A general analytical expression for the radiation source function of emitting and scattering media within the matrix operator method

We derived an analytical expression for the radiation source function for a thermally emitting and scattering medium within the Matrix-Operator-Method (MOM). The final formulation is equivalent to the one found by Aronson and Yarmush (1966), who applied the transfer matrix to gamma-ray and neutron penetration (Aronson and Yarmush, 1%6) and to transpon problems in slab geometry (Aronson, 1970). For the thermal infrared case, the general analytical expression reduces to a simple formula , which depends only on the zenith angle. The formula is incorporated in the MOM together with analytical expressions of the transmission and reflection operators following Liu (1990). With the aid of these formulations, expressions are derived as parameterizations of the scattering effects of clouds in non·scattering radiative transfer models by a modification of the emissivity and transmittance of clouds. The accuracy is beller than 0.5 % in the 11.5 IJ.m window region for clouds of arbitrary optical depths. Zusammenfassung Allgemeine Formulierung der Strahlungsquellfunktion cineremittiercnden und streucndcn Atmosphare mitt cis der Matrix-Operator-Methodc Eine analytische Formulierungder Strahlungsquellfunktion einer cmittiercnden und streuenden Atmos­ phare wird zur Anwendung in der Matrix-Operator-Methode (MOM) hergeleitet. Eine ahnliche Formu­ lierung findet man bei Aronson und Yarmush (1966), die den Durchgang von Gammastrahlen und Neutronen behandeln. FOr das thermische Infrarot reduziert sich der allgemeine Ausdruck auf eine einfache Form, nur abhangig vom Zenitwinkel. Die Gleichung wird zusammen mit den analytischen AusdrOcken fOr die Transmission und ReOexion nach Liu (1990) in die MOM eingefOhn. Damit ki)nncn die Streueffekte in Wolken durch eine Modifikation der Emissivitat und TransmissiviUit der Wolken parametrisien werden. Das hat zur Folge, daG mit einem Modell fOr nicht-streuende Wolken gerechnet werden kann. Die Genauigkcit dieser Parametrisierung wird an mehreren Beispielen gezeigt.

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