Radar shadowing

[Jeff Duda]

Can someone explain to me the concept of radar shadowing? By that, I mean the fact that an actual real feature may not be colocated with its radar signature. The best example is how an actual tornado track matches up with the core of a gate-to-gate or center-of-a-velocity-couplet signature as it appears on radar. I know that the two paths will not coincide exactly, and that has something to do with the elevation of the radar as well as other things.

 

[rdale]

I'm not sure I understand the term you are using... However the radar is scanning aloft, at 3000 ft or whatever depending on the distance from the radarsite. When you watch a tornado, you'll notice it doesn't form a perfectly vertical appendage, i.e. it snakes and ropes around. So it's not going to show up directly underneath the radar signature. The the "gates" are usually much larger than the tornado itself, so it doesn't have the resolution to give you the exact center of the rotation aloft even.

 

[Jeff Duda]

True, but sometimes I've seen mis-estimates of tornado paths from radar imagery by as much as a mile. Most, but not all, tornadoes don't extend 1 mile in the horizontal below the mesocyclone (or maybe I'm wrong and many do). Maybe it's a combination of that and the size of the bins.

 

[Chip Redmond]

Tornadoes can also develop in any part of the mesocyclone that could be many miles wide therefore making assumption of the exact location of the tornado nearly impossible. Also, keep in mind the radar is always behind. It could have a certain timestamp, but the image is already several seconds old...perhaps this could be a shadow, the radar is behind the actual rotation on the ground...?

 

[rdale]

Also, keep in mind the radar is always behind. It could have a certain timestamp, but the image is already several seconds old

It doesn't take that long to do a single elevation scan, it's not "behind" really. It's a snapshot in time.

perhaps this could be a shadow, the radar is behind the actual rotation on the ground...?

No, the radar is an actual image of how the storm looked at that time.

 

[Chip Redmond]

No, the radar is an actual image of how the storm looked at that time.

If you are looking at the raw data as it comes in from the radar itself. However most secondary sources don't show an image until the entire scan is finished making the radar actually old by the amount of time it took to finish the scan.

 

[Jeff Duda]

I'd like to revisit this thread. Since I'm taking a radar meteorology class at OU this semester, I have the tools to solve this problem myself. I've done so graphically and would like to share it with others.

To my knowledge, radar products in the GRLevelX software are given in range/azimuth coordinates, where range is the distance from the radar along a conical or angled surface defined by the elevation angle. This is a point which I continually forget. While ignoring this is usually not a problem, for small scale features such as tornadoes, ignoring it may result in significant errors at large distances from the radar. I made some plots to illustrate this using two methods:

Method 1 (straight-up right triangle trigonometry)
By drawing a right triangle with one leg representing horizontal distance from the radar (rh), the hypotenuse representing the radar rage (rr), and the elevation angle above radar horizontal (theta), the difference between radar range (which is what's displayed in GRX) and actual horizontal distance from the radar is given by rh = rr*cos(theta). The difference between rh and rr is thus |rh - rr| = rr*|1-cos(theta)| (here I chose to use absolute value to remove negative numbers...only the difference between rh and rr is important). The result is plotted below for radar ranges of 0 - 300 km and elevation angles from 0 - 20 deg. (those most commonly covered by 88Ds):

The color bar has been adjusted so that one color range represents 1 km of difference. It's pretty obvious that there isn't much significant difference for any reasonable range for the lowest tilts, but there is some variation of nearly 1 km as you get out pretty far from the radar.

Method 2 (adjusting for beam bending)
In the real atmosphere, the beam is bent vertically by vertical changes in the index of refraction due to temperature, pressure, and water vapor changes (and also due to the concentration of electrons, but that's of secondary significance). Using the method in Chapter 2 of Doviak and Zrnic's book Doppler Radar and Weather Observations, the difference between radar and horizontal range is only slightly different if you assume the "4/3 effective Earth's radius model":

In fact, there is almost no effective difference between using this method and the one from above, but I'm still checking the equations in the textbook for representativeness (in other words, I might've screwed something up in method 2). Also, it should be noted that I assumed the 4/3 effective Earth's radius model in this problem which assumes standard vertical refractivity gradients under standard conditions, which bends the beam downward from straight out. In a thunderstorm, standard conditions hardly apply, so the actual vertical profile of refractive index likely deviates from the standard used to make this model. Qualitatively, the difference between radar range and horizontal direction is directly proportional to the effective Earth's radius, which is inversely proportional to the vertical refractive index gradient. Thus, a weaker (smaller) vertical gradient in the refractive index will result in less beam bending, which results in less difference. So it appears that what I earlier called "radar shadowing" which is this range difference, really isn't all the big even at large distances from the radar (as long as you're at low tilts). Tornado track errors using the center of a gate-to-gate shear couplet therefore would be on the order of the size of one or two pixels at far distances, and even less than that at closer distances.

 

[Jeff Snyder]

To my knowledge, radar products in the GRLevelX software are given in range/azimuth coordinates, where range is the distance from the radar along a conical or angled surface defined by the elevation angle. This is a point which I continually forget. While ignoring this is usually not a problem, for small scale features such as tornadoes, ignoring it may result in significant errors at large distances from the radar. I made some plots to illustrate this using two methods:

Are you sure that GRx shows PPIs as a function of slant-range and not true ground-relative range? Having GIS features, I'd think that GRx would display horizontal range on the PPIs (to help avoid the errors that you've mentioned). For what it's worth, it looks like Solo II uses slant range for displaying PPIs, but it calculates the ground-relative range in the "Data Widget" readout. Solo II doesn't really have any GIS features, though.

The difference between horizontal ("ground") range and slant range is still often important to keep in mind when high elevation angles are used. For example, we've been collecting mobile radar data up to 40-50 degrees on some storms. As you can imagine, the differences between slant range and horizontal-equivalent range is very large when these elevation angles are used (cos(45 deg) = 1/sqrt(2) = 0.7); it's also important to keep this in mind when using Kdp and Zdr for quantitative purposes at these angles (though the beam often quickly goes above the freezing level at such elevaiton angles).

Aside from deviations from a true vertical alignment (I'm sure most of us have seen tornadoes that rope out at pretty extreme horizontal displacements), differences in the location of vortex centers in radial velocity data compared to ground assessments may also arise from the resolution being too low to properly sample the vortex. If a significant storm is located at a large distance from the radar, the actual center of a vortex may be anywhere within a range bin. In other words, the radials may not perfectly dissect the tornado/vortex. In these cases, the center of the couplet on radar will not be coincident with the center of the vortex. For example, at a horizontal range of 200 km, the "width" of 1 degree radial is ~3.5 km.

 

[Jeff Duda]

Jeff,
Yep. You are right about everything. In fact, I was wrong in assuming that Mike did not account for standard refraction when making the GR software. I tested the range/height output given on a single elevation scan and found my computational numbers to be within < 0.1% of what came up in the GR display, which is precise enough to be a result of rounding errors in the conversions and in truncation errors in the readout. Initially I brought this up because if GR was displaying radar data essentially over the true ground location, then one would expect to see the radar bins change shape with height. If you look up through the tilts you will see pretty much no change in bin shape/size through the lower few tilts. But I just checked it again and when you get far from the radar and get high enough tilts, indeed the bins do change shape slightly. Thus it really does appear that curvature and slant range (what I was calling "radar range" in my post above) is being accounted for in the GR software. Heh...go figure, I guess.

I also should correct in my post above yours that the bending of the radar beam is incremental as it passes through the atmosphere instead of happening only at the target range. Thus, having one storm at the target will only curve the beam so much, whereas the rest of the clear atmosphere between the radar and the storm will contribute just as much, if not way more, to the actual bending. So it is important to include all of the atmosphere.