A Method to Calculate the Temperature Error Amplitude in Temperature/Emissivity Retrieval

3. Departamento de Engenharia Elétrica/UFRGS, Brazil

4. Programa de Pós-Graduação em Engenharia Civil (PPGEC), Laboratório de Saneamento Ambiental, UNISINOS, Brazil

Abstract: Thermal infrared (TIR) data are collected by orbital sensors in order to analyze targets on the Earth’s surface in a local or global scale. The radiance captured by the sensors in the TIR is dependent on two physical quantities: temperature (T) and emissivity (ε) of the surface. In this case, Planck’s law, which describes the relation between these variables, leads to an equation without a single solution. Many methods have been proposed in the last decades and each technique has a set of restrictions that must be observed in order to generate reliable results. In this paper, we present the Cauchy-Schwarz Inequality Temperature Based (CSI-TB) algorithm to T and ε retrieval. This new approach allows calculating both the range and the standard deviation of the errors related to temperature estimation. Initially, the problem of separating the T and ε from radiance data will be treated as a comparison between vectors. The Cauchy-Schwarz inequality (CSI) is applied to sort the spectral similarity between vectors. The vectors are formed by radiance data from the sensor and reference data. The reference data are formed by a database (DB) of a given target with spectral signatures of radiance measured at different temperatures. Thus, the first estimate for T will be the temperature corresponding to the most similar spectrum of the DB, with proportional error to the differences between the temperatures of the DB. In the second step of the algorithm, linear regression is applied in the parameters for a 2nd degree polynomial between the results from CSI (ordinate axis) and temperatures (abscissa axis). In this case, the final estimate for T will be the abscissa of the vertex of the 2nd degree polynomial generated by the regression. The inclusion of this step allows obtaining more accurate estimates for the T when comparing the estimates of the first stage. The algorithm was tested in simulated radiance, temperature and emissivity data in which the target of interest is the quartz mineral, since it has a known spectral signature, associated with the Si-O bond in the TIR region. The simulated sensor was the TIR/ASTER subsystem onboard the EOS-Terra satellite. The results of the simulations obtained performance within the theoretical limits predicted by the method.