Last Updated on March 15, 2025 by John Berry
Scattering of radio waves is complex, with boundary conditions describing when the mechanism is reflection, and when it is scattering. Many radio hams mis-categorise. That matters when trying to understand signal returns from solid objects like the Moon, particularly in allowing for reflection or scattering loss in the path budget. So, does the Moon reflect or scatter our EME/moonbounce signals? And how should we account for the Moon as a sphere, rather than a flat plate?
Here’s my attempt to conclude.
Nature of reflection
Energy incident as a wave-front hits the particles of the scattering or reflecting surface. The particles absorb the energy and re-radiate it. It’s the Huygens-Fresnel principle dating from 1818. Long wavelengths will be absorbed (without re-radiation), while short wavelengths (high frequencies) will be re-radiated, either resulting in reflection or scattering.
If the surface is smooth, the reflection is specular, and the mechanism is described by Fresnel’s reflection laws. In this case, the local reflections are in phase. This specular component is referred to as the coherent scattering component. It’s coherent because the reflection is along one coherent vector (back to Earth, for EME). As the surface becomes rougher, the incident wave becomes partly reflected and partly scattered in all directions. As these scattered, incoherent components, rise with roughness, the coherent component reduces. Losses in the desired direction back to Earth rise. Generally, therefore, for flat surfaces it’s better if signals reflect.
The idea is shown below.
Relative contributions of coherent (solid line) and diffuse scattering components (dotted lines) for different surface-roughness conditions: (a) specular, (b) slightly rough, (c) very rough. Solid lozenges describe the envelope of all possible scattered vectors (in (b) and (c)).
From Ticconi F., et.al. referenced in bibliography.
In some experiments, a slightly rough surface gave a 0.7dB attenuation of the wave. A rough surface gave 11dB loss. And a very rough surface gave 45dB loss. This loss is the incident energy over the scattered energy in the desired direction.
Surface roughness
Surface roughness is assessed in terms of wavelength. A surface that appears very rough to an optical wave (at 1015Hz), may appear very smooth to a microwave (at 1010Hz).
There are then some limit conditions which, if met, lead us to determine a surface as near-smooth, slightly rough, rough etc. Now we’re getting to the crux. As radio hams we need some guidance about what is and is not rough at the wavelengths (frequencies) we use.
The analysis starts by considering the Rayleigh roughness criterion, an optical construct.
The Moon’s surface is characterised by large plains pock-marked with meteor collision craters. On the one hand, it has macro variations in height from a few centimetres up to kilometres from crater bottom to rim. On the other hand, it has vast local areas of little undulation. The Indian Chang’e-4 lander reported on this, determining surface roughness for different radar frequencies.
Does the Moon reflect or scatter?
Continuing the analysis, the Fraunhofer criterion suggests that:
A surface is smooth if s, the RMS height, s < λ / 32cos θ. [By using the RMS value for heights we’re saying that most of the heights across an area are around a particular value.]
So, for reflections at 10GHz, a wavelength of 3cm, the Moon will be smooth if the RMS heights are around 1mm or less. Otherwise, it is rough or very rough.
For reflections at 432MHz, a wavelength of 70cm, the Moon will be smooth if the RMS heights are around 20mm or less. Otherwise, it is rough or very rough.
For reflections at 144MHz, a wavelength of 2m, the Moon will be smooth if the heights are around 60mm or less. Otherwise, it is rough or very rough.
The Chang’e-4 lander reports the Moon’s variation in surface heights locally to be in the range 1mm to 10mm. So, the Moon is smooth to all those frequencies considered above. At 10GHz though, it’s on the edge of the roughness criteria and could be considered rough. EME signals at above 10GHz will experience a rough or very rough surface and signals will be scattered. As noted above, when signals are scattered, additional losses occur.
So does the Moon reflect or scatter? Simply, it depends on the signal frequency.
Reflections from a sphere
The above would apply if the Moon were a flat plate. But it’s not. It’s a sphere. Only the very centre, out to a radius of maybe 10% could approximate to a plate. The curve of the Moon’s surface suggests that specular reflections will be lost to space. In that case, scattering is useful!
Radio amateurs researching EME moonbounce suggest that the inner 70% is to be considered useful in returning signals to Earth. The above analysis suggests that this is optimistic and that maybe this should be less than 30%.
Ultimately, there are two parts to assessment of reflection/scattering: the efficiency of the re-radiation (Huygens-Fresnel principle) estimated at ρ=6.5%, and the usefulness of the reflection and scattering (described by Rayleigh and Fraunhofer) covered by the area of useful surface. The two must be considered separately in the path budget, unless a purely empirical reflection coefficient is to be developed.
This suggests that maybe to get a compromise between reflection and scattering, and path loss and antenna gain, the best frequencies for EME/moonbounce are in the low GHz region.
Finally, we must also consider that the Moon is pock-marked with craters. There will be varying angles of arrival, and hence varying angles of reflection on the crater sides. Any coherent component will therefore seldom be towards the Earth and that adds loss proportional to cosθ. Luckily the Moon has more plains than crater sides!