Why signals bend (and why your 10 GHz signal sometimes goes through a hill)

A line-of-sight microwave link is easy to reason about. Two antennas, straight line, if nothing touches the line the path works.

Ham radio doesn't cooperate with that story. People working the ARRL 10 GHz and Up contest log contacts over hundreds of kilometers where a string between the two stations would go straight through an Oklahoma mesa. The path profile looks broken. The contact happens anyway. The universe has, as usual, not consulted the path profile.

Below, one layer at a time, with scenes you can interact with. We start from the straight-line-and-obstacle model and break it four times.

Start simple: a line between two antennas

Two antennas, flat ground, a hill in the middle. If the hill sticks above the line, you get nothing.

That model is how nearly every commercial point-to-point microwave link is designed. WISPs, utility SCADA, carrier backhaul, all of it. The model is accurate enough that entire industries have been built on top of it, which is a minor miracle, because it rests on two small lies it would prefer nobody mention. The Earth isn't flat, and the signal isn't a line.

Earth gets in the way too

Zoom out far enough and the ground curves. From an antenna of height h the geometric horizon sits at about √(2Rh). Two 30-meter towers can each see a horizon roughly 20 km away. Push them 80 km apart and the Earth bulges more than 120 meters up between them.

No mast is tall enough to beat that on its own. The Earth has spent four and a half billion years being round and is unlikely to stop on anyone's account.

The signal is fatter than a line

One thing helps NLOS along a little even in the geometry-only model. Radio energy doesn't ride a pencil-thin ray. It fills an ellipsoid between the antennas called the first Fresnel zone, and you want about 60% of that zone clear for the path to behave like free space.

The ellipsoid is fattest at the middle of the path. Its radius there is √(λ*d/4), so it grows with wavelength. A 40 km path on 50 MHz has a midspan radius around 24 meters. Same path on 10 GHz is 1.7 meters.

This is why microwave links behave so much worse than HF over the same obstructed path. Higher frequency, skinnier ellipsoid, easier to clip with a ridge that doesn't look like much on the path profile. The ridge does not care how it looks on the path profile. The ridge is, in fact, indifferent to every document that has ever been written about it.

Ridges leak signal

When the line of sight is fully blocked, a sharp ridge doesn't eat everything. The wavefront hitting that edge keeps going, and some energy diffracts into the shadow. The usual way to describe it is the Fresnel-Kirchhoff parameter ν, which mashes together the geometry and the wavelength into a single dimensionless number with the virtue of fitting on one axis of a graph. ITU-R P.526 gives the loss curve. ITU publications are not known for brevity, but they do, eventually, tell you what you need to know.

A knife edge grazing the LOS already costs 6 dB. Stick it one Fresnel zone above and you're down 20+. It's cheap power on VHF. On 10 GHz the same geometry is almost never workable on its own.

Real hills are rounded, which costs more

Knife edges are a convenient fiction. A rounded summit scatters more energy away from the shadow zone, so actual loss is worse than the knife-edge baseline. P.526 tacks on an extra term based on the crest's radius of curvature.

This is where path profile software starts lying to you, politely and with conviction. If it says a ridge clears by 5 meters and you'll be +3 dB over the noise floor, you can usually subtract another 5–10 dB once you account for the rounding. Most of the time that subtraction puts the link under water. Except the link works. So something else is going on.

Air isn't empty

Air, it turns out, cares quite a bit about radio waves. Microwaves travel a hair slower through it than through vacuum, and the slowness changes with pressure, temperature, and humidity. Radio engineers express it as N-units:

N = (n − 1) * 10⁶

where n is the refractive index. At sea level N is usually around 315. It falls off with altitude, and the slope of that fall is what bends rays.

The dashed line is the international standard atmosphere. Drag the slope and the regime label changes with it. At VHF and UHF, useful tropo propagation starts happening as soon as you leave standard. At 10 GHz and above it takes a much steeper gradient to do useful work, and the gradient also has to sit on a thick enough inversion to trap the wavelength. Many of the atmospheres that ducting-detection code flags as "ducting" are only strong enough to carry 1 to 5 GHz, and the shape the slider has to make for a reliable 24 GHz contact is rare enough that you can count the nights per year it happens.

Rays follow the gradient

A horizontal ray in a medium whose refractive index falls with altitude bends downward. The tighter the gradient, the tighter the curve. Under the standard atmosphere the bend is gentle, about a quarter of Earth's own curvature, and engineers handle this by quietly pretending the Earth is 4/3 its real size and the rays are straight. Nobody points out that this is a slightly strange thing to do. That's the familiar k = 4/3 effective-Earth factor.

The slider below is labeled dN/dh, which reads as "the change in N per unit of altitude." It's just the slope of the refractivity curve from the previous scene, in N-units per kilometer. If N drops from 315 at the surface to 275 at 1 km up, dN/dh is −40 N/km. That's the international standard atmosphere, and it's also the zero point for everything that follows. More negative numbers mean refractivity is falling faster with altitude, which is what bends rays harder:

• around −40 N/km, standard. k ~ 4/3, mildly bent rays, ordinary radio horizon.
• below −79 N/km, super-refraction. Rays bend meaningfully more than normal. A lot of VHF/UHF tropo propagation lives here.
• below −157 N/km, the asymptotic trapping boundary. Rays curve tighter than the Earth itself curves away. Textbooks draw the line here. In practice a usable duct needs the gradient to be stronger still. The actual step from "rarely ducting" to "almost always ducting" in real atmospheric soundings sits closer to −200 to −300 N/km, because the duct also has to be thick enough to hold the wavelength you care about.

Drag dN/dh through those thresholds and watch the traced ray. Standard air, ray exits into space eventually. Super-refractive air, the ray stays close to the surface far longer. Trapping air, the ray never gets out at all, which is where the next scene picks up.

There's a second way to write the same thing that propagation engineers use, called the modified refractivity:

M = N + 0.157 * h

where h is altitude in meters. The 0.157 term is chosen so that M flattens Earth's curvature into the math: a ray in a medium with dM/dh = 0 travels parallel to Earth's surface, and a layer where dM/dh < 0 is, by definition, a duct. It's the same statement as "dN/dh below −157 N/km" said in nicer coordinates, and most duct papers use it because the condition becomes a single sign check rather than a threshold to remember.

The signal gets trapped

Propagation models talk about "rays" because the math is a lot easier when you treat a radio wave as a bundle of straight (or curving) lines, each representing the direction the wavefront is moving at that point. In practice, those lines are the actual paths your RF follows out of the antenna. When the atmosphere bends them, your signal goes where the lines go.

A duct is a layer of atmosphere where the refractivity drops steeply enough that any signal launched into the layer at a shallow enough elevation angle can't escape upward. Picture a slice of your antenna's radiation pattern between about −0.5° and +0.5° elevation. Normally that energy just keeps rising into space and is lost. Under a duct, it climbs into the layer, curves back down, hits the ground (or a lower boundary of the duct) and reflects, climbs again, curves back down again, and keeps repeating. The signal skips along inside the duct for hundreds of kilometers, losing very little on each bounce because the reflections are shallow and the layer is keeping the wavefront collimated.

Green paths in the scene are the rays (ray = the direction the wavefront is traveling) that stay trapped inside the duct. Orange ones escape. The "critical angle" is the elevation window that stays trapped, and it's tiny, usually a fraction of a degree. That's why ducting is so fickle. Your antenna has to be radiating useful power inside that window, and the duct has to exist above your station, and the station on the other end has to be inside it or close enough for its energy to enter, and the duct has to be thick enough in meters to actually trap whatever wavelength you're running. When all four line up, your 10 GHz rig works a contact 400 km away and you feel briefly invincible. When any one of them is off, the path behaves the way the terrain profile says it should, which is to say, not at all. The atmosphere is happy to support this arrangement, briefly, before rearranging itself into something less interesting, usually right around the time you reach for the microphone.

The reframing buried in that list is the one I keep coming back to. "Is there a duct, yes or no" is a surprisingly bad predictor of whether a 10 GHz contact will work. In the contact logs we've collected, the median distance on days with a duct and the median on days without one are almost identical. What actually correlates with distance is the magnitude of the gradient, the thickness of the trapped layer, and how well that thickness matches the wavelength. The same inversion that supports 1 GHz beautifully might do nothing at 24 GHz because it's a few meters too thin.

Flavors of duct

Four show up in the wild, same physics, different mechanisms for setting them up.

Radiation (nocturnal) ducts are the ones that matter inland. Clear calm sky, ground radiates to space faster than the air can cool, an inversion forms 50 to 300 meters thick, gradient runs anywhere from −100 to −300 N/km. Forms one to three hours after sunset, peaks right before dawn, collapses an hour or two after sunrise when mechanical mixing chews it up. The dataset says about 70% of June-July pre-dawn mornings in Texas have one. Almost every inland 10 GHz DX contact you will ever make is riding a radiation duct.

Subsidence (elevated) ducts sit aloft under the ridge axis of a high-pressure system. Sinking air warms and dries, producing a sharp temperature rise and dewpoint drop at 800 to 900 hPa. The trapped layer sits 500 to 2000 meters up and can stay there for days. Both ends of the link usually need to be inside it, or aimed high enough to couple into it. This is the "an elevated duct parks over North Texas" situation people mean when they say a good weekend is forecast.

Advection ducts form when warm dry air moves over a cooler surface. Coastal and marine. Gulf Coast operators know these well. Not a mechanism inland North Texas sees.

Evaporation ducts exist almost continuously in the first 5 to 40 meters over warm oceans, driven by the humidity gradient at the sea surface. Every over-water microwave link above a few GHz lives inside one. Also irrelevant for anyone operating from Oklahoma.

A real path

This is the 65 km path I've been staring at: my QTH at 205 m above sea level to the W5HN/B beacon on top of the TWU dorms in Denton. Earth's curvature (using the standard k = 4/3 effective Earth) is added on top, which is why the middle of the profile bulges up about 62 m above the endpoints. The first Fresnel zone at 10 GHz has a midspan radius of about 22 m. The terrain is unambiguously in the way. On the right days the path works anyway.

Figuring out whether it's going to work on any given day is the whole point of https://prop.w5isp.com. The app pulls HRRR at its 50 native hybrid-sigma levels (not the standard 13 pressure levels, which are spaced about 250 m apart and completely miss the 50 to 100 m thin ducts we care about), corrects against ASOS surface observations and twice-daily radiosonde profiles, and computes the refractivity gradient along every tracked path. It also polls seven commercial 11, 24, and 68 GHz microwave links around Princeton TX every five minutes as an inverse sensor, because the same refractivity anomaly that helps beyond-LOS paths causes multipath fades on LOS ones. My QTH happens to be one endpoint of that network, so the commercial-link telemetry becomes a direct real-time signal on whether the atmosphere over this specific path is doing anything interesting.

A few things I didn't expect going in, now painfully clear from the data.

The path is almost always blocked. The numbers in this section come from the ARRL 10 GHz and Up Contest logs I've been able to collect so far. In that dataset, 97 percent of logged amateur microwave contact paths are terrain-blocked at an average diffraction loss of 36 dB. Beyond-LOS is not the exceptional case, it is the default case. Clear LOS paths are short and rare. The long contacts that show up in contest logs are all riding atmospheric enhancement of some kind. The sample is as good as it currently is; if you have logs I don't, submit them so the distributions sharpen instead of drift.

Binary duct detection is a coin flip. The median contact distance when HRRR says "there is a duct" is statistically identical to when it says "there is not." What separates a working 10 GHz morning from a silent one is the strength of the gradient (how far past −200 N/km it gets), the thickness of the layer in meters, and whether that thickness matches the wavelength. Ducting is a shape, not a switch.

Low pressure beats high pressure. The folklore says high pressure is a ham microwave operator's friend. The data says otherwise: on 10 GHz, contacts made under 970 mb average 243 km; over 1020 mb average 103 km. Subsidence ducts under a ridge are real and matter, but on average the ridge itself is not actually the best time to be on the radio.

Shallow boundary layer beats deep. HPBL (planetary boundary layer height) under 200 m averages 230 km; over 2000 m averages 100 km. A calm cool morning with a shallow boundary layer is what you want.

The useful window is centered on sunrise, not midnight. Radiation ducts are strongest right before dawn and for an hour or two after. The contacts cluster in a window about an hour and a half either side of sunrise, and that's where the app focuses its scoring.

Pre-dawn matters more the higher you go. At 10 GHz, time-of-day enhancement is about +4% over the daily median. At 24 GHz, +28%. At 75 GHz, +360%. If you are pointing a 3 mm dish at someone, you are almost certainly doing it before breakfast.

Humidity is a trade. Precipitable water, or PWAT, is the total amount of water vapor in a column of atmosphere above you, expressed as the depth you would get if you condensed all of it into liquid and let it puddle on the ground. A PWAT of 25 mm means that if every water molecule above your antenna fell out of the sky at once, you would be standing in an inch of water. It is the number forecasters use to describe how wet the air is in bulk, and for microwave propagation it is the variable that matters, because the refractivity gradient that bends signals depends on water vapor more than any other single thing.

At 10 GHz, 20 to 30 mm of PWAT is the sweet spot. Too dry and the surface refractivity stays low, too wet and the column destabilizes. At 24 GHz, drier is always better, because the 22.235 GHz water vapor line happily eats whatever signal the atmosphere just delivered.

Spring is the worst season. Summer is the best. I went into this thinking winter would be the dead period, but March and December turn out to be the two worst months of the year, and June and July are by far the best. The dominant Texas radiation-duct season runs May through August.

Troposcatter is the floor under all of this, and fortunately the floor is higher than the arithmetic suggests it has any right to be. The average blocked path is 36 dB down from free space, and realistic duct enhancement caps out around 14 dB. On paper these numbers add up to the paths not working, which would be a compelling argument for abandoning the hobby, except that the paths work anyway. The gap gets made up by forward-scattered energy bouncing off small-scale refractive irregularities in the troposphere, which are present all the time whether you need them or not. Ducting is the spike on top of the troposcatter baseline, not the only mechanism in play.

The part that keeps me working on it is how often the path surprises me. People who have been operating these bands for forty years still show up on the post-contact mailing lists asking each other how a contact made it through, which is both humbling and a strong argument against ever finishing this project. The more verified contacts we log with atmospheric data behind them, the better the model gets.

If you run 50 MHz or up, submit your contacts or beacons at https://prop.w5isp.com. A continuous, known-distance path is worth a lot more to the model than a pile of contest contacts.