeds the Nyquist frequency, the aliased signal will have a frequency given by

𝑎𝑙𝑖𝑎𝑠𝑒𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = −𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 + (𝑎𝑛𝑙𝑎𝑜𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 − 𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦)

= −𝑋

2+ (𝑌 −

𝑋

2) = 𝑌 − 𝑋

Where the folding comes in is that basically the frequency excess of the analog signal over the Nyquist frequency is added onto the negative of the Nyquist frequency. Negative frequencies are not unphysical. The negative sign just implies a change in direction of the signal. This is the difference between an inbound and an outbound Doppler radial velocity. Let’s see this in action with our example above. The 7.5 Hz analog signal exceeds the Nyquist frequency by 2.5 Hz. So we add that onto the negative of the Nyquist frequency to get -2.5 Hz. Visual inspection confirms that the aliased signal has a frequency of 2.5 Hz. The 9 Hz analog signal exceeds the Nyquist frequency by 4 Hz. Therefore the frequency of the aliased signal is -1 Hz, which is again confirmed by visual inspection.

In real radar meteorology, we do not know what the analog signal is. We measure the analog signal, but we only obtain a discretized version of it. It may be aliased. It is up to us to determine if the returned signal is folded or not. However, we can solve for the analog frequency:

𝑎𝑛𝑎𝑙𝑜𝑔 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑎𝑙𝑖𝑎𝑠𝑒𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 + 2 ∗ 𝑁𝑦𝑞𝑢𝑖𝑠𝑡 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 So if we suspect our -1 Hz signal above is aliased, we can obtain the analog signal using the formula above. We compute the analog frequency as 9 Hz. You might be wondering how we would know from visual inspection that the aliased frequency is -1 Hz as opposed to +1 Hz. Typically we don’t know. However, corrections to aliased signals are not made by visual inspection, so this is not really an issue. The end product, velocity, is obtained via computation of the first moment of the Doppler spectrum. However, aliasing will appear even there. We will get to that.

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 =𝑐

𝑃𝑅𝐹

𝑟𝑚𝑎𝑥 =𝑐

2𝑃𝑅𝐹

Sidebar: The Doppler Dilemma One of the gives-and-takes in radar meteorology is called the Doppler Dilemma. It describes how the PRF impacts two important parameters associated with reflectivity and velocity. The two parameters are called the maximum unambiguous range (rmax) and the maximum unambiguous velocity (vmax). The maximum unambiguous range The maximum unambiguous range is the farthest away from the radar a scatterer can be detected as a first-trip echo. This distance is determined by the PRF. Since a new pulse is transmitted every 1/PRF seconds, then the distance one pulse can travel before the next pulse is emitted is given by

Since the pulse must be able to travel out to a scatterer and back to the radar in that time, the maximum unambiguous range is thus

The possible ranges a radar is able to detect is [0,rmax]. The radar receiver is stupid and does not know which pulse a given return came from. It assumes the returned energy is coming from the pulse that was just emitted. Let’s see what happens to scatterers farther away than rmax. Let a scatterer be at a range R such that rmax < R < 2rmax (you will see why I used such a strict range soon). The energy from that scatterer will return AFTER the next pulse is emitted, and its range will be indicated by X < rmax < R by the radar. We call this a second-trip echo since it arrived after the second pulse was emitted. Since we know it is a second-trip echo, we also know its true range since we know rmax. Simply add rmax to the indicated range to get the true range. We use the term range-folded to describe this echo, since its range will be depicted as less than rmax by the radar, even though its true range is greater than rmax. See below for an illustration. But what happens when a scatterer is really far out, past even 2rmax? Is it even detectable? If there is enough power in the pulse and the radar beam is low enough to still intercept the scatterer at such a long range (due to trigonometry and Earth’s curvature, the beam continually rises with range), it is certainly possible for third-trip and fourth-trip echoes to be detected! The apparent range will then be the true range minus however many multiples of rmax the scatterer is past rmax. (continued)

distance between rmax

and scatterer

same distance between

radar and apparent (i.e.,

range-folded) range

The scatterer will appear at a range less than rmax corresponding to the exceedance of the true range over

rmax. Since it was between rmax and 2rmax, this is a second-trip echo.

A scatterer (orange oval) is located farther out than rmax. A pulse is emitted. Due to the regularity of the

pulses (i.e., the PRF), a second pulse is emitted before the first pulse is reflected back.

Radar

location first-trip range

rmax

second pulse

(outgoing)

first pulse

(returning)

𝑓𝐷 =2

𝜆𝑣𝐷

𝑓𝑠 > 2𝑓𝑁

𝑃𝑅𝐹 > 2 (2𝑣𝐷

𝜆)

𝑣𝐷 = 𝑣𝑚𝑎𝑥 =𝜆

4𝑃𝑅𝐹

𝑟𝑚𝑎𝑥 ∗ 𝑣𝑚𝑎𝑥 =𝑐𝜆

8

𝑟𝑚𝑎𝑥 =𝑐𝜆

8𝑣𝑚𝑎𝑥

𝑣𝑚𝑎𝑥 =𝑐𝜆

8𝑟𝑚𝑎𝑥

Sidebar: The Doppler Dilemma (continued) The maximum unambiguous velocity (the folding or aliasing velocity) The maximum unambiguous velocity is the same thing as the aliasing velocity described above the the section regarding aliasing. The Doppler frequency is the part of the echo voltage that contains the meteorologically relevant signal. It has the form

where λ is the radar wavelength and vD is the Doppler radial velocity. According to the Nyquist sampling theorem, we must sample at least twice per cycle. In other words, if fs is the sampling interval (basically, the PRF), then

where fN is the Nyquist frequency. Substituting the value of fD for fN and PRF for fs, we get

Notice how the PRF is in the numerator of vmax and in the denominator of rmax. In a perfect world, we could have a very long rmax (so there are no ambiguous echoes) and a very high vmax (so we could capture velocities from even the strongest of tornadoes). To get a long rmax we want to use a small PRF. However, decreasing the PRF causes vmax to decrease also. This is the Doppler Dilemma, since we can’t improve one aspect without bound without degrading the other aspect. The limit is summarized by putting the two equations together (eliminating PRF between them):

which gives us the relations

or

which shows the inverse proportionality between them. We want both to be as large as possible, but as one increases, the other decreases.

The (weather) radar equation The radar equation is very powerful because it enables us to extract the meteorologically relevant information from the signal that returns to the radar. This is especially the case with reflectivity, which is a meteorologically relevant variable. However, it must be derived from the returned power to the radar. Note that the returned power itself is generally derived from the Doppler spectrum, so the radar equation does not technically take the direct input to the radar and convert it directly to reflectivity. The only thing the radar knows is the frequency of the signal it emitted, the amount of power it emitted and how long it has been since it last emitted a pulse. With a few assumptions, we can find out a lot more. Let’s assume the total power transmitted by the radar in the pulse is Pt. We will initially assume the power radiates isotropically since it is simpler and easier to understand. Therefore, at any given range from the radar, the power density (expressed in power per unit area) is given by

𝑆(𝑟) =𝑃𝑡

4𝜋𝑟2

where r is the range, and the denominator is the surface area of a sphere of radius r. In the real world, the energy is not emitted isotropically. It is directed by the antenna. We can define how well the antenna focuses the beam using two parameters from antenna theory: 1) gain; 2) directivity (or antenna pattern). The gain is simply the ratio of the power density at a point along the most intense portion of the beam to that at the same point from an isotropic radiator. We will use the symbol g to represent gain. For WSR-88Ds, the gain is on the order of 10000 to 100000. The antennas do a great job at focusing the power emitted into a small beam. The directivity or antenna pattern relates how well focused the beam is around the antenna. Antennas are designed to be parabolic (Fig. 4). This is because parabolas have a useful property in that any ray parallel to the axis of the parabola (the line going through the focus and the vertex) will reflect off the parabola and pass through the focus. Conversely, any ray emitting from the focus will reflect off the parabola and move straight out parallel to the axis. This implies parabolas will make very focused beams if energy is emitted from the focus (which is where the transmitter is placed).

Figure 4. Annotated diagram of a parabola and its elements and properties. Note how all rays coming into the parabola parallel to the axis reflect off the parabola and into the focus. You can think of the spread of the red rays as how the beam might look if equal

power was emitted from the transmitter in all directions. What is actually done is something called “tapering” where more energy is transmitted at small angles so that the distribution of energy is more focused along the axis of the parabola.

Due to irregularities in parabolic dish antennas (coming from imperfections in construction or the presence of screws, bolts, or rivets in the dish due to connecting pieces together, paint, stress from connecting the waveguide and transmitter, and just wear and tear of the material due to changes in heat, moisture, and light), the antenna pattern is not the perfect ray engineers seek. Instead, we get a pattern that features a main lobe that indeed centers on boresight (the axis of the parabola), where the majority of the power is focused, and side lobes where some energy is transmitted well off boresight. There are even back lobes that emit energy behind the antenna! We use the notation f

2(θ,φ) to denote the antenna pattern to denote the dependence of the power on horizontal

angles θ and vertical angles φ from boresight (at boresight, θ = φ = 0). The antenna pattern is normalized by the maximum power value so that 0 ≤ f

2 ≤ 1. At that point, we can use base-10 logarithms to depict the pattern graphically. The beamwidth of the radar is

defined as twice the off-boresight angle at which the power drops to one-half its max value, or 3 dB below the peak (since log10(0.5) ≈ -3). Now that we have some technical stuff out of the way, we can use these properties to refine the equation. It now is

𝑆(𝑟) =𝑃𝑡𝑔𝑓2(𝜗, 𝜑)

4𝜋𝑟2

This is the power density that will reach a scatterer at range r. The amount of energy incident on the scatterer depends on its backscatter cross-sectional area, denoted by Aσ. The backscatter cross-sectional area is the area of the scatterer as the radar sees it. Because electromagnetic waves of different frequencies interact with objects differently, the backscatter cross sectional area is not necessarily the same as the physical cross sectional area as it appears to your eyes (visible light is just a different form of electromagnetic radiation). So the power incident on the scatterer at range r is

𝑆𝜎(𝑟) =𝑃𝑡𝑔𝑓2(𝜗, 𝜑)

4𝜋𝑟2∗ 𝐴𝜎

Although the scatterer need not reflect energy isotropically (and many scatterers do not), we will assume the energy is reradiated isotropically, so that the power density returning to the radar is

𝑃𝑟 =𝑆𝜎

4𝜋𝑟2∗ 𝐴𝑒

where Ae is the equivalent area of the antenna, which is not necessarily its physical cross sectional area. The effective area of the antenna is given by

𝐴𝑒 =𝑔𝜆2

4𝜋𝑓2(𝜃, 𝜑)

Putting this all together, we get

𝑃𝑟 =𝑃𝑡𝑔2𝜆2𝑓4𝜎

(4𝜋)3𝑟4

where the (θ,φ) notation has been dropped for simplicity, and Aσ has been simplified to just σ. We need to include one additional term to represent losses due to attenuation of the beam as it travels through space and interacts with scatterers. Unfortunately, it is impossible to have one single formulation of this term that always applies, so we leave it alone. With that term added, we have the radar equation for a single particle:

𝑃𝑟 =𝑃𝑡𝑔2𝜆2𝑓4𝜎

(4𝜋)3𝑟4𝑙2

where the l

2 term has been added to account for two-way attenuation of the beam.

To get the radar equation as it is used in meteorology, we need to account for the fact that there will be multiple targets illuminated by a single radar pulse. To do that, we start by introducing the resolution volume, or pulse volume. The resolution volume is the three-dimensional volume of space illuminated by a radar pulse at a given time. Given a roughly Gaussian (bell-shaped) antenna pattern with no rotational asymmetry about the center of the beam, the beam can be thought of as a truncated circular cone, or just a cylinder given the small angle of the point of the cone. We assume there is a range-dependent weighting in the resolution volume so that scatterers (at this point, since we are dealing with meteorology, we will switch over from “scatterers” to “hydrometeors” – water particles formed by meteorological processes) near the center of the volume preferentially receive energy. The range weighting is not critically important, but it helps to visualize the process by which returns from a large number of hydrometeors in a

small space are converted into a single number. These assumptions allow us to convert the antenna pattern, f4, into something

specific. We also need to account for the backscattering cross section, σ. It has been determined that for liquid water spheres, σ is given by

𝜎 =𝜋5

𝜆4|𝐾|2𝐷6

where K is the complex index of refraction of the substance (|K|

2 ≈ 0.92 for liquid ice and |K|

2 ≈ 0.18 for ice spheres) and D is the

diameter of the drop. The fact the backscatter cross section is proportional to the sixth power of the diameter is significant: it indicates that large rain drops (and hailstones) will return a significantly greater amount of power than will small rain drops or cloud drops. It is important to note that this relationship is only valid when the ratio of D/λ – the relative size of the hydrometeors to the wavelength – is small (on the order of 10

-1 and smaller). This range of ratios of D/λ is said to be in the Rayleigh regime of the Mie

scattering theory, where σ increases monotonically as D increases. As D/λ becomes larger (closer to 1 and greater), the monotonic relationship between σ and D breaks down. In fact, in that general range, σ may either increase or decrease as D increases. This makes determination of meteorologically relevant parameters difficult, and also demeans the parameter values. This generally explains, for example, why you never see reflectivity values greater than the low 80s dBZ. Very large water-coated hailstones are much too large to be in the Rayleigh scattering regime and do not necessarily return more power than smaller water-coated hailstones. Thus the returned energy effectively has an upper bound for meteorological applications. We also introduce the drop-size distribution (DSD), which is the statistical distribution of sizes of rain drops per unit volume. We use the symbol N(D) to represent the total number of rain drops having diameters ranging from D to D + dD (some arbitrarily small size increment). Much research has been done on DSDs for rain drops. The most common type of model distribution is the Marshall-Palmer DSD, which is just an inverse exponential function:

𝑁(𝐷) = 𝑁0𝑒−𝜆𝐷 Here N0 and λ are the intercept and slope parameters, respectively, indicating where the distribution would intersect the D = 0 axis, and how quickly the number drops off with increasing diameter. More recent research has indicated that for heavy rainfall, a more realistic DSD is given by the generalized gamma distribution:

𝑁(𝐷) = 𝑁0𝐷𝛼𝑒−𝜆𝐷 where the factor D

α has been added, and where α is a shape parameter which impacts how much more large drops there are. For

both distributions, the vast majority of mass is located on the small end of the diameter range since most rain drops are very small. In rain processes, the larger drops cause spectral broadening through drop brekaup, collision, and coalescence (autocollection and accretion). We take advantage of these known DSDs by introducing the radar reflectivity:

𝜂 = ∫ 𝜎𝑁(𝐷)𝑑𝐷

∞

0

This is otherwise known as the expected backscattering cross section per unit volume. This function finds the mean backscatter cross section according to the DSD. Note that this is not the same thing as the meteorological reflectivity! This particular parameter is more radar relevant, although it is quite similar to the meteorological parameter soon to be introduced. This is a key link because it gives us a way to represent all of the hydrometeors in the resolution volume using a single value. We also need to integrate this over our resolution volume including our weighting function and the antenna pattern. When we do that, our radar equation becomes the following:

𝑃𝑟 =𝑃𝑡𝑔2𝜆2

(4𝜋)3𝑟2𝑙2∗

𝜋𝜃2

8 ln 2∗

𝜂𝑐𝜏

2=

𝑃𝑡𝑔2𝜆2𝜂𝑐𝜏𝜋𝜃2

(4𝜋)3𝑟2𝑙216 ln 2=

𝑃𝑡𝑔2𝜆2𝑐𝜏𝜗2𝜂

1024𝜋2 ln 2 𝑟2𝑙2

To finalize this, we need to introduce the meteorological reflectivity factor. We do this by expanding the radar reflectivity term η, as so:

𝜂 = ∫𝜋5

𝜆4|𝐾|2𝐷6𝑁(𝐷)𝑑𝐷

∞

0

We can pull out the factors that do not depend on the diameter D to get

𝜂 =𝜋5

𝜆4|𝐾|2 ∫ 𝐷6𝑁(𝐷)𝑑𝐷

∞

0

=𝜋5

𝜆4|𝐾|2𝑧

where we have used z to represent the meteorologically relevant reflectivity factor that you see plotted on radar images. In the real world, we do not have a continuous DSD, so we discretize the integral and normalize by the volume. If you do not understand why we would normalize by volume, consider the difference in size of the resolution volume for a pulse 10 km from the radar versus one 100 km from the radar. The approximate ratio of resolution volumes is the square of the ratio of the ranges, so the volume ratio is (100/10)

2, or 100. It is much easier to pack more hydrometeors in a volume that is 100 times larger to get greater reflectivity, but

that does not say much about the content within the volume. So, in discretized form

𝑧 =1

𝑉∑ 𝐷𝑖

6

𝑖

where we sum over all hydrometeors (i) in the volume (V). Since z typically spans several orders of magnitude, we typically convert the linear reflectivity into a logarithmic form using

𝑍 [𝑑𝑏𝑍] = 10 log10 𝑧 This is what is plotted in radar maps. When we take our expanded form of η and put it into the radar equation, we get the following form of the weather radar equation:

𝑃𝑟 =𝑃𝑡𝑔2𝑐𝜏𝜗2𝜋3|𝐾|2𝑧

1024 ln 2 𝜆2𝑟2𝑙2

It still needs some manipulation since we want the meteorological reflectivity, z. So we rearrange and group a bunch of variables into one concise form:

𝑧 =𝑃𝑟𝑟2𝑙2

𝐶

We can convert this into the logarithmic form by taking log10 of both sides and multiplying by 10 to get

𝑍 = 10 log 𝑃𝑟 + 20 log 𝑟 + 20 log 𝑙 − 10 log 𝐶 We have grouped the transmitted power, antenna gain, beamwidth, wavelength, and other scientific and numeric values into the calibration constant C, which is dependent only on the radar system, and on no portion of the meteorology. Two final important notes: 1) We have assumed that the scatterers are liquid water spheres whose size is very small compared to the wavelength. Because of these assumptions, it is technically more accurate to use the “effective reflectivity”, Ze, instead of Z, to represent reflectivity. The “effective” wording is generally dropped with the understanding of the underlying assumptions that went into computing the reflectivity. 2) The loss term remains. Losses due to attenuation of the beam through atmospheric gases, clouds, rain, and snow are well known. However, the problem remains that we are trying to use a measurement that is affected by attenuation to try to measure the attenuation itself. Usually, some sort of assumed constant is used in place of that term for the atmospheric gas effects, whereas other attenuation items are simply ignored. Thus you must always be watchful of attenuation impacting the final reflectivity measurements. Typically, attenuation is quite weak for S-band (10 cm wavelength) radars like the WSR-88Ds. In fact, unless the beam is pointed along the convective portion of a strong and long squall line, or unless heavy rain or hail are falling directly at the radar site, attenuation is negligible.

The Doppler spectrum

We now return to the Doppler spectrum introduced earlier to delve deeper. Once all samples are collected, we can take the autocorrelation and the Fourier transform to obtain the power-weighted distribution of radial velocities within the resolution volume of the radar. While not always valid, we assume that the Doppler spectrum has a Gaussian shape, so that it looks something like this:

Figure 5. Schematic diagram of a Doppler spectrum using a Gaussian model.

Note that the spectrum is only defined between positive vmax (marked va in Fig. 5) and negative vmax since any sampled velocities more extreme than the maximum unambiguous velocity will be folded onto velocities that are below vmax. There is also the presence of a noise floor, which is the result of all the stochastic (random) noise that is not removed from the signal. This noise comes from a multitude of sources, including from the radar system electrical components as well as from energy received by the transmitter from non-meteorological sources such as ground objects (trees, buildings, etc), the ground itself, and even nearby humans. After all, everything in the universe with a temperature greater than 0 K (in other words, everything) emits radiation at all frequencies, even if it is only a very small amount. It is just about impossible, as well as impractical, to try to account for this additional energy received by the receiver. The noise is useful in determining the signal-to-noise ratio (SNR), which is the ratio of power due to legitimate signal to that due to noise. The signal will be more robust and representative of meteorological targets when the SNR is large, or when the noise floor is low compared to the amount of space under the “hump” in the Doppler spectrum. An ideal Doppler spectrum with noise is shown in Fig. 5. In reality, there will be a lot of variability in the spectrum. A more realistic example is shown in Fig. 6 (below).

Figure 6. A more realistic, but fabricated, Doppler spectrum. Lines are the same as those marked in Fig. 5.

Even with variation in the spectrum, it is still fairly easy to spot the mode of the spectrum which, for Gaussian distributions, is also the mean of the spectrum. This dashed line going through the peak is the mean Doppler velocity (vd), and this is what is plotted in the base velocity radar product. Likewise, the standard deviation of the distribution is defined as the spectrum width (σv), or SW, and indicates the amount of variability in the movement of scatterers within the resolution volume. Spectrum width is one way of quantifying the uniformity of the velocity of the scatterers within the resolution volume. Certain phenomena are known to have greater or smaller spectrum width. Ground clutter, for example, has a very small spectrum width, and the Doppler spectrum is typically narrow around a near-zero mean velocity. If a tornado is entirely confined to one radial (or single azimuth), on the other hand, the Doppler spectrum will look flat, nearly uniform (Figs. 7 and 8). There are two causes of this. One is that the radar will sample scatterers with inbound velocities on the left side of the tornado (assuming the tornado is cyclonic and we are looking from the point of view of the transmitter) and outbound velocities on the right side, with near-zero velocities at the center of the tornado. Obviously, such variance will show up in the spectrum. The other cause is that there will likely be velocities exceeding vmax in the tornado, so those velocities will fold back into the unambiguous velocity range (see example below).

Figure 7. An example Doppler spectrum containing strong outbound (positive) velocities such that some aliasing (folding) is occurring.

Notice the increased power at strong inbound (negative) velocities as a result of positive velocities greater than the folding velocity folding over to the negative side.

In fact, since vmax is known, so is the maximum SW. The maximum SW occurs in a perfectly flat (uniform) distribution. From the variance of a uniform PDF, the maximum SW is given by

𝑆𝑊𝑚𝑎𝑥 =𝑣𝑚𝑎𝑥

√3

Therefore, the three basic moments, or products, plotted in radar images – reflectivity, velocity, and spectrum width – can be computed based off the following aspects of the Doppler spectrum:

Reflectivity: use the weather radar equation given the zeroth moment of the spectrum, or the area under the spectrum from –vmax to +vmax, which is the returned power (Pr in the radar equation)

Velocity: compute the first moment of the spectrum, which gives the mean velocity. NOTE: for all moments greater than the 0th moment, you must normalize by the 0th moment since the Doppler spectrum is not truly a probability distribution function (because it violates the property that the total area under the PDF is 1).

Spectrum width: compute the second moment of the distribution, which is the variance. Then take the square root. Normalization should be done before taking the square root.

While it is nice to visualize Doppler spectra, realize that there is one spectrum for every single bin of radar data. There may be a few million bins that contain signal worth processing in a single volume scan! Thus it would be impossible to do any meaningful analysis via visual inspection of Doppler spectra. Even computing the moments would be time consuming. There are still simpler methods to computing the moments. Recall that the Dopler spectrum is the Fourier transform of the autocorrelation function of the timeseries of echo voltages. Since we have assumed a Gaussian form of the Doppler spectrum, we know it has a concise mathematical formula:

𝑆(𝑣) =𝑆

𝜎𝑣√2𝜋exp (

−(𝑣 − 𝑣𝑑)2

2𝜎𝑣2

) +𝑁

2𝑣𝑎

Figure 8. A Doppler spectrum typical of a bin containing a tornado. where we have switched symbolism so that SW is now σv, S represents signal power, and N represents noise power. We can apply the inverse Fourier transform to S(v) to get the autocorrelation function (ACF) model:

𝑅(𝑙𝑇𝑠) = 𝑆𝑒𝑥𝑝 (−8 (𝜋𝜎𝑣𝑙𝑇𝑠

𝜆)

2

) 𝑒𝑥𝑝 (−𝑗4𝜋𝑣𝑑

𝜆𝑙𝑇𝑠) + 𝑁𝛿𝑙

where lTs represents the lag index and pulse repetition time (the reciprocal of the PRF), j is the imaginary number √−1 and δl is a special “delta function” that has the property that it is zero at most values, but it is non-zero at arbitrarily small values and has an integral of 1 over all values of the input. It turns out that if we let l = 0, we will get just S + N out of the function above. This is the returned power! It also turns out that

𝑅(𝑙 = 0, 𝑇𝑠) =1

𝑀∑ 𝑉∗(𝑚)𝑉(𝑚)

𝑀−1

𝑚=0

where the superscript asterisk represents the complex conjugate. This can be quickly calculated straight from the time series of echo voltages! Here, M represents the number of pulses. To estimate velocity, we can use R(0) and R(1) to get the estimate for velocity as

𝑣𝑑 = −𝜆

4𝜋𝑇𝑠

∗ 𝑎𝑛𝑔𝑙𝑒(𝑅(𝑙 = 𝑡, 𝑇𝑠))

where

𝑅(𝑙 = 1, 𝑇𝑠) =1

𝑀∑ 𝑉∗(𝑚)𝑉(𝑚 + 1)

𝑀−1

𝑚=0

Spectrum width can be estimated by

𝜎𝑣 =𝜆

2𝜋√2𝑇𝑠

(ln𝑆

|𝑅(𝑙 = 1, 𝑇𝑠)|)

12⁄

What these estimates based on the ACF model tell us is that we can technically get estimates of the power (reflectivity) using just one pulse, and estimates of the velocity and spectrum width with a pair of pulses. For that reason, the estimates of velocity and spectrum width are typically called “pulse-pair processors”. While you can get estimates much faster using these methods, they are also more error-prone. Uncertainty estimates are also known for these methods, and all of them feature reduced uncertainty the more pulses are used. This goes back to the idea that the more independent samples you can obtain, the more accurate the estimate.

Sweet spots (aka, turning knobs) It has been previously noted that there are many aspects of radar meteorology that have a “sweet spot” in the middle of a range of values of some parameter rather than unlimited improvement by turning the knob on that parameter all the way to one end or the other. One big one – the Doppler dilemma – illustrating the give-and-take between maximum unambiguous range and velocity due to the PRF, has already been covered. In this section several other examples will be listed and discussed. The reason these sweet spots exist is because, when considering one component of a radar configuration isolated from the rest, the sky is the limit. We want to have the most accurate (lowest uncertainty/error), finest spatial and temporal resolution, and greatest areal coverage of radar data as possible. However, turning a knob to improve one parameter frequently results in a degradation of some other parameter. This behavior of one changing parameter countering changes in another is generally unavoidable since it boils down to physical laws. Getting more accurate estimates Obtaining more accurate estimates requires obtaining more independent samples. This means sending out more pulses. This increases the dwell time, which would require a slower antenna rotation rate in order to maintain the same spatial resolution. This causes an increase in the time between scans. There are other ways, however, of obtaining more independent samples without increasing dwell time. These include -Azimuth averaging: average bins at the same range but neighboring azimuths -Range averaging: average bins at the same azimuth but at neighboring range gates. Azimuth and range averaging are actually at the heart of super-resolution radar data. Super-resolution data, released publicly in 2008, were nothing new. Prior to their release, level 2 and 3 NEXRAD data were being range and azimuth averaged; the shift to super-resolution data removed the averaging. Thus, the estimates on reflectivity and velocity decreased following the upgrade. This is evinced in the speckled nature of super-resolution reflectivity data. Check any level 2 data prior to the upgrade for a comparison. -Range oversampling: you can sample the echo voltage more frequently than that supported by the range resolution. There will obviously be some issues with spatial correlation between nearby bins, though. -Frequency hopping: you can insert a small frequency change between pairs of pulses so that the receiver will be able to distinguish between pairs. Thus you can increase the PRF without reducing the maximum unambiguous range, although only a little bit. Refined spatial and temporal resolution Spatial resolution is controlled by two factors: beamwidth and range resolution. For most antennas, the beamwidth is directly proportional to the wavelength and inversely proportional to the physical size of the antenna (not the effective area used in the radar equation). A narrower beam requires either a decreased wavelength or an increased antenna size. Since the wavelength of the radar usually determines the purpose of the radar, changing the wavelength is usually not an option. Therefore it usually boils down to getting a bigger antenna. However, a larger antenna requires a rotating pedestal that can support it. So a larger antenna usually incurs additional mechanical costs. A possible way around this issue is to use a dual-wavelength radar. However, design complexity increases and there will likely be increased costs from mechanical or electrical/signal processing issues. Range resolution can be improved by using shorter pulses. However, shorter pulses reduces Pt in the radar equation, which subsequently leads to a reduction in returned power. The sensitivity of a radar refers to the lowest returned power level it can detect. The less power transmitted, the poorer the sensitivity becomes, therefore limiting either how many echoes the radar can detect or the maximum distance at which even powerful echoes can be detected. Temporal resolution (the time between scans) can be increased by rotating the antenna faster and sending fewer pulses. However, fewer pulses leads to less accurate estimates. Faster antenna rotation is rough on the mechanical aspects of the radar, and it also causes increased spectrum width due to the beam spreading component. However, another way to increase the temporal resolution is to reduce the sampling volume by either scanning only azimuthal sectors or scanning over fewer elevation angles. However, doing

this would violate the National Weather Service’s duty of weather surveillance, since much more air space would go uninspected by radar. New algorithms such as the Automated Volume Scan Elevation and Termination (AVSET) and Supplemental Adaptive Intra-volume Low-level Scan (SAILS) have been developed to improve on this. Despite these competing issues, researchers and engineers continue to come up with innovative ways to nudge closer to the optimization limit of these problems or to bypass them entirely. Phased-array radar is one such way of bypassing accuracy and temporal resolution issues. Phased-array radars can use something called “beam multiplexing” where pulses are sent out at radically different angles at very closely spaced times so that multiple portions of the atmosphere are being sampled at the same time. Phased-array antennas are also fixed and have no moving parts. Another improvement using WSR-88Ds (i.e., rotating dish antennas) is split-cut, batch-cut, and staggered-PRT scanning strategies. Some of these are already in use. Basically they involve doing elevation scans using a batch of pulses at one low PRF that allows for greater rmax, then switching to a higher PRF at the same azimuth to get a higher vmax, thus bypassing the Doppler dilemma. Another way this is done is by making two elevation scans at the same elevation angle, one with each PRF. The staggered-PRT approach is similar to the batch cut approach, except the PRF is switched every pulse, with the end effect that the pulse pattern is not as regular. More techniques are developed every day.

Application – scans from 31 May 2013 El Reno tornado in Canadian County, Oklahoma Let’s use this information to analyze reflectivity, velocity, and spectrum width scans from the tornado that struck rural Canadian County, Oklahoma between 6 and 7 PM CDT on 31 May 2013. This tornado was rated EF3 by the National Weather Service by damage, but research teams from the University of Oklahoma and the Center for Severe Weather Research each detected winds approaching or exceeding 300 mph over short time scales in this tornado. The tornado was initially rated EF5 based on these measurements, but was later downgraded for consistency with prior tornado rating procedure. First, let’s note that the radar was in VCP 212, which is a VCP used to sample widespread severe convective storms (where high velocities are expected to be measured and storms may be quite distant from the radar). The NOAA Warning Decision Training Branch website offers a very detailed online course offering detailing all VCPs. Check Topic 3, Lesson 1 at this URL: http://www.wdtb.noaa.gov/courses/dloc/outline.html. By evaluating Tables 13, 17, and Figure 23, we can get an idea of the structure of this rather complicated VCP. VCP 212 is the same as VCP 12, except the SZ-2 algorithm, which is used to obtain velocity data for a larger portion of overlaid echoes, is used. We will focus on the lowest three elevation angles (0.5°, 0.9°, and 1.3°), since they all use the same parameters. We can see this is a VCP for which the split cut strategy to mitigate the Doppler dilemma is used. Two scans are actually performed at all three elevation angles, one called contiguous surveillance (SZCS) – where the goal is a large rmax so that the detection and ranging part of scatterers is accomplished – and the other called contiguous Doppler (SZCD) – where the goal is accurate velocity estimation (large vmax). For the SZCS cuts, there are 15 long pulses, whereas for the SZCD cuts, there are 64 pulses. Unfortunately, I don’t know which PRI number KTLX was installed to have. It could be one of five, with the PRF differing slightly between each PRI number. However, we can use information given by the radar software GRLevel2 Analyst Edition to get an estimate. Unfortunately, we will see that the estimates don’t quite match. For example, the software lists vmax as 54.8 kts (28.19 m/s), which is consistent with a PRF of 1127.7 Hz. However, we also find the maximum SW is 32.1 kts (16.51 m/s), which gives a vmax of 55.60 kts (28.60 m/s), for a PRF of 1144.1 Hz. Also, by reading the maximum range of the first-trip echoes (73.52 nmi), we can derive a PRF of 1100.8 Hz. Therefore, there may be some frequency hopping going on or the precision of the reported values is enough to cause the difference. Either way, we can assume that for the SZCD cuts, the PRF is about 1125 Hz give or take 2%. Let’s analyze a particular scan where interesting things are happening. The following is the 2323 UTC scan from KTLX on 31 May 2013. The image covers most of central and eastern Canadian County, Oklahoma, including the county seat of El Reno. The tornado tracked in an arc across the bottom-left to bottom-center part of the image and at this scan is located a few miles southeast of the intersection of U.S. Highway 81 and Interstate 40 southeast of El Reno. The presence of a large ρhv hole at the location of the ball of higher reflectivity (not shown) that is the tornado indicates this is actually a debris ball.

Figure. 0.5° base reflectivity scan valid 2323 UTC 31 May 2013 from KTLX (located about 30-35 nmi to the ESE of the center of this

image. Follow the radials – they converge on the radar location.)

Figure. Same scan as reflectivity figure above, except for 0.5° base velocity. The white arcs east of El Reno denote the approximate

location of the forward flank downdraft gust front. This may or may not play a significant role in tornadogenesis or maintenance, and can be quite subtle on radar images. It marks a transition from mostly ESEly flow to the southeast of the arcs to mostly Ely or ENEly

flow northeast of the arcs.

Figure. Same as above except for 0.5° spectrum width.

We are going to analyze data from two bins. They are highlighted in the images below. One bin is located within very heavy precipitation (rain and hail) in the forward flank on the northeast side of El Reno, while the other is located towards the tail end of the RFD region with lighter precipitation, but stronger winds, located a few miles south-southwest of El Reno. The bins are outlined in white.

Figure. Doppler spectrum computed for the two bins outlined in thin white. Reflectivity is shown.

Figure. Doppler spectrum computed for the two bins outlined in thin white. Folded velocity is shown.

Figure. Doppler spectrum computed for the two bins outlined in thin white. Spectrum width is shown.

The reflectivity, mean Doppler velocity, and spectrum width of the bin northeast of El Reno is, respectively: Z = 61.5 dBZ V = +1.9 kts SW = 7.8 kts whereas those quantities for the bin to the south-southwest of El Reno are: Z = 37.5 dBZ V = -19.4 kts SW = 11.7 kts Given these moments, we can reconstruct what the Doppler spectrum might look like. Since we don’t have the Level I data, we cannot fully recover the true Doppler spectrum. Some noise is added for realism, but it probably doesn’t match the actual noise present in the signal.

Above: Spectrum for bin with Z = 61.5 dBZ, v = +1.9 kts, and SW = 7.8 kts

Above: Spectrum for bin with Z = 37.5 dBZ, v = -19.4 kts, and SW = 11.7 kts.

However, one thing really stands out in the base velocity image. Exactly what is going on in the region of the tornado? The simple answer is that there is a lot of velocity aliasing/folding occurring. Fortunately, the GR2 Analyst Edition software has the ability to dealias or unfold most of the velocities. When that is performed, the result is the following.

Let’s go in closer.

Above: The white line denotes a constant radial along which dealiasing is explicitly explained. The white rectangle indicates the likely

center of the tornado.

It is clear that the velocities of most of the pixels have been corrected to values that appear more realistic when looking 2000 ft above ground. There are about two dozen bins with velocities more extreme than -120 kts (inbound, this is base velocity, not storm relative velocity, so given the storm and tornado were moving towards the radar at this time, the velocties are expected to be larger on the right side of the tornado’s path, in this case, on the south side of the tornado), with one bin at -194.8 kts! You may wonder how the dealiasing algorithm was able to do this. Let’s investigate. I begin with the disclaimer that I do not know how the GR2 Analyst Edition software dealiasing algorithm works. But if I were manually dealiasing velocities, here is what I would do. It is also important to note that manual dealiasing requires a lot of logic and is not perfect. Knowing the folding velocity is 54.8 kts, I would look for areas where velocities of nearly +54.8 kts and -54.8 kts are juxtaposed. The color table makes this easy since the color of inbound velocities in that range is bright green and that of the outbound velocities in that range is bright pink. This leads the eye immediately to the area around the tornado. It seems quite unlikely that such strong convergence would occur in that location (what meteorologically would cause divergence on the order of 0.2 s

-1? Such a value is 4 to 5 orders of magnitude larger than values typically seen in synoptic or mesoscale scenarios. Even the

convergence along the leading edge of the RFD gust front is at least one order of magnitude smaller), and the values are still very close to folding velocities. So let’s see what picture develops if we attempt to unfold such velocities. To unfold, we need to find out

how much the measured velocity exceeds the folding velocity. We know this by calculating the difference between –vmax and the measured velocity. Along one radial on the west side of the tornado there are two bins with velocities of +50.5 kts and -51.5 kts in order going away from the radar. I suspect the +50.5 kt bin is folded. It should be more negative than -51.5. 54.8 – 50.5 = 4.3 kts. This is how much the measured velocity exceeds the folding velocity. Add that back onto the folding velocity with the other sign. This leaves us with -54.8 – 4.3 = -59.1 kts, which is indeed more negative than -51.5 kts. The unfolded velocity value is given as -59.8 kts. I assume there is some precision error, or the missing information regarding the precise PRF is causing the difference from what we calculated. We can move upradial (closer to the radar along a constant radial) to unfold velocities that are progressively more strongly folded. Closer to the center of the tornado there is a bin with a velocity of -35.9 kts. If we assume that is folded, we can unfold using the same method as for the other bin: Amount of exceedance: 54.8 - -35.9 = 90.7 kts. -54.8 – 90.7 = -145.5 kts. The software gives an unfolded velocity of -145.7 kts, which means our method performed well. This works pretty well for dealiasing most of the tornadic region. However, there are about 10 bins that appear very noisy and for which the velocities differ greatly from those of neighboring bins, both along the same radial and along neighboring radials. While noise or debris could contribute to significant error in these estimates, it’s also possible these are extreme velocities that may be double-folded. The maximum velocity that can occur for a single fold is 3*vmax, which in this case is 164.4 kts. Given the presence of measured EF5 winds, velocities of > 164.4 kts are certainly possible. We can unfold these noisy values once or twice and see which one appears more likely. For reference, we will convert to imagery here. The radial along which the unfolding occurs is marked by the white line in the images to follow.

Figure. Same as above, except the velocities are folded.

Figure. Approximated dealiased velocities along the radial marked by the white line in the velocity images above.

Doing this requires making assumptions about what bins are folded and whether they are folded once or twice. However, a glance at the strip of unfolded velocities should reveal which possibility is most likely for most of the velocities. Again, it is quite unlikely there is a sudden jump in velocities around range -34 nmi. A single unfold makes the resulting series of velocities much more believable since the big jump is removed. Another big jump occurs around the -32.5 nmi range. If we assume that area is doubly folded and unfold accordingly, we see a resulting series that seems a bit more plausible as we approach the most intense flow of the tornado. However, in assuming a double fold at this one bin, we’ve turned a positive jump into a negative one. The bins upradial continue to bounce around. However, the single fold appears most plausible, as a double fold would result in velocities of 230 to 260 kts inbound. While such winds are possible in portions of a tornado, the double unfolding would imply a signficant stretch along the radial of such velocities. That is less likely. It’s possible one or two of these other bins are doubly folded, but a single fold makes more sense. What about in the center of the tornado? There are eight bins arranged in a 4-by-2 array (2 radials, 4 range gates along each radial) showing weak inbound velocities. The dealiasing algorithm in GR2 Analyst Edition does not change these. Given the record size of the tornado (2.6 miles wide at about this point), it may be the case that these bins were in/near the center of the tornado so that the velocities are legitimate. These bins are approximately at the center of the analyzed tornado track (updated map available at

http://www.srh.noaa.gov/oun/?n=events-20130531). Spectrum width is high to extreme in these bins, which is consistent with

being in/near the center of the tornado. This lends additional credence to the legitimacy of these values. Immediately up- and downradial of the center of the tornado are additional noisy looking bins. It is unlikely that the tornado was so asymmetric as to appear more elliptical than circular, but it is possible that the radar is capturing suction vortices here. However, it seems just as likely that some of these bins were improperly unfolded. In particular, along the radial immediately north of the one previously discussed, and immediately downradial of the center of the tornado, the progression of the velocities suggests another fold is occurring. Moving upradial towards the tornado from the WNW, the velocities are +31.1, -4.9, -52.5, +38.9, and +26.2 kts (the bin just outside the center of the tornado). These values are consistent with single and double folded velocities, and when unfolded would become -78.5, -114.5, -162.1, -180.3, and -193 kts, which is a monotonically increasing wind as one approaches the tornado center.