For microwave radiation propagating vertically through the Earth€™s atmosphere, the delay to a pulse as a result of the dry component of the atmosphere, expressed as an equivalent distance, is 2.33 meters irrespective of frequency. The corresponding figure for the water vapour component of the atmosphere is 7.10 meters per meter of precipitable water, again irrespective of frequency.
(i) Explain why the dry atmosphere delay can be expressed €˜per atmosphere€™ while the water vapour delay is expressed per meter of precipitable water.
(ii) Showthat a pulse of microwave radiation with a carrier frequency f will be delayed by its passage through the ionosphere by a time equivalent to a free-space distance of

where e and m_e are respectively the charge and mass of the electron,N_t is the total electron content (TEC) of the ionosphere and e₀ is the permittivity of free space.
(iii) A dual-frequency satellite radar altimeter measures the range to the Earth€™s surface at 3.2 GHz and 13.6 GHz. After averaging, the travel times of the pulses are found to be equivalent to free-space ranges of 793 125.20 ± 0.10 m at 3.2 GHz and 793 123.35 ± 0.05 m at 13.6 GHz. The atmospheric pressure at the Earth€™s surface is 0.970 atmospheres and the water vapour content is 0.15 m of precipitable water. Calculate the true range to the surface and its uncertainty. Also calculate the TEC and its uncertainty.

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