Why can’t a single-fan-beam radar determine the elevation angles of its primary targets?

Question

A while back, I asked about the reason for one of the well-known limitations of most ATC primary radars (their inability to determine target altitudes).¹ The answers boiled down to “single-beam fan-beam² primary radars can determine the azimuth of targets (from what direction the radar beam was pointing when the return popped up) and the slant distance to targets (from the amount of time it took for the beam to travel from the radar to the target and back), but they have no way of knowing the elevation angle of a target, since they can’t know what part of the beam bounced off the target”.³⁴

But hang on - why can’t the radar determine what part of the beam hit the target? An aircraft reflecting a radar beam behaves (at first approximation, anyway) as a point source (or, at least, a very-small-angle-coverage source), and, while finding the direction of point sources emitting in the radio spectrum is harder than finding the direction of shorter-wavelength point sources (such as those in the infrared, visible, or gamma-ray spectra), it is by no means impossible, and the technology for doing so (involving a spinning directional antenna, and known, descriptively, as ADF - Automatic Direction-Finding) has been mature for a very long time - otherwise, one of the mainstays of air navigation in remote or poorly-developed areas, the NDB, would be utterly useless!

It should not be very hard, at least conceptually, to use a similar system (essentially an ADF receiver turned 90 degrees onto one side [so as to scan vertically, rather than horizontally] and resized for PSR⁶ rather than NDB frequencies) to determine the elevation angles of primary targets, allowing the altitudes of these targets to be calculated, so why isn’t this generally done in practice?

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¹: In contrast, military air-defence radars generally do display meaningful altitude information for primary targets (the better to intercept boogers with), as do a few other specialised types of radar (such as precision approach radar, which is designed to provide altitude and azimuth information about an incoming aircraft to an air traffic controller to allow the execution of an accurate ground-controlled approach down to low minima).

²: Radars that illuminate an entire vertical slice of sky at once.

³: In contrast, pencil-beam radars (which illuminate only a small spot of sky at any one time) and dual-beam radars (which have two radars, mounted one on top of the other, and use trigonometry and the time-of-flight difference between the two radars to calculate the exact location of a target in space) are able to determine the elevation angles of targets, but come with their own drawbacks: pencil-beam radars take far longer to scan the whole sky than fan-beam radars do, while dual-beam radars require twice as much radar per radar (and radar equipment is expensive).

⁴: In addition to denying ATC altitude information about primary targets, this also introduces a considerable amount of uncertainty into the horizontal position (and, thus the ground track) of such a target, forcing the use of greater separation distances between aircraft than would be necessary in an all-secondary-target⁵ environment.

⁵: Secondary surveillance radar (SSR), unlike primary radar, is not a true radar system; instead of bouncing radio waves off an aircraft’s skin, it sends an interrogation signal to the aircraft’s transponder, which then replies with information about the aircraft’s altitude (among many other things). Newer transponders also provide the aircraft’s horizontal location; for aircraft with older transponders, secondary radar falls back to determining the target’s azimuth and slant distance using antenna-orientation and signal-time-of-flight information, and adds the transponder-provided altitude to this to calculate the aircraft’s exact position in three-dimensional space. Secondary radar provides far more information than primary radar, and provides more reliable target tracking than primary radar (especially for aircraft at low heights above terrain, which would otherwise be hidden by ground clutter, and for small, nonmetallic, and\or stealth aircraft, which reflect less primary-radar beam than big metal non-stealth aircraft), but only works with targets with an operating transponder (whereas primary radar can detect sufficiently-reflective targets with nonfunctional, or no, transponders, and even non-aircraft targets, such as weather, flocks of birds, or parts detached from aircraft).

⁶: Primary Surveillance Radar.

Answer 1

What you propose is called a monopulse radar. Monopulse antannas have two feeds and beams, slighly to the left and right of boresight or above and below. In transmit mode, both feeds are connected in parallel to create centered beam and a single pulse is transmitted this way. In receive mode, the phase or amplitude difference in the two channels is used to find the exact direction. Many civilian primary and secondary radars use the monopulse technique with a left and right channel. Many military radars also have an upper an lower channel for height finding.

However, height finding is simply too inaccurate for civilian air traffic purposes. It happens that all airplanes operate at pretty much the same altitude and that barometric altimeters are extremely accurate, which led to uncomfortably low vertical separation minima, often as low as 500'/150m. There is no way to achieve such accuracy with radio direction finding from 50km away, which is why they don't even try.