Abstract
Radiative transfer theory is formulated to permit a meaningful definition of emissivity for bulk emitting media such as snow. The emissivity in the Rayleigh-Jeans approximation is then the microwave brightness temperature T
B divided by an effective physical temperature 〈T〉. The 〈T〉 is an average of the physical temperature, T(z), weighted by a radiative transfer function ƒ(z). Similarly,
where e(z) is the local emittance. An approximate ƒ(z) is used to determine analytically the effects of various absorption coefficients, of scattering coefficients that vary with depth, and of the seasonal variation of T(z). It is shown that a mean emissivity, which is equal to the mean annual T
B divided by the mean annual surface temperature T
m, is a useful quantity for comparing theory and observations. Snow-crystal size measurements, r(z), at seven locations in Greenland and Antarctica are used to determine the Mie/Rayleigh scattering coefficient γs
(z and to calculate the mean emissivities. The observed mean emissivities are determined by a
which is the average of 12 monthly Nimbus-5 (1.55 cm) microwave observations, and the Tm measured at the same locations. The calculated emissivities are about one-half of the observed values. The assumption that each snow crystal is an independent and equally effective scatterer, and the use of an approximation to ƒ(z), tend to over-estimate the effect of scattering. Therefore, a parameter multiplying γs
(z) is used. The emissivities calculated with a single value of this empirical parameter for all seven locations agree well with the observed emissivities, showing that the microwave emissivity variations of dry polar urn can be characterised as a function of the crystal sizes. One optical depth corresponds to a typical fini depth of 5 m, but significant radiation emanates from up to 30 m. Since r(z) depends on the snow accumulation rate A and T
m. the sensitivity of the emissivity to changes in T
m or A are estimated using this semi-empirical theory. The results show that a one degree change or uncertainty in Tm is approximately equivalent to a 10% change in A, and that such a change will affect the emissivity by 0.003 to 0.014 or the T
B by about 0.6 K to 3 K, depending on the location.
Monte Carlo simulations of the microwave emissivity of the sea surface
