Planck's law describes the rate of energy emitted by a blackbody as a function of frequency or wavelength. A blackbody absorbs all the radiation it receives and emits radiation at a maximum rate for its given temperature. Planck's law gives the intensity of radiation Lx emitted by unit surface area into a fixed direction (solid angle) from the blackbody as a function of wavelength (or frequency). The Planck Law can be expressed through the following equation:

where T is temperature, c the speed of light (2.99 • 10-8 m s-1), h the Planck's constant (6.63 • 10-34 J • s), k the Boltzmann's constant (1.38 • 10-23 J K-1) and LA the spectral radiance per unit of wavelength and solid angle in W m-3 sr-1.

The Planck law gives a distribution that peaks at a certain wavelength; the peak shifts to shorter wavelengths for higher temperatures. The Wien displacement law and the Stefan-Boltzmann law are two other useful radiation laws that can be derived from the Planck law. The Wien law gives the wavelength of the peak of the radiation distribution (hmax= 3 • 107/T) while the Stefan-Boltzmann law gives the total energy E being emitted at all wavelengths by the blackbody (E=a • T4). Thus, the Wien law explains the shift of the peak to shorter wavelengths as the temperature increases, while the Stefan-Boltzmann law explains the growth in the height of the curve as the temperature increases. Notice that this growth is very abrupt, since it varies as the fourth power of the temperature.

The Rayleigh-Jeans approximation (Lx=2kcT/XA) holds for wavelengths much greater than the wavelength of the peak in the black body radiation form. This approximation is valid over the microwave band.

Most bodies radiate less efficiently than a blackbody. The emissivity e is defined as the ratio of graybody radiance to the blackbody. It has a non dimensional unit and is comprised between 0 and 1. The emissivity generally depends on wavelength (A) and polarization and has a directional dependence. e can be considered as a physical surface property and is a key quantity for ocean remote sensing. A blackbody absorbs all the energy it receives. A graybody absorbs only part of it and the remaining part is reflected and/or transmitted. The absorptivity is equal to the emissivity as a surface in equilibrium must absorb and emit energy at the same rate (Kirchoff's law). Similarly the reflectivity is equal to 1 - e.

The brightness temperature (BT) is defined as BT=e • T where T is the (physical) temperature. In the microwave band, it is proportional to the radiation LA.

2.3.3.3 Retrieval of Geophysical Parameters for Microwave Radiometers

The brightness temperature is an integrated measurement that includes all surface and atmosphere emitted power. Depending on frequency, it is more sensitive to a given parameter. Physical retrieval algorithms for geophysical parameters, such as the sea surface temperature, sea surface wind speed, sea ice or sea surface salinity are derived from a radiative transfer model (RTM), which computes the brightness temperatures that are measured by the satellite as a function of these variables. The RTM is based on a model for the sea surface emissivity and a model of microwave absorption in the Earth's atmosphere. The ocean sea surface emissivity (or reflectivity see above) depends on the dielectric constant s (which is a function of frequency, water temperature and salinity), small scale sea surface roughness, foam as well as viewing geometry and polarization. The retrieval of a given parameter is possible through the inversion of a set of brightness temperatures measured at different frequencies and/ or at different incidence angles. Inversion methods minimize the difference between measured and simulated (through a RTM) brightness temperatures. Statistical or empirical inversions are also often used given uncertainties in RTMs. They use a regression formalism (e.g. parametric, neural network) to find the best relation between brightness temperatures and the geophysical parameter to be retrieved.

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