New estimation methodologies for well logging problems via a combination of fuzzy Kalman filter and different smoothers

New smoothers are proposed in this article to estimate the spectrum of radioactivity of Gamma ray in well-log data problems. Natural gamma-ray tool (NGT) is one of the most important instruments in well logging. The natural gamma-ray tool detects changes in natural radioactivity emerging from the variations in concentrations of elements as uranium (U), thorium (Th), and potassium (K). A computer is utilized to convert the data obtained by NGT to show concentrations of the elements in various depths. The aim of this research is to estimate the concentrations of these elements. Regular techniques have some uncertainties in the estimation of the elements of reservoir evaluation. To solve this problem, a fuzzy Kalman filter is designed to overcome uncertainty in comparison with the ordinary Kalman filter. Although the estimation result is better than Kalman filter, there is still a high level of noise on the estimation. Therefore, two types of smoothers including Rauch-Tung-Striebel and wavelet smoothers are used to reduce the noise level. Results show the better performances of the suggested smoothers in comparison with fuzzy Kalman filters.