基于模式识别算法的高频瑞雷波频散曲线非线性反演研究
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摘要
利用瑞雷波推断地下介质结构和岩土力学参数,已在研究地球内部结构、地壳及地幔物质组成、大地构造和地震灾害预测等区域及全球地震学领域、近表面地球物理工程和超声无损检测等领域中获得了广泛应用。特别是在近表面地球物理研究中,由于高频瑞雷波勘探具有快速、轻便、无损、衰减小、抗干扰能力强、浅层分辨率高且不受各地层速度关系的影响等优点,已在实际生产和科研中得到了广泛应用。由于瑞雷波理论的复杂性及重要性,近十年来,参与瑞雷波勘探技术研究的人越来越多。尤其是最近几年,国内外已掀起了瑞雷波勘探技术的研究热潮。可以预料,瑞雷波勘探将会成为本世纪浅层或超浅层地球物理勘察和工程质量无损检测的重要手段之一。
瑞雷波频散曲线的反演解释是瑞雷波勘探技术的关键步骤之一,通过反演频散曲线可直接进行地质分层和获取重要的岩土力学参数——横波速度。和大多数其它地球物理反演问题一样,瑞雷波频散曲线的反演是一典型的高度非线性、多参数、多极值的地球物理反演问题和最优化控制问题。目前现有的局部线性化方法,其反演结果的可靠性严重依赖于初始模型的选取,只有当初始模型接近真实模型时才能获得较好的反演结果,否则将很容易陷入局部极小值中得出错误的地质解释;同时该方法还需要计算偏导数信息,雅可比矩阵的求取精度也将直接影响反演结果的质量。现有的非线性全局优化算法,如遗传算法(GA)和模拟退火(SA),用于瑞雷波反演是相当有用的,但由于它们本质上是对可行解空间进行随机取样,执行的是非确定性的随机搜索过程,全局最优解的获取需要多次成功的独立试验后(例如20次)加以平均,且并不一定保证每次反演总能搜索到全局最优解。所以计算工作量大、收敛速度慢、计算时间长,在实际应用中受到了限制。
鉴于此,本文围绕瑞雷波反演技术中存在的主要问题,结合代数拓扑学、应用数学和最优化理论等前沿交叉学科的最新研究成果,研究了基于模式识别算法的高频瑞雷波频散曲线非线性反演方法。本文研究的模式识别算法是一类确定性的全局优化算法,其反演机理可以概括为:(1)在每一次迭代时,算法通过当前模式搜索足够的方向以确保一个全局最优下降方向最终能被识别。(2)算法采用合理的反向追踪机制可对试验步长进行有效控制,避免了模式识别过程中过长或过短的试验步长,从而更加有利于算法向最有前景的全局最优点运动。(3)算法特有的隐含弹性网格结构,使其具有隧道效应特征,从而可使算法有效地逃逸局部极小值的吸引。因此,模式识别算法可有效地克服上述瑞雷波反演问题。
本文以“基于模式识别算法的高频瑞雷波频散曲线非线性反演研究”这一策略为核心问题进行了深入地研究与探讨,取得了以下主要研究成果:
1、调研、总结了瑞雷波勘探技术和模式识别算法的研究现状;指出了当前用于瑞雷波反演的局部线性化方法和非线性全局优化算法所存在的不足;分析了模式识别算法用于高频瑞雷波频散曲线非线性反演的可行性和优越性。
2、深入研究了一种新的确定性全局优化算法——模式识别算法。深入研究并总结了模式识别算法的基本原理、反演机制、算法实施及反演的一般流程;针对其在实际反演中的具体情况,探讨了算法实施过程中的一些关键技术问题。如初始网格步长和模式的选取、扩展因子和收缩因子的选取、完全预测和完全识别策略的执行,以及算法终止准则等问题。
3、执行了有效的数值模拟测试。本文首先通过一个高度非线性、多参数、多极值的数学函数“Rastrigin”(有100个极大值和90个极小值)具体测试、分析了模式识别算法的执行效率,并通过反复试验获得了最优的反演参数。然后用另外18个著名数学函数进一步检验了反演参数选取的合理性、算法的有效性和程序的正确性。这些测试函数不仅具有高度非线性、多参数、多极值的特点,而且目标函数形态各异,因此能有效地模拟高频瑞雷波频散曲线非线性反演问题。测试结果表明:模式识别算法是一种不依赖于初始模型的确定性全局优化算法,不仅计算精度高,而且由于算法执行的是确定性的模式识别过程,最优解一次计算便可完成,而不需要多次独立试验后取平均值,所以计算时间短。
本文数值模拟测试后建议:在最优化反演过程中,初始网格步长取为1、扩展因子取为2、收缩因子取为0.5、选取最大正基模式、同时采用完全预测和完全识别策略,则在多次模式识别后最终迭代网格步长将会近似收敛到零,此时一般将会获得较好的反演结果。
4、实现了快速稳定的多模式瑞雷波频散曲线正演模拟算法。由于瑞雷波反演的大部分时间都将用在正演模拟计算上,因此正演是反演的前提和瓶颈,成为提高反演速度和反演精度的关键因素之一。而以往的瑞雷波频散曲线正演模拟算法(如Haskell算法),不仅计算速度慢,而且易出现高频数值精度丢失和高频数值溢出等不稳定现象。为此,本文分析并实现了快速矢量传递算法,该方法是建立在快速δ矩阵算法基础之上的,不仅表达形式简单、易于编程实现,而且能有效地克服上述问题。例如,利用本文给出的模型测试结果表明:同等条件下快速矢量传递算法比Haskell方法计算速度约快40%左右;Haskell算法在频率4000Hz左右出现了高频数值精度丢失问题;Menke算法在频率24kHz左右即出现了高频数值溢出问题:而快速矢量传递算法在频率高达40kHz仍未出现高频数值精度丢失和高频数值溢出等不稳定现象。该算法的高频数值稳定性不仅对近表面研究具有重要的意义,而且对超声表面波在无损检测中的实际应用(如混凝土质量检测、裂缝探测、超声探伤等)也具有重要的研究价值。
5、研究并探讨了多模式表面波叠加耦合机理。多层介质正演模拟结果表明瑞雷波往往表现为多个模式,在实际工作中利用表面波谱分析方法提取的频散曲线往往是一条,而不是理论计算的多条,且在某些公路型等特殊地质结构中频散曲线有时还会出现“之”字形回折现象。为此,本文分析并实现了多模式瑞雷波频散曲线叠加计算方法,并通过三个常见的典型地质模型(递增模型、软夹层模型和硬夹层模型)进行了叠加频散曲线的正演模拟计算,探讨了多模式表面波的叠加耦合机理。其重要的理论与实际意义在于:
(1)能够在理论上对多模式表面波共同叠加作用的耦合机理有一个深入地了解和认识,并能对实测频散曲线中出现的“之”字形回折现象的成因予以解释。利用本文模型研究结果表明:在大多数情况下,基阶波与其它各模式波相比,其能量仍占主导地位,叠加频散曲线实际上就是基阶波频散曲线。而对于某些地下含有明显的软夹层或硬夹层等特殊地质结构,基阶波的能量则可能在某些频段(例如高频段40～100Hz或中频段13～23Hz)处于次要地位,此时叠加频散曲线是基阶波与各高模式波在相应频段相互叠加共同耦合作用的结果,即会出现“之”字形回折现象,从而为前人在实践中总结的利用频散曲线的拐点进行地质分层提供了重要的理论依据。
(2)在已知地质信息较少的情况下,通过叠加频散曲线正演模拟计算能够有利于指导我们对实测频散曲线进行合理的地质分层,尤其是对于“之”字形频散曲线。
(3)为多模式表面波的反演提供了另一个有潜力的、可能有效的反演途径。使用叠加频散曲线进行反演可能会避免多模式表面波反演中易犯的正常模式误识别的错误和需要指定各模式数据权重的困扰。
6、成功进行了大量理论模型试算。(1)为了进一步检验模式识别算法程序的正确性和该算法反演瑞雷波频散曲线的有效性(计算精度和计算时间),本文针对瑞雷波在实际近表面应用中经常遇到的典型地质结构,设计了大量有代表性的地质模型,并用模式识别算法进行了理论模型试算,测试了该算法反演瑞雷波频散曲线的可靠性。(2)由于实测数据不可避免地会含有噪声,为了验证算法的稳定性,作者还对其抗噪能力进行了检验。(3)考虑到实际表面波相速度反演还常受许多因素干扰,本文进一步探讨了频带范围、纵波速度和密度中的误差、频点数和初始模型、层数和层厚度对模式识别算法执行效率的影响。(4)为了进一步演示模式识别算法的优越性,本文还将其与两种常用的非线性全局优化算法(GA和SA)进行了对比测试。本文理论模型试算结果表明:模式识别算法是一种不依赖于初始模型的确定性全局优化算法,不仅计算精度高、具有较强的抗干扰能力(稳定性好),而且由于算法执行的是确定性的模式识别过程,最优解一次计算便可完成,所以计算速度快。例如,在同等条件下,模式识别算法计算速度一般比GA快90倍左右,比SA快80倍左右,能有效地用于高频瑞雷波频散曲线非线性反演。
7、成功基于模式识别算法执行了多模式表面波联合反演研究。考虑到在某些含低速夹层等特殊地质结构中,高阶波较基阶波在某些频段(如高频段)有时可能会更加发育,此时利用多模式表面波联合反演是必要的。为此,本文首先通过一个典型的地质模型,分析了多模式瑞雷波的敏感性;然后通过模式识别算法进行了多模式表面波联合反演研究,校验了理论模拟结果的正确性,并探讨了充分利用多阶表面波联合反演的优点。研究结果表明:(1)对于本文给出的地质结构,深层介质基阶波敏感性一般仅分布在11Hz左右较窄的频带范围内,且在深层(例如第三层和第四层)还会出现敏感性重叠现象,从而降低了基阶波深层敏感性和分辨率,易使反演出现多解性。(2)高阶波敏感性则分布在较宽的频带范围内,且基本不会出现深层敏感性重叠现象。(3)随着模数的增高,表面波峰值敏感性会向高频方向移动,且敏感性分布得更宽、相互分离得更好。(4)高阶波较基阶波对地质结构具有更强的敏感性,充分利用高阶波不仅可使反演过程稳定、提高横波速度的反演精度和计算结果的可靠性,而且还能够提高模型纵向分辨率。(5)利用的高阶波越多,计算结果的可靠性就越高,分辨能力就越强(高阶模式数据比低阶模式数据具有更强的分辨能力)。
8、执行了有效的典型实例分析研究。通过一个来自某高速公路路基上的典型实例反演研究,进一步检验了模式识别算法的有效性和实用性。
9、实现了与本课题相关的重要程序源代码。具体包括:多模式瑞雷波正演模拟算法(如Haskell算法、Schwab-Knopoff算法、Menke算法和快速矢量传递算法):多模式表面波叠加频散曲线正演模拟算法;广义模式识别算法(GPS),遗传算法(GA),模拟退火(SA),人工神经网络(SWIANN,Surface Wave Inversion by Artificial Neural Networks);粒子群优化算法(PSO);Occam算法;f-k变换频散曲线提取算法;基于相移法的多模式表面波频散曲线成像工具(Dispersion Curves Imaging Tools,DCIT)及其它相关程序,并编写了相应的显示程序模块。
本文的主要创新点为:
1、在国内外首次将基于最大正基模式的广义模式识别算法(GPS)应用于高频瑞雷波频散曲线非线性反演研究,并获得了成功。本文大量典型的数值试验、理论模型试算、抗噪能力检验以及与其它全局优化算法对比分析等均证明了GPS算法具有明显的优势,不仅反演精度高,而且计算速度快。本文的研究同时也为地球物理非线性反演和其它全局优化控制领域提供了另一个重要的研究方向——开发研究确定性的全局优化算法,以提高算法的执行效率。
2、在国内外首次提出了基于GPS算法的多阶表面波联合反演策略,并通过广义模式识别算法证明和演示了多模式瑞雷波联合反演不仅能使反演过程稳定、提高横波速度的反演精度和计算结果的可靠性,而且还能够提高模型纵向分辨率;同时利用的高阶波越多,计算结果的可靠性就越高,分辨能力就越强。
3、深入分析了多模式表面波的相互叠加耦合机理,从而为前人在实践中总结的利用频散曲线的拐点进行地质分层提供了重要的理论依据,并为多模式表面波联合反演提供了另一个有潜力的发展方向。
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 model
3. 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.
瑞雷波频散曲线的反演解释是瑞雷波勘探技术的关键步骤之一,通过反演频散曲线可直接进行地质分层和获取重要的岩土力学参数——横波速度。和大多数其它地球物理反演问题一样,瑞雷波频散曲线的反演是一典型的高度非线性、多参数、多极值的地球物理反演问题和最优化控制问题。目前现有的局部线性化方法,其反演结果的可靠性严重依赖于初始模型的选取,只有当初始模型接近真实模型时才能获得较好的反演结果,否则将很容易陷入局部极小值中得出错误的地质解释;同时该方法还需要计算偏导数信息,雅可比矩阵的求取精度也将直接影响反演结果的质量。现有的非线性全局优化算法,如遗传算法(GA)和模拟退火(SA),用于瑞雷波反演是相当有用的,但由于它们本质上是对可行解空间进行随机取样,执行的是非确定性的随机搜索过程,全局最优解的获取需要多次成功的独立试验后(例如20次)加以平均,且并不一定保证每次反演总能搜索到全局最优解。所以计算工作量大、收敛速度慢、计算时间长,在实际应用中受到了限制。
鉴于此,本文围绕瑞雷波反演技术中存在的主要问题,结合代数拓扑学、应用数学和最优化理论等前沿交叉学科的最新研究成果,研究了基于模式识别算法的高频瑞雷波频散曲线非线性反演方法。本文研究的模式识别算法是一类确定性的全局优化算法,其反演机理可以概括为:(1)在每一次迭代时,算法通过当前模式搜索足够的方向以确保一个全局最优下降方向最终能被识别。(2)算法采用合理的反向追踪机制可对试验步长进行有效控制,避免了模式识别过程中过长或过短的试验步长,从而更加有利于算法向最有前景的全局最优点运动。(3)算法特有的隐含弹性网格结构,使其具有隧道效应特征,从而可使算法有效地逃逸局部极小值的吸引。因此,模式识别算法可有效地克服上述瑞雷波反演问题。
本文以“基于模式识别算法的高频瑞雷波频散曲线非线性反演研究”这一策略为核心问题进行了深入地研究与探讨,取得了以下主要研究成果:
1、调研、总结了瑞雷波勘探技术和模式识别算法的研究现状;指出了当前用于瑞雷波反演的局部线性化方法和非线性全局优化算法所存在的不足;分析了模式识别算法用于高频瑞雷波频散曲线非线性反演的可行性和优越性。
2、深入研究了一种新的确定性全局优化算法——模式识别算法。深入研究并总结了模式识别算法的基本原理、反演机制、算法实施及反演的一般流程;针对其在实际反演中的具体情况,探讨了算法实施过程中的一些关键技术问题。如初始网格步长和模式的选取、扩展因子和收缩因子的选取、完全预测和完全识别策略的执行,以及算法终止准则等问题。
3、执行了有效的数值模拟测试。本文首先通过一个高度非线性、多参数、多极值的数学函数“Rastrigin”(有100个极大值和90个极小值)具体测试、分析了模式识别算法的执行效率,并通过反复试验获得了最优的反演参数。然后用另外18个著名数学函数进一步检验了反演参数选取的合理性、算法的有效性和程序的正确性。这些测试函数不仅具有高度非线性、多参数、多极值的特点,而且目标函数形态各异,因此能有效地模拟高频瑞雷波频散曲线非线性反演问题。测试结果表明:模式识别算法是一种不依赖于初始模型的确定性全局优化算法,不仅计算精度高,而且由于算法执行的是确定性的模式识别过程,最优解一次计算便可完成,而不需要多次独立试验后取平均值,所以计算时间短。
本文数值模拟测试后建议:在最优化反演过程中,初始网格步长取为1、扩展因子取为2、收缩因子取为0.5、选取最大正基模式、同时采用完全预测和完全识别策略,则在多次模式识别后最终迭代网格步长将会近似收敛到零,此时一般将会获得较好的反演结果。
4、实现了快速稳定的多模式瑞雷波频散曲线正演模拟算法。由于瑞雷波反演的大部分时间都将用在正演模拟计算上,因此正演是反演的前提和瓶颈,成为提高反演速度和反演精度的关键因素之一。而以往的瑞雷波频散曲线正演模拟算法(如Haskell算法),不仅计算速度慢,而且易出现高频数值精度丢失和高频数值溢出等不稳定现象。为此,本文分析并实现了快速矢量传递算法,该方法是建立在快速δ矩阵算法基础之上的,不仅表达形式简单、易于编程实现,而且能有效地克服上述问题。例如,利用本文给出的模型测试结果表明:同等条件下快速矢量传递算法比Haskell方法计算速度约快40%左右;Haskell算法在频率4000Hz左右出现了高频数值精度丢失问题;Menke算法在频率24kHz左右即出现了高频数值溢出问题:而快速矢量传递算法在频率高达40kHz仍未出现高频数值精度丢失和高频数值溢出等不稳定现象。该算法的高频数值稳定性不仅对近表面研究具有重要的意义,而且对超声表面波在无损检测中的实际应用(如混凝土质量检测、裂缝探测、超声探伤等)也具有重要的研究价值。
5、研究并探讨了多模式表面波叠加耦合机理。多层介质正演模拟结果表明瑞雷波往往表现为多个模式,在实际工作中利用表面波谱分析方法提取的频散曲线往往是一条,而不是理论计算的多条,且在某些公路型等特殊地质结构中频散曲线有时还会出现“之”字形回折现象。为此,本文分析并实现了多模式瑞雷波频散曲线叠加计算方法,并通过三个常见的典型地质模型(递增模型、软夹层模型和硬夹层模型)进行了叠加频散曲线的正演模拟计算,探讨了多模式表面波的叠加耦合机理。其重要的理论与实际意义在于:
(1)能够在理论上对多模式表面波共同叠加作用的耦合机理有一个深入地了解和认识,并能对实测频散曲线中出现的“之”字形回折现象的成因予以解释。利用本文模型研究结果表明:在大多数情况下,基阶波与其它各模式波相比,其能量仍占主导地位,叠加频散曲线实际上就是基阶波频散曲线。而对于某些地下含有明显的软夹层或硬夹层等特殊地质结构,基阶波的能量则可能在某些频段(例如高频段40～100Hz或中频段13～23Hz)处于次要地位,此时叠加频散曲线是基阶波与各高模式波在相应频段相互叠加共同耦合作用的结果,即会出现“之”字形回折现象,从而为前人在实践中总结的利用频散曲线的拐点进行地质分层提供了重要的理论依据。
(2)在已知地质信息较少的情况下,通过叠加频散曲线正演模拟计算能够有利于指导我们对实测频散曲线进行合理的地质分层,尤其是对于“之”字形频散曲线。
(3)为多模式表面波的反演提供了另一个有潜力的、可能有效的反演途径。使用叠加频散曲线进行反演可能会避免多模式表面波反演中易犯的正常模式误识别的错误和需要指定各模式数据权重的困扰。
6、成功进行了大量理论模型试算。(1)为了进一步检验模式识别算法程序的正确性和该算法反演瑞雷波频散曲线的有效性(计算精度和计算时间),本文针对瑞雷波在实际近表面应用中经常遇到的典型地质结构,设计了大量有代表性的地质模型,并用模式识别算法进行了理论模型试算,测试了该算法反演瑞雷波频散曲线的可靠性。(2)由于实测数据不可避免地会含有噪声,为了验证算法的稳定性,作者还对其抗噪能力进行了检验。(3)考虑到实际表面波相速度反演还常受许多因素干扰,本文进一步探讨了频带范围、纵波速度和密度中的误差、频点数和初始模型、层数和层厚度对模式识别算法执行效率的影响。(4)为了进一步演示模式识别算法的优越性,本文还将其与两种常用的非线性全局优化算法(GA和SA)进行了对比测试。本文理论模型试算结果表明:模式识别算法是一种不依赖于初始模型的确定性全局优化算法,不仅计算精度高、具有较强的抗干扰能力(稳定性好),而且由于算法执行的是确定性的模式识别过程,最优解一次计算便可完成,所以计算速度快。例如,在同等条件下,模式识别算法计算速度一般比GA快90倍左右,比SA快80倍左右,能有效地用于高频瑞雷波频散曲线非线性反演。
7、成功基于模式识别算法执行了多模式表面波联合反演研究。考虑到在某些含低速夹层等特殊地质结构中,高阶波较基阶波在某些频段(如高频段)有时可能会更加发育,此时利用多模式表面波联合反演是必要的。为此,本文首先通过一个典型的地质模型,分析了多模式瑞雷波的敏感性;然后通过模式识别算法进行了多模式表面波联合反演研究,校验了理论模拟结果的正确性,并探讨了充分利用多阶表面波联合反演的优点。研究结果表明:(1)对于本文给出的地质结构,深层介质基阶波敏感性一般仅分布在11Hz左右较窄的频带范围内,且在深层(例如第三层和第四层)还会出现敏感性重叠现象,从而降低了基阶波深层敏感性和分辨率,易使反演出现多解性。(2)高阶波敏感性则分布在较宽的频带范围内,且基本不会出现深层敏感性重叠现象。(3)随着模数的增高,表面波峰值敏感性会向高频方向移动,且敏感性分布得更宽、相互分离得更好。(4)高阶波较基阶波对地质结构具有更强的敏感性,充分利用高阶波不仅可使反演过程稳定、提高横波速度的反演精度和计算结果的可靠性,而且还能够提高模型纵向分辨率。(5)利用的高阶波越多,计算结果的可靠性就越高,分辨能力就越强(高阶模式数据比低阶模式数据具有更强的分辨能力)。
8、执行了有效的典型实例分析研究。通过一个来自某高速公路路基上的典型实例反演研究,进一步检验了模式识别算法的有效性和实用性。
9、实现了与本课题相关的重要程序源代码。具体包括:多模式瑞雷波正演模拟算法(如Haskell算法、Schwab-Knopoff算法、Menke算法和快速矢量传递算法):多模式表面波叠加频散曲线正演模拟算法;广义模式识别算法(GPS),遗传算法(GA),模拟退火(SA),人工神经网络(SWIANN,Surface Wave Inversion by Artificial Neural Networks);粒子群优化算法(PSO);Occam算法;f-k变换频散曲线提取算法;基于相移法的多模式表面波频散曲线成像工具(Dispersion Curves Imaging Tools,DCIT)及其它相关程序,并编写了相应的显示程序模块。
本文的主要创新点为:
1、在国内外首次将基于最大正基模式的广义模式识别算法(GPS)应用于高频瑞雷波频散曲线非线性反演研究,并获得了成功。本文大量典型的数值试验、理论模型试算、抗噪能力检验以及与其它全局优化算法对比分析等均证明了GPS算法具有明显的优势,不仅反演精度高,而且计算速度快。本文的研究同时也为地球物理非线性反演和其它全局优化控制领域提供了另一个重要的研究方向——开发研究确定性的全局优化算法,以提高算法的执行效率。
2、在国内外首次提出了基于GPS算法的多阶表面波联合反演策略,并通过广义模式识别算法证明和演示了多模式瑞雷波联合反演不仅能使反演过程稳定、提高横波速度的反演精度和计算结果的可靠性,而且还能够提高模型纵向分辨率;同时利用的高阶波越多,计算结果的可靠性就越高,分辨能力就越强。
3、深入分析了多模式表面波的相互叠加耦合机理,从而为前人在实践中总结的利用频散曲线的拐点进行地质分层提供了重要的理论依据,并为多模式表面波联合反演提供了另一个有潜力的发展方向。
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 model
3. 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.
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