Last Updated on July 13, 2026 by John Berry
Most amateur propagation in the troposphere relies on diffraction. There will seldom be a free-space path between the two stations, even from hilltop to hilltop. Typically some terrain objects intrude into the path. The object with the greatest effect is the Earth’s bulge.
A typical path is shown below. The Earth bulge is shown.
Were it not for diffraction, the signal loss between transmitter and receiver would be total.
Diffraction works and is explained using Huygen’s principle.
Huygen’s principle
Huygen’s principle has it that as a wave spreads radially from the transmitting antenna, every point on every radius out into the troposphere launches a secondary wavelet. Each secondary wavelet spreads and each point on each sets up another wavelet, and so on. The received signal at the receiver is the vector sum of all signal arrivals captured by the receiving antenna from all participating wavelets.
This principle means that it is impossible to block all energy from the receiving antenna. Some wavelets will always get through no matter how severe the obstruction. Some wavelets will get through regardless of how far off the direct path they have to be to get over the hill or whatever. Of course, the energy received may be infinitesimally small and unusable. But it does mean that reception is possible behind hills and in the shadow of the Earth’s bulge.
Diffraction loss due to spherical Earth
The Earth is a sphere. No surprises there. The horizon is for most a few tens of kilometres distant, so unless we use very high antennas, the Earth quickly becomes the dominant intrusion into the path.
The path loss between two stations is given by four terms:
Loss = F(FSL) + F(X) + G(Y1) + G(Y2) dB
FSL is the Free Space Loss as discussed elsewhere on this site. X is a term including primarily the distance but also the type of ground and polarisation. Y1 and Y2 are antenna height gain terms. Both X and Y terms also consider obstruction due to the Earth’s bulge and hence the Earth’s effective radius.
This page is about calculating the excess loss over long paths due to diffraction. I discuss height gain on another page.
Calculating loss
The excess loss over Free Space can be calculated using formulae from Recommendation ITU-R P.526-15. X, the normalised path length is given by:
Where:
d : path length (km)
ae : equivalent Earth’s radius (km). For k = 4/3, ae is approx 8495 km.
h : antenna height (m)
f : frequency (MHz).
β = 1 (valid for horizontal polarisation at all frequencies, and vertical polarisation above 300 MHz over sea or above 20 MHz over land).
F(X) is then given by:
For X≥1.6,
For X≺1.6,
The result is a negative value of decibels with respect to Free Space Loss – a loss over Free Space. I’ve had Gemini AI do a calculator using these formulae.
Diffraction Path Loss
This is valid for frequencies between 30 MHz and 10 GHz, and for path lengths of 10 km to 1000 km. I have included a dropdown for the percentage of time, currently locked at 50% (k=4/3). This will be opened in the adjacent page discussing tropospheric lifts and in the tropospheric propagation path budget tool.
You will see some big numbers appear as the Earth’s bulge takes effect, severely limiting normal tropospheric communications. And remember that this loss has to be added to the Free Space Path Loss to get a final value to compare with the system value. As a result, it’s only during lift conditions that long distance tropospheric paths work.
Using a nomogram
In the above formula, X, the excess loss due to diffraction, can be calculated using a nomogram. Nomograms are graphs that enable the independent variables (location, polarisation, distance and frequency) to be set and the dependent variable (X) to be read off. The nomograms in Rec. ITU-R P. 526 are given for k=1 and for k=4/3. k=4/3 represents the normal troposphere when k is exceeded for 50% of the time.
See the page on tropospheric lifts for more on how k changes with time.
An edited nomograph is shown below.
Figure 3, Rec. ITU-R P.526-15)
The nomogram is read like this. For 50% of the time, select the line for k=4/3. Draw a line from the operating frequency through the likely path length and extend the line to the signal level scale. Read off the loss over free space in decibels. Then calculate the Free Space Loss for that path length and frequency and add the two figures.
Using the loss value
You can use that aggregate loss (diffraction loss plus Free Space Path Loss) in a path budget to see what the likely received signal would be and hence whether or not the link will work. Alternatively, subtract the aggregate loss from the System Value and that will show how much antenna gain (at each end) you’ll need to make the link work.
Application
So, what is this telling us?
Referring to the page on this web site on system value, we see that at 144MHz for SSB, there’s a system value of 166dB available.
The free space loss over a 300km path, for example, is:
FSL = 32.4 + 20log f + 20log d, or 125dBi
As an example, and for 50% of time with 125dBi Free Space loss plus 125dB excess loss, that’s 250dBi in total. The system value is 166dB. That means that we need an extra 84dBi. Simply, for an SSB QSO, a path length of 300km for 50% of the time is improbable.
For k=3 representing around 5% of time, the excess loss drops to 65dB. The total path loss is therefore 125dBi plus 65dB, or 190dBi. At a system value of 166dB, we need to find a mere 24dBi to make the path work! That’s easily found if both stations are using Yagi antennas with gains of at least 12dBi – not a great problem considering that the typical radio amateur Yagi has 11 elements with a gain of 15dBi. There’s then 6dB in hand!
So, this shows why radio amateurs can communicate using SSB at 144MHz over distances of more than 300km for small percentages of time. I discuss such lift conditions further on an adjacent page.
Greater distances
If the mode is changed to FT8, with its system value of 186dB, the same 300km could be achieved for greater percentages of time or for low percentages of time using dipole antennas.
And likewise, if one or other of the stations went onto a hilltop with full Fresnel Zone clearance in the foreground and significant height compared to the foreground terrain, the resulting height gain would reduce the excess loss. A height of 300m at one end might give a height gain there of 25dB. This moves SSB success over a 300km path to around 50% of the time.
Finally, if both stations have height gain of around 25dB, this ‘extra’ 50dB extends the range to around 550km for 50% of time. This is the state typically existing during VHF NFD.
So the story is simple. For small percentages of time, paths can be long. Great DX is possible. And to promote tropospheric DX, height and antenna gain is needed.
Numerical methods
The diffraction modelling methods discussed so far are empirical. with accuracies better than 10dB. For shorter paths in search of greater accuracy, numerical methods can be used.
A path profile or slice through the earth, like the one at the top of this page, can be drawn using high resolutions terrain and land use models. At the extreme, it is possible to build a profile using centimetric resolution. In the frequency range to 1300MHz, and for amateur applications, a resolution of 100 metres is probably adequate. Even at this coarse resolution, the profile will be constructed with thousands of points.
Once the profile has been captured, it can be analysed. Analysis methods are set out in RECOMMENDATION ITU-R P.526-10, Propagation by diffraction. While the nomograph methods given above is an estimate that’s adequate for point-to-area assessment, Rec. P.526-10 gives highly accurate point to point loss.
In the above diagram, hills, buildings and trees can be assessed by drawing equivalent cylinders, and then calculating the loss from each. In the lower half of the diagram, and once cylinders have been assessed, loss due to obstruction of the lower half of the Fresnel zone can be calculated.
These are methods used by spectrum regulators and wireless operating companies. There are however a few terrain-model based path loss calculators available to the radio amateur online. Despite higher accuracy, they offer little by way of enhanced understanding about likely ranges for varying conditions.