NDT: The Booker-Gordon Formulae

The basic equation which defines the received power when radio waves are transmitted from one site and reflected from an object before reaching the receiver is

The Bistatic Radar Equation

where P is power (t for transmitted, r for received), g is the antenna gain (t and r again), lambda is the wavelength, r is the outward or return path length (assuming they are of similar length), sigma is the scattering cross-section, and f is a propagation factor. (You will see variations on the theme of this equation in the literature - I use this particular form of the equation to relate to the theme of this page.) Note how important the square of the wavelength is in this, suggesting that we could make better use of NDT on lower VHF frequencies.

Booker and Gordon's model uses a modified version of this, essentially integrated over all the reflections/refractions taking place in the 'blobs' within the common volume,

The Booker-Gordon Version

where V is the scattering (common) volume. To this they added a complex formula for sigma, the scattering cross section, in terms of two characteristics of the scattering medium: the variance of the index of refraction fluctuation, and the correlation distance. Typical values for these are given as a variance of refractive index of one-millionth (10 to the power of -6) and a correlation distance of 20 to 130 metres. If the correlation distance is much greater than the wavelength, which it is expected to be for the 2 m amateur band and above, then the scattering cross section is largely independent of frequency, and mainly dependent on the variance of the refractive index and the angle of scattering. However, the 6 and 10 m amateur bands have wavelengths much closer to typical correlation distances.

It would therefore be very interesting to study NDT across a range of wavelengths over the same paths. Experience is that the quality of NDT varies on different bands at different times: sometimes, 6 m or 4 m will be enjoying a 'lift', whilst 2 m is not. In general, though, it appears that 2 m enjoys such improved conditions more commonly than longer wavelengths. Examining this in more detail would be particularly interesting, as it may give insight into the relationship between middle and upper tropospheric weather and the factors determining the scattering cross section.

NDT Inner Sanctum

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Last updated 2 Oct 1998
Howard Oakley
Mail howard@quercus.demon.co.uk