Investigation of lithological heterogeneities from velocity logs using EMD-Hölder technique combined with multifractal analysis and unsupervised statistical methods

Well logging data are the main support of petrophysical information in petroleum engineering, and as many natural signals, they have deterministic component and noisy component. Conventional denoising methods depend on various filtering parameters, which increase the possibility of error and losing useful information in the signal. This research provides a well-controlled method to reduce noise, based on empirical mode decomposition (EMD) and regularity analysis indexed by the Hölder exponent (in short, EMD-Hölder). First, well velocity logs are handled as noisy signals, and decomposed into intrinsic mode functions (IMFs) for fast oscillations to slow oscillations via EMD, then regularity exponent is computed for each IMFs using wavelet leaders (WL) algorithm. The Hölder exponent (h) is one of the best metrics to quantify the singularity. The value of the Hölder exponent h = 0.5 characterizes the white noise of the analyzed signal, and any value h calculated from an intrinsic modal function IMF less than or equal to a predefined specific threshold, this IMF function is considered as noise, to be discarded while the reconstruction of the new denoised signal. To determine the multifractal properties of P- and S-wave velocities denoted Vp and Vs, respectively, a multifractal analysis was performed using the wavelet leaders (WL). The estimated multifractal parameters: scaling exponent τ (q), multifractal spectrum D(h), singularity strength h, and Hausdorff dimension D are used to quantify the non-stationarity and non-linearity of velocity, followed by an application of unsupervised statistical methods (a hierarchical clustering analysis, HCA, and principal component analysis, PCA) to establish a possible relationship between multifractal parameters and type of lithology (sandstone and clay). It is shown that the width of the multifractal spectra (Δh) estimated from well logs can be used as a lithological indicator of the studied geological formations.