Pattern Search Algorithms for Surface Wave Analysis

In recent years Rayleigh waves have been used increasingly as a nondestructive tool to obtain near-surface S-wave velocity, one of the key parameters in environmental and engineering applications (Miller et al., 1999a, 1999b; Park et al., 1998, 1999; Xia et al., 1999, 2003; Foti et al., 2002; Beaty and Schmitt, 2003; Tian et al., 2003; Zhang and Chan, 2003; Lin and Chang, 2004; O’Neill et al., 2003; Ryden et al., 2004; and Song et al., 2006). Utilization of surface wave dispersive properties may be roughly divided into three steps: field data acquisition, dispersion-curve picking, and inversion of phase velocities (Xia et al., 2004). Inversion of Rayleigh wave dispersion curves is one of the key steps in surface wave analysis to obtain a shallow subsurface S-wave velocity profile (Rix, 1988; Lai et al., 2002). However, inversion of high-frequency Rayleigh wave dispersion curves, as with most other geophysical optimization problems, is typically a nonlinear, multiparameter inversion problem. Dal Moro et al. (2007) have demonstrated the high nonlinearity and multimodality by a synthetic model. Consequently, local optimization methods, e.g. matrix inversion, steepest descent, conjugate gradients, are prone to being trapped by local minima, and their success depends heavily on the choice of a good starting model (Boschetti et al., 1996) and the accuracy of the partial derivatives. Thus, global optimization methods that can overcome this limitation are particularly attractive for surface wave analysis (e.g., Meier and Rix, 1993; Yamanaka and Ishida, 1996; and Beaty et al., 2002). Pattern search algorithms are a direct search method well suitable for the global optimization of highly nonlinear, multimodal objective functions (Lewis and Torczon, 1999). Pattern search algorithms have recently been used and tested to optimize complex mathematical problems characterized by the large numbers of local minima and/or maxima (Lewis and Torczon, 2000). However, few attempts have been made to address real-world geophysical problems, especially for the inversion of surface wave data. In this chapter, In this chapter, I first implemented and tested a Rayleigh wave dispersion curve inversion scheme based on GPS Positive Basis 2N, a commonly used pattern search algorithm. Incorporating complete poll and complete search strategies based on GPS Positive Basis 2N into the inverse procedure greatly enhances the performance of pattern search algorithms because the two steps can effectively locate the promising areas in the solution space containing the global minima and significantly reduce the computation cost, respectively. The calculation efficiency and stability of the inversion scheme are tested on three synthetic models. Secondly, effects of the number of data points, the reduction of the frequency range