Shear-Wave Splitting Analysis Using Optimization Algorithms

AbstractShear-wave splitting (SWS) analysis is used to predict fractures in subsurface media. Specifically, two parameters relevant to SWS analysis (the azimuth of the fast shear wave and the time delay between the fast and slow shear waves) are used to quantify the main azimuth and degree of the fracture development, respectively. However, the algorithms of SWS analysis using a grid search have relatively low computational efficiency, as they need to calculate the objective function values of all grid points. To improve the efficiency of SWS analysis, we proposed new algorithms using the gradient descent, Newton, and advance-retreat methods. The new methods use the direction of the fastest gradient descent, the intersection points of the tangent plane of the first-order objective function with the zero plane, and narrowing the range of extremum points to determine the search path. Therefore, this removes the necessity to compare all grid points in the value region. We compared the three methods and the rotation-correlation method, and both synthetic and field data tests indicated that all three methods had higher computational efficiency than the traditional grid search method. Among the proposed methods, the gradient-descent method obtained the most accurate results for both synthetic and field data. Our study shows that SWS analysis combined with the gradient-descent method can accurately and efficiently obtain SWS parameters for fracture prediction

Local Convergence of a Family of Weighted-Newton Methods

This article considers the fourth-order family of weighted-Newton methods. It provides the range of initial guesses that ensure the convergence. The analysis is given for Banach space-valued mappings, and the hypotheses involve the derivative of order one. The convergence radius, error estimations, and results on uniqueness also depend on this derivative. The scope of application of the method is extended, since no derivatives of higher order are required as in previous works. Finally, we demonstrate the applicability of the proposed method in real-life problems and discuss a case where previous studies cannot be adopted