Last Updated on July 13, 2026 by John Berry
If we draw a path profile with an effective Earth radius of 4/3, we can draw a straight line between stations and this will be the state, or better, for 50% of the time. We can then term the resulting loss over this path with this geometry as the median loss. Path losses, and corresponding signal strengths received, are then normally distributed in time about this median. When the signal falls below the median, we term this a fade.
Several pages here discuss incidence of k-factor fading – slow fading about the median caused by changes in the refractivity of the troposphere. In a fade, the signal level falls and is unusable. k-factor fading is governed by the troposphere’s refractivity, or strictly, the troposphere’s refractivity gradient. Typically, in UK and western Europe, the refractivity gradient, dN/dh (∆N), is -40 N/km. N is in units of refractivity. A dN/dh of -40 corresponds to an Earth radius factor, k, of 4/3 and an effective radius of 8500km. For reference, the actual Earth radius is 6731km.
Just as a fade is an increase in path loss above the median, so it’s possible to have a significant drop in path loss with corresponding increase in signal level. Radio amateurs refer to this as a lift. Literally, the signal level is lifted above the median (for that path).
k exceeded
Professionals talk of ‘k exceeded’ for 50% of the time, and indeed some design systems for the ninety-percentile – k exceeded for 90% of time. The signal received will be the corresponding (high) level or better.
Mostly however, k-factor fading is an issue of interference. If k rises, and the Earth flattens, interference might be incident from distant stations that would not normally be heard. Interference degrades wanted services. This is the core of frequency planning – ensuring that interference will only be experienced for small percentages of time.
But radio amateurs want to know the value of k for low percentages of time – perhaps at the five percentile, the value exceeded for 5% of the time or less.
Recommendation ITU-R P.453-14 gives the monthly median value of ∆N at around -40. Rec. P.453 also gives values for 1% (-300), 5% (-100), 90% (-20) and 99% (+30) of time, allowing the frequency distribution below to be assembled. Simply, the bigger the k, the better.
Then ke, k effective in the diffraction nomograph, can be calculated from the formula, ke = 157/(157-∆N). As noted in Recommendation ITU-R P.452-16, ∆N is a positive quantity in this formula (otherwise the result is nonsense). And of course, things go a bit haywire as ∆N approaches -157 and the Earth effectively goes flat!
Application
I’ve now modified the calculator I used in discussion of excess loss due to diffraction on long paths to allow for a modified ke to give loss exceeded for 5% of time. Select 50% or 5% of time and note the huge different in excess loss.
I’ve given the formula use in this calculator on the page about diffraction. All that changes is the k-factor which changes from 4/3 to 3.
Diffraction Path Loss
ke can be also be used in the calculation of diffraction losses and distance estimated in the nomogram in the page on Diffraction.
At VHF and above, this flattened Earth as a result of change in tropospheric refractivity is exciting because it leads to DX working. For 50% of time, the distance achievable might only be 50km. For 5% of time this range might increase to 300km, and for 1% of time it might be 1,000km. To work stations beyond the normal requires radio amateurs to watch the weather and get on the radio when conditions suggest a lift.
These ideas are also applied in other pages. If the path loss for 5% of the time is to be estimated, for example, the corrected frequency used in the nomogram is f/ke2 or about 16MHz for a 144MHz operational frequency. This gives a diffraction loss over Free Space Loss of about 65dB. For reference, the diffraction loss over Free Space Loss for 50% of the time is around 130dB. A loss of 65dB will allow a 300km path to work. At 130dB it won’t work.
In summary then, a lift occurs when k, the Earth radius factor rises above 4/3, to values like 3. At that point, on that day and time, the path loss drops and stations can be worked hundreds of kilometres away. This state seldom occurs – perhaps only on 5% of days in a year.
This is not to be confused with ducting. Ducting and tropospheric lifts are different.
The extremes of super-refraction and inverse beam bending are extensions of the refractivity discussed here. Under these conditions, a path that suffers huge loss due to the Earth’s bulge can end up with a free-space or near-free-space path between the stations. This is illustrated in the left hand tail of the above graph. Such conditions are, however, rare.