Mathematical Assessment of the Effects of Substituting the Band Radiative Transfer Equation (RTE) for the Spectral RTE in the Applications of Earth’s Surface Temperature Retrievals from Spaceborne Infrared Imageries

Abstract

The Planck’s thermal emission function, the reflectivity-emissivity decoupled Kirchhoff’s law and the associated atmospheric radiative transfer equation (RTE) is a theoretical base for Earth surface temperature (ST) retrievals from spaceborne infrared imageries. The infrared (IR) instruments generally collect band averaged radiance which are usually different from the RT codes simulated spectral one. Although IR band RTE is widely used, the effects of substituting the band-averaged RTE for the corresponding spectral one for those broadband observations (e.g., the Moderate Resolution Imaging Spectroradiometer (MODIS) thermal IR bands) have not been evaluated. In this paper, mathematical analysis and numerical experiments have been conducted to clarify the uncertainties arising from this substitution treatment. Firstly, we present the IR spectral RTE in a concise manner, and then, based on the law of conservation of energy and the integral assumption, a detailed mathematical derivation of the commonly-used IR band RTE has been derived. The significant improvement of the derivation is the validation of the integral assumption, which states that over a small spectral region, the integral of a product is approximately equal to the product of integrals. In the IR spectral region, taking the most significant term of the IR band RTE as an example (i.e., the surface emission term), we confirmed that, for the satellite collected IR signals emitted from the Earth’s surface, over any bandwidth at any band-location and under any instrument spectral response function (SRF), the integral approximation (IA) is a well-founded approximation and thus the IR band RTEs are good approximations for the corresponding spectral ones. Furthermore, in the ST, especially the land ST, product validation investigations, the ST errors introduced by the substituting treatment are negligible and do not need to be taken into consideration.