Focus

# MATHEMATICS MEETS SATELLITES

"Satellites are invading our skies". At a first sight this sentence could
represent only a common sense; actually it summarizes a powerful reality
that permeates (sometimes in an inquietant way) everyday life of each
person and that involves both strongly industrialized countries and those
that hope to become such. The main reason is that satellite is the
cheapest tool able to "photograph" or "inform" the whole terrestrial globe
(including oceans, isolated areas, poles and everything most inaccessible
one
can imagine on the planet; see Figure 1).
One of the most important satellite applications is observation of Earth
geophysical parameters, such as temperature and water content over the
atmospheric column: their accurate measure is considered an important tool
to improve and extend over the time reliability of meteorological
forecasts, at moment limited to about 3 days. The crucial point of
satellite applications is that by its very definition satellite cannot
make DIRECT measurements of geophysical quantities (for instance it is not
possible extend a wire from the satellite down to the Earth surface along
which a hypothetic thermometer could record temperature). Therefore it is
necessary to rely on INDIRECT measurements of the quantities: from
satellite other parameters are measured, from which the required
geophysical quantities depend, and from those one tries to infer the value
of the geophysical quantities. From the mathematical point of view one
says that an INVERSE problem is solved (by the way, the reversed order of
the characters in the title testifies this kind of mathematical problem).
The Istituto per le Applicazioni del Calcolo 'Mauro Picone' has been
engaged since several years in the solution of this problem under a
cooperation that involves the Dipartimento di Ingegneria e Fisica
Ambientale of the Basilicata University and the Istituto per le
Metodologie di Analisi Ambientale CNR on projects that are funded or
endorsed by main International Space Agencies (Italian Space Agency,
European Space Agency, EUropean organization for the exploitation of
METeorological Satellites, NAtional Space Development Agency of Japan).
The instrument that flies on the satellite is a Michelson interferometer,
that measures the Fourier (cosine) transform of the radiance emitted by
Earth, that on its own depends on the sought geophysical parameters. The
mathematical model that describes the whole phenomenon is the radiative
transfer equation in the atmosphere, whose inverse problem has been the
main subject of the research. A particular formulation of the model that
makes least numerical approximations and development of effective and
accurate methods for the numerical solution have led to the development of
an accurate and fast enough software (Figure 3 shows the logo of the fast
forward radiative transfer model developed). The method has been validated
successfully on data measured by interferometer IMG of the NASDA (see
Figure 2) and is the reference for the design of new generation
interferometers, like IASI, initially developed by Italian industry and
research and after become an international project.

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