The one-dimensional radiative transfer equation: Derivation of a crosswise layer solution
Sievers, Uwe J.; Panhans, Walter-Georg K.
Meteorologische Zeitschrift (2026)
published online: Sep 1, 2026
manuscript accepted: Jun 22, 2026
final revised version received: Jun 19, 2026
manuscript revision requested: Mar 17, 2026
manuscript received: Jan 13, 2026
DOI: 10.1127/metz/20263
Open Access (paper may be downloaded free of charge)
Abstract
This paper deals with the one-dimensional spectral Radiative Transfer Equation (RTE) of the atmosphere in its discrete approximation. The aim is, to contribute to the efficiency of its numerical calculation. This intention shall be reached by investigating the upward-downward symmetry of the equation and making use of its consequences. First, it is shown that the matrix exponentials of the analytic solution decompose into a 2 × 2 block structure with the four submatrix functions forming two pairs connected by upward-downward symmetry relations. The interesting aspect thereby is that they all can be expressed analytically in terms of the hyperbolic sine and cosine functions. Thanks to symmetry properties of the decomposed matrix exponentials it is possible to construct special layer solutions where the upward and downward radiation parts at the boundaries of the considered layer are linked crosswise. These Crosswise Layer Solutions (CLS) formally fit the basic equation of the MOM (Matrix Operator Method) Doubling procedure. This agreement opens the possibility to replace the Transmission and Reflection matrices, calculated with the Doubling procedure, by the corresponding analytic expressions of the CLS. The expectation is that such an eventual replacement would result in an accuracy gain by less computational effort.
Keywords
RTE • Radiative Transfer • Doubling • MOM • Transmission • Reflection