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.
Highlights
Surface temperature retrieval, including LST and SST, from space-based imagery began at the dawn of 1970s [1,2,3]
The fundamental physics for surface temperature (ST) retrieval from space-based imagery, in the scope of this paper, is limited to the local thermal equilibrium (LTE) assumption of the concerned Earth-surface and atmosphere. i.e., the IR absorption is balanced by the spontaneous thermal emission and the spontaneous emission of an object above 0 K is determined by its temperature
Specifications on the Surface-Leaving Radiance In Equation (2), comparing with surface thermal emitted radiance, RT simulations indicates that at the surface level, the solar incident radiance is negligible in the TIR range 8–14 μm [26], while in the Medium Wavelength Infrared (MWIR) window 3.5–4.2 μm, the solar incident radiance is of the same order in magnitude as the surface thermal radiance [26]. i.e., for clear sky solar radiance in the radiative transfer equation (RTE) (1) and (2) over 3–14 μm, only the MWIR range should be considered
Summary
Surface temperature (hereinafter abbreviated to ST in text and denoted by Ts in formulae and use LST and SST to distinguish land ST and sea ST respectively) retrieval, including LST and SST, from space-based imagery began at the dawn of 1970s [1,2,3]. The correlated k-distribution method works well under the conditions of LTE but fails to accurately reproduce band radiance in the case of non-LTE, whereas some of the bands (3.3 μm , 4.3 μm , 6.3 μm , 9.6 μm ) in the considered spectral domain of this paper (3–14 μm) are subject to this phenomenon (the effects for at least 4.3 μm are noticeable in nadir geometry). This problem was first solved for stellar atmospheres [10] and transferred to planetary atmospheres [11].
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