Pattern Search Algorithms For Nonlinear Inversion Of High-frequency Rayleigh Wave Dispersion Curves

Inverting subsurface geologic structures and geotechnical parameters by Rayleigh waves, has played an important role in studying regional and global seismology (e.g., earth's internal structure, crust-mantle composition, crustal movements, and prediction of earthquake harzards), near-surface geophysical engineering, and ultrasonic nondestructive testing. Because of their portable, flexible, nondestructive, smaller attenuation, stronger immunity of interference, cost effective, and high-resolution characteristics, as well as not limiting by subsurface velocity properties, high-frequency Rayleigh waves have been used increasingly as an appealing tool to service near-surface site investigations for academic reasearch and practical engineering applications. In the last decade numemous researchers have been devoted to exploiting and utilizing Rayleigh wave techniques. Especially, in recent a few years there is a great deal of interest in surface wave studies worldwide. Predictably, Rayleigh waves will have a prevailing trend as one of the most powerful approaches in shallow geophysical surveys and non-invasive testing in the near future.Inversion of Rayleigh wave dispersion curves is one of the key steps in surface wave analysis to obtain a shallow subsurface shear (S)-wave velocity profile. However, inversion of high-frequency surface waves, as with most other geophysical optimization problems, is typically a highly nonlinear, multiparameter, and multimodal inversion problem. Consequently, traditional local search methods, e.g. 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 and the accuracy of the partial derivatives (Jacobian matrix). Existingly commonly used global optimization techniques, such as genetic algorithms (GA) and simulated annealing (SA), have proven to be quite useful for determining S-wave velocity profiles from dispersion data but are computationally quite expensive due to their randomness and slower convergence in GA and SA nondeterministic search procedures.In order to effectively overcome the above described difficulties, the author proposed a high-frequency Rayleigh wave dispersion curve inversion scheme based on pattern search algorithms. The proposed approaches, based on algebraic topology, applied mathematics, and optimization theory, are a class of deterministic algorithms for global optimizations. Their inversion mechanisms can be summarized briefly as follows: (1) At each iteration, they look in enough directions to ensure that a suitably good descent direction will ultimately be considered. (2) They possess a reasonable back-tracking strategy that avoids excessively long or short trial steps, which helps exploratory moves implemented by the proposed algorithm to discover the most promising domain in the solution space for a good valley. (3) Pattern search methods have exciting features of tunneling effects because of their underlying elastic lattice structure, which make them less likely to be trapped by spurious local minimizers.Main contributions of PhD Dissertation "Pattern search algorithms for nonlinear inversion of high-frequency Rayleigh wave dispersion curves" are as follows:1. An overview of Rayleigh wave exploration techniques and pattern search algorithms is presented in this dissertation, together with disadvantages and drawbacks of local search methods and global optimization techniques currently used for surface wave analysis. Feasibilities and advantages of pattern search algorithms for nonlinear inversion of high-frequency Rayleigh wave dispersion curves are further analyzed.2. A class of novel deterministic algorithms for global optimization, called pattern search algorithms, is proposed. The principles, inversion mechanisms, and inversion flows of pattern search algorithms are described in detail here. The settings of the key control parameters of the proposed methods, such as initial mesh size and pattern, expansion factor and contraction factor, complete poll and complete search strategies, and termination criteria are further discussed in this dissertation for practical applications.3. The performance of the pattern search algorithm is tested through extensive numerical experiments on some well-known functions characterized by high nonlinearity, multiparameter, and multimodality, such as "Rastrigin" characterized by 100 maxima and 90 minima and other functions listed in the appendix. Also, the behavior of tested objective functions varies, which is proven to be very efficient to simulate nonlinear inversion of high-frequency Rayleigh wave dispersion curves. Modeling results demonstrate that pattern search algorithms applied to highly nonlinear inversion problems should be considered good not only in terms of accuracy but also in terms of computation effort due to their global and deterministic search process.Numerical experiments of this study suggest that during the optimization procedure, an optimal solution can be achieved when the final mesh size converges to approximately zero by setting the initial mesh size at 1, expansion factor at 2, contraction factor at 0.5, and by choosing maximal positive basis pattern, as wll as complete poll and complete search strategies.4. A fast and stable algorithm for computing multimode surface wave dispersion functions has been successfully developed. Since most of the computational time is spent in the calculation of those forward problems in function evaluations, forward modeling has a great effect on Rayleigh wave dispersion curve inversion not only in terms of accuracy but also in terms of computational cost. However, traditional approaches for dispersion function computations, such as Haskell's scheme, not only undergo the high computational effort, but also suffer from computational difficulties associated with numerical overflow and loss of precision at high frequencies. By extracting the best computational features of the various methods, the author develops a simple and efficient algorithm, called the fast vector-transfor algorithm, to overcome the above described difficulties. The approach developed in this research, based on the fast delta matrix algorithm, is a significant improvement over traditional approaches. As a bonus, the fast vector-transfor algorithm is easy to program and slightly faster than others. For example, the algorithm described here is about 40 per cent faster than the Haskell's method for computing dispersion images of size 200 x 200 pixels on a PC586 using the model designed in this dissertation. Catastrophic precision loss occurs in the Haskell's scheme at a frequency 4000 Hz or so, and the Menke's approach suffers from disastrous numerical overflow at a frequency 24k Hz or so. However, the fast vector-transfor algorithm puts to rest numerical instability problems associated with numerical overflow and loss of precision over a high frequency 40k Hz, which plays a significant part in both near-surface investigations and ultrasonic nondestructive testing, such as testing of high strength concrete and detection of surface breaking cracks in concrete.5. The coupling mechanism of multimode surface waves is theoretically analyzed by three synthetic models commonly encountered in near-surface site investigation to compare between effective phase velocities extracted by the Spectral Analysis of Surface Waves (SASW) method and modal phase velocities modeled by three multilayer models. The main lessons learned from the numerical simulations are as follows:(1) Modeling results from this research provide a powful insight into mode superposition of multimode phase velocities and zigzag dispersion curves. For a normally dispersive profile (a model with stiffness increasing with depth), the fundamental mode is strongly predominant on higher modes at every frequency, and hence the effective phase velocity (the superposed dispersion curve) is practically coincident with the fundamental mode one. However, for an inversely dispersive profile (such as, a model with a soft layer trapped between two stiff layers or a model with a stiff layer sandwiched between two soft layers), the fundamental mode is not dominant at high frequencies (such as 40-100 Hz) or at middle frequencies (such as 13-23 Hz), and more modes participate to the definition of the wavefield with increasing frequency. In this case the effective phase velocity is not monotonically decreasing with increasing frequency but is a combination of the individual mode phase velocities at correspongding frequencies, which accounst for the mechanism of zigzag dispersion curves, that is to say, zigzag dispersion curves arise from mode superposition, and provides a theoretical support for previous rules of thumb.(2) In practice, geologic information may not be always a priori known in surface wave studies. In such situations utilization of superposed dispersion curves will help us to reasonably interpret the picked dispersion data, especially for zigzag dispersion curves.(3) Using the superposed dispersion curve to invert surface wave data may provide an efficient strategy for joint inversion of multimode Rayleigh wave dispersion curves, which may potentially avoid a possible pitfall associated with misidentification of normal modes and difficulties associated with assigning different weighting dependent on multimode data accuracy.6. Testings and analyses of synthetic earth models: (1) To examine and evaluate the calculation efficiency and stability of the pattern search algorithm for nonlinear inversion of high-frequency Rayleigh wave dispersion curves, a variety of synthetic earth models are used. These models are designed to simulate situations commonly encountered in shallow engineering site investigations. (2) Since the real data are inevitably noisy, the proposed inverse procedure is applied to nonlinear inversion of fundamental-mode dispersion curves with different levels of random noise for evaluating its immunity of noise. (3) Effects of the reduction of the frequency range of the considered dispersion curve, estimated errors in P-wave velocities and densities, the number of data points and the initial S-wave velocity profile, as well as the number of layers and their thicknesses on inversion results are also investigated in the present study to further evaluate the performance of the proposed approach. (4) A comparative test with two commonly used global optimization techniques (such as, GA and SA) is also undertaken, to further highlight the merit of the proposed algorithm. Results from synthetic models demonstrate that pattern search algorithms applied to nonlinear inversion of high-frequency Rayleigh wave dispersion curves should be considered good not only in terms of accuracy but also in terms of computation effort due to their global and deterministic constructions. It is important to point out that all of the final solutions in the current tests are determined by one computation instead of the average model derived from multiple trials because pattern search process is deterministic. This advantage greatly reduces the computation cost. For example, the proposed method is nearly 90 times faster than GA, and 80 times faster than SA using a five-layer earth model. The approach described here is shown to be very efficient and robust for surface wave analysis.7. A joint inversion of multimode surface waves is successfully implemented by the pattern search algorithm. Considering that higher modes possess significant amounts of energy at higher frequencies and contribution of higher modes tends to become more significant especially in the presence of a low velocity layer, only by combining the fundamental and higher-mode surface waves, can a final desired model be accurately obtained. Sensitivities of multimode surface waves are first analyzed by a typical geologic model. Theoretical results and advantages of fully exploiting intrinsic multimodal properties of Rayleigh waves are then tested and investigated by a joint inversion of multimode surface waves based on the proposed algorithm. Our modeling results support at least five quite exciting surface wave properties: (1) The sensitivities of the fundamental mode are concentrated in a very narrow frequency band around 11 Hz. In particular, the sensitivities for the third and fourth layers overlap to a considerable extent, which indicates that there would be much ambiguity of uniqueness when reconstructing model parameters of these two layers in an inversion that employs the fundamental mode data alone. (2) The sensitivities of higher modes are distributed over a wide frequency band with a better separation for sensitivities of the third and the fourth layers. (3) The peak sensitivity shifts to higher frequency with increasing mode number. In addition, the sensitivities are distributed over a wider frequency band, and the separation between peaks becomes better. (4) Higher modes are relatively more sensitive to fine changes in S-wave velocity than the fundamental mode. The inversion process can be stabilized; the ambiguity can be reduced, as well as the accuracy and resolution of the S-wave velocity model can be improved when simultaneously inverting the fundamental and higher mode data. (5) The more exploited higher mode, the higher the accuracy and resolution of the deduced model (Higher mode data generally possess higher data resolving power than the lower mode data).8. The effectiveness and practicability of the proposed pattern search algorithm are verified through a representative real example from some pavement system.9. All of significant source codes involved in this research have been developed successfully. These codes include: Algorithms for forward modeling of multimode Rayleigh wave dispersion curves (such as the Haskell's approach, Schwab-Knopoff's method, the Menke's algorithm, and the fast vector-transfor algorithm); Algorithms for forward modeling of superposed dispersion curves; Generalized Pattern Search (GPS), Genetic algorithms (GA), Simulated annealing (SA), Surface Wave Inversion by Artificial Neural Networks (SWIANN), Particle Swarm Optimization (PSO); Occam's algorithm; frequency-wavenumber domain analysis for reconstruction of dispersion curves; Dispersion Curves Imaging Tools (DCIT) based on the phase-shift approach, and so on.The original contributions in this dissertation are as follows:1. The proposed inverse scheme is the first successful attempt to invert high-frequency Rayleigh wave dispersion curves for near-surface site characterization by pattern search algorithms (GPS) worldwide. Results from both numerical simulations and synthetic models demonstrate that pattern search algorithms applied to nonlinear inversion of high-frequency Rayleigh wave dispersion curves should be considered good not only in terms of accuracy but also in terms of computation effort due to their global and deterministic constructions. In particular, the scheme described here provides an encouraging direction for geophysicalnonlinear inversion and global optimizations in other fields-to develop deterministicalgorithms for global optimizations.2. The present study is the first attempt to invert multimode surface waves by pattern search methods. Results show that the inversion process can be stabilized, the ambiguity of the deduced model can be reduced, as well as the accuracy and resolution of the S-wave velocity model can be improved when simultaneously inverting the fundamental and higher mode data. Moreover, the more exploited higher mode, the higher the accuracy and resolution of the deduced model3. This research provides quite valuable insights into the coupling mechanism of multimode surface waves. Results not only give a theoretical support for previous rules of thumb but also propose a potential scheme for joint inversion of multimode Rayleigh wave dispersion curves.