A Grey System Approach for Estimating the Hölderian Regularity with an Application to Algerian Well Log Data

Abstract

The Hölderian regularity is an important mathematical feature of a signal, connected with the physical nature of the measured parameter. Many algorithms have been proposed in literature for estimating the local Hölder exponent value, but all of them lead to biased estimates. This paper attempts to apply the grey system theory (GST) on the raw signal for improving the accuracy of Hölderian regularity estimation. First, synthetic logs data are generated by the successive random additions (SRA) method with different types of Hölder functions. The application on these simulated signals shows that the Hölder functions estimated by the GST are more precise than those derived from the raw data. Additionally, noisy signals are considered for the same experiment, and more accurate regularity is obtained using signals processed using GST. Second, the proposed technique is implemented on well log data measured at an Algerian exploration borehole. It is demonstrated that the regularity determined from the well logs analyzed by the GST is more reliable than that inferred from the raw data. In addition, the obtained Hölder functions almost reflect the lithological discontinuities encountered by the well. To conclude, the GST is a powerful tool for enhancing the estimation of the Hölderian regularity of signals.