Basin-Edge Effects in Earthquake Modelling

The section establishes the importance of site effects in modifying incoming seismic wave fields. Geological irregularities, particularly soft sedimentary deposits, can significantly amplify and trap seismic energy through processes like bending, scattering, and focusing. When these sediments are laterally confined by a more rigid basement, as in alluvial basins or valleys, the seismic behavior becomes highly multi-dimensional. The interaction of the incoming wave field with basin edges generates horizontally propagating surface waves, increasing the duration of shaking and creating frequency-dependent resonance. This emphasizes the importance of understanding basin geometry and material properties in predicting ground motion. The study highlights how subsurface irregularities locally distort incoming wave fields, creating site effects that are often as crucial, if not more so, than the source processes themselves. Soft sedimentary deposits, especially when laterally confined by rigid basement rock (e.g., in alluvial basins), play a significant role, amplifying and trapping seismic energy via internal reflections. This multi-dimensional behavior significantly increases the duration of ground shaking and creates frequency-dependent resonance.

Early research on the seismic response of sedimentary basins relied heavily on analytical methods. Exact and approximate solutions for displacement response in the time domain were developed for simple basin configurations. The work of Trifunac (1971), providing a closed-form solution for a two-dimensional semi-circular valley in a homogeneous elastic half-space, is highlighted. While Trifunac didn't explicitly identify the response as resonant, subsequent research by Wirgin (1995) derived the associated resonant frequencies. Extensions of this work by Wong and Trifunac (1974) and Sánchez-Sesma and Esquivel (1979) explored more complex basin shapes, though closed-form solutions weren't always attainable. This foundational work emphasizes the importance of simple analytical models for validating more complex numerical techniques used to study basin response. The section highlights the evolution of analytical models in understanding seismic response in sedimentary basins, starting with Trifunac (1971)'s work on semi-circular valleys, and extending to more complex shapes. These analytical solutions, though limited to simplified geometries, are crucial benchmarks for verifying the accuracy of contemporary numerical modeling techniques for large-scale ground motions.

The study delves into multi-dimensional basin resonance, where basins bounded by bedrock exhibit resonance in both vertical and horizontal orientations. This occurs when incoming seismic energy excites appropriate basin frequencies. A substantial body of analytical work exists to determine eigenfrequencies and mode shapes, verified by numerical methods. Rial and Ling (1992) and Wirgin (1995) provide good overviews of this research. Rodríguez-Zúñiga et al. (1995) provide an exact solution for a 2-D rectangular Helmholtz resonator, extending it to 3-D configurations. Other studies, such as that by Zhou and Dravinski (1994), demonstrate that resonant frequencies are independent of impedance contrast, contradicting earlier assumptions. Numerical modeling has been applied to various sedimentary basins worldwide, using both strong motion and weak motion recordings for validation. Early work by Seed et al. (1970) on the Caracas basin exemplifies this approach. More recent studies, like that by Semblet et al. (1999), confirm the presence of two-dimensional amplification patterns. The section explores the complexities of basin resonance, emphasizing both analytical and numerical approaches to understanding the phenomenon. The interplay between basin geometry, material properties (impedance contrast), and input frequencies is highlighted, along with the use of numerical models to simulate observed ground motions.

The research focuses on the basin edge effect, a phenomenon where constructive interference between horizontally propagating surface waves and direct shear waves causes significant ground motion amplification near basin edges. Kawase (1996) first described this effect in Kobe, Japan, attributing damage to interference between Rayleigh waves and direct shear waves. Subsequent studies, including those by Graves et al. (1998) on the Santa Monica basin, further support this hypothesis. Researchers like Motosaka and Nagano (1995, 1996, 1997) and Higashi (2000) have used various numerical methods (finite-element and pseudospectral) to demonstrate this interference effect. Further research investigated the specific surface wave modes involved in this constructive interference using methods like the Thin Layer Method (TLM). The studies highlighted in this section focused on specific events such as the Kobe earthquake (1995) and the Northridge earthquake (1994), illustrating the significant role of basin edge effects. The importance of constructive interference between different wave types and the broad-band nature of the resulting amplification are discussed. The concept of 'focussing effect' is mentioned, suggesting a more general phenomenon. These studies demonstrate the significant role of the basin edge effect in shaping ground motion amplification patterns and causing damage during earthquakes.

The section explores analytical solutions for SH wave propagation at simple geometrical corners, providing insights into amplification at shallow dipping sedimentary basin edges. The focus is on a wedge model approximating a dipping basin edge, with one side fixed to a rigid boundary and the other a free surface. Hudson (1963) provided an approximate solution for a line source in the dipping layer, showing the existence of Love waves for small apex angles. Sánchez-Sesma and Velázquez (1987) offered a more refined analytical solution for a moving rigid lower boundary. These studies demonstrate that maximum amplification occurs a short distance from the edge due to constructive interference, a key finding for ground motion above shallow dipping edges. This section provides valuable theoretical background linking simplified models to the more complex basin edge geometries studied in other parts of the thesis. These analytical solutions provide valuable insights into the mechanics of ground motion amplification at basin edges with varying geometries. The studies of simple wedges illustrate the formation of Love waves and the localization of amplification at basin edges due to constructive interference. The results highlight the importance of SH wave propagation analysis in understanding this phenomenon.

This section introduces the extensive use of finite element methods (FEM) throughout the research to understand seismic response due to subsurface geology. The focus isn't on developing FEM techniques, but using them as a tool to analyze wave propagation at basin edges. The software used for two-dimensional elastic analyses of anti-plane SH wave propagation is Archimedes. This software is applied to both theoretical geological models and real geological cross-sections of the Lower Hutt Valley. The text mentions that the software is used for two-dimensional elastic analyses of anti-plane SH wave propagation, emphasizing the specific type of wave and modeling dimension used. The use of Archimedes, a specific software for finite element analysis, is highlighted, along with its application to theoretical and real-world geological models. The methodology emphasizes the application of existing computational tools rather than the development of new ones, focusing on using FEM to better understand wave propagation at basin edges.

The section discusses two key challenges in using finite element methods (FEM) for modeling seismic motions in unbounded domains. The first is the need to limit the computational domain with an artificial boundary to prevent spurious reflections. Several boundary techniques are mentioned: viscous boundaries (spring-dashpot model by Lysmer and Kuhlemeyer, 1969), transmitting boundaries (Lysmer and Waas, 1972), non-reflecting boundaries (Smith, 1974), and Kosloff's method (Kosloff and Kosloff, 1986; Seki and Nishikawa, 1988). The second challenge involves effectively incorporating seismic excitation into the finite-element mesh. Solutions mentioned include the effective force method, domain decomposition (Bielak and Christiano, 1984; Cremonini et al., 1988), and the use of an internal source (Hisada et al., 1998). This highlights the computational complexities involved in accurately simulating seismic wave propagation using FEM, and demonstrates the utilization of established methods to overcome these obstacles. The section details the use of various boundary conditions to limit reflections in finite element models of seismic waves, including viscous boundaries, transmitting boundaries, and non-reflecting boundaries. Methods for incorporating excitation into the model are discussed, focusing on effective force methods, domain decomposition, and the use of internal sources. These are all crucial for accurate seismic simulation with FEM.

This section describes the mesh generation process within the finite element modeling (FEM) framework, using Triangle, a two-dimensional triangular mesh generator specifically designed for FEM. It utilizes Delaunay triangulation algorithms to create efficient unstructured meshes. The choice of mesh parameters—geometry of layer boundaries, free surfaces, computational boundaries, maximum triangle area, minimum internal angle—is highlighted, along with the option of creating quadratic elements for enhanced computational efficiency. Mesh generation is crucial for achieving accuracy in finite element models. Node spacing is optimized relative to local shear-wave velocities to maintain accuracy and computational efficiency. A maximum node spacing of one-eighth to one-fifth of the shortest wavelength is considered desirable (Kuhlemeyer and Lysmer, 1973; Lysmer et al., 1975), and the use of six-noded quadratic triangular elements is described. This discussion focuses on ensuring model accuracy through careful mesh design and node spacing relative to the shortest wavelength, demonstrating a crucial aspect of computational model design and optimization. The use of Triangle software for mesh generation is highlighted, emphasizing its features like Delaunay triangulation and the option to create quadratic elements for improved computational efficiency. Optimal node spacing relative to shear-wave velocities and wavelength is discussed, ensuring numerical accuracy.

Model verification is performed by comparing the finite element method (FEM) results to analytical solutions for resonant frequencies in a homogeneous rectangular inclusion within a rigid half-space (Bard and Bouchon, 1985). The analytical solution, initially derived for a rigid-base closed system (Helmholtz resonator), is shown to hold for open systems with elastic half-spaces (Zhou and Dravinski, 1994). Although the positions of extremum points are approximate for the open system, they are expected to be close for high impedance contrasts. A specific example is given: a 1000m wide by 100m deep 2-D rectangular basin with a shear-wave velocity contrast of 5, excited by a vertically incident Ricker pulse at 2.0 Hz. The FEM results are compared with analytical predictions from Bard and Bouchon (1985), focusing on the resonant frequencies and verifying the accuracy of the computational method employed. This section demonstrates the validation of the FEM technique by comparing its predictions of resonant frequencies in a simple 2D rectangular basin against a well-established analytical solution. The use of Bard and Bouchon's (1985) work as a benchmark highlights the importance of validating numerical simulations with existing analytical solutions.

This section details the initial investigation of elastic SH-wave propagation at the edge of a two-dimensional semi-infinite homogeneous layer above a half-space. The approach utilizes both geometrical ray-path analysis and wavefront analysis to understand wave behavior. This theoretical approach precedes the use of numerical methods, providing a foundational understanding of wave interactions before moving to more complex numerical models. The analysis reveals that Love waves, generated at the layer edge, create distinctive amplification patterns in both the time and frequency domains. These patterns are categorized into three classes based on their development mechanisms: the Airy-phase edge effect (input frequency near the layer's fundamental frequency) and the wedge effect (above shallow sloping edges). The wavefront analysis provides a crucial theoretical framework for understanding the generation and propagation of Love waves, laying the groundwork for subsequent numerical simulations. The study emphasizes the use of both ray path and wavefront analysis as analytical tools to understand the behavior of SH waves before employing more computationally intensive numerical simulations. The generation of Love waves at the layer's edge and their resulting amplification patterns are identified as key aspects requiring further investigation.

Building upon the wavefront analysis, this part develops mathematical expressions to calculate the transient positions of constructive interferences between refracted wavefronts. The analysis simplifies to vertical incidence and a rectangular edge, and it compares the effects of pulse and harmonic inputs. The resulting expressions help determine the timing and location of constructive interference. Characteristic amplification patterns are proposed, stemming from the edge-generated Love waves. The discussion addresses how the angle of incidence and edge slope significantly influence the wave propagation. The impact of refraction, diffraction, and total internal reflection on wave behavior is detailed, especially in generating the horizontally propagating Love waves. Mathematical expressions for wavefront positions and timing are derived, simplifying the analysis for specific cases of vertical incidence and rectangular edges, providing a more tractable solution to complex wave behavior. The focus on understanding constructive interference helps to predict amplification patterns arising from the edge-generated Love waves, highlighting the theoretical basis for observed amplification.

This subsection analyzes the interaction between edge geometry and wave input characteristics. Constructive interference between undispersed wavefronts is shown to be limited to specific combinations of incidence angles and edge-slope angles. Conversely, constructive interference between dispersed wavefronts (diffracted or totally reflected) is predicted for all incidence angles and edge-slope angles. The analysis highlights the limitations of using simple geometric expressions for predicting constructive interference. It clarifies that such predictions work only for undispersed wavefronts; the more likely scenario of dispersed wavefronts makes accurate predictions far more complex. The role of dispersive effects in modifying constructive edge amplification is highlighted, indicating a need for more comprehensive models. The analysis differentiates between constructive interference of undispersed and dispersed wavefronts, highlighting the limitations of simplistic models. The influence of edge slope angle and the angle of incidence of the incoming wave on wave behavior is investigated. The significant influence of dispersive effects on the observed amplification patterns suggests that a more complete understanding requires the inclusion of dispersive characteristics within the theoretical models.

This section examines the results of finite element modeling (FEM) of a semi-infinite layer subjected to a high dimensionless pulse frequency (ηc = 5). The simulations show the arrival of a discrete Ricker pulse at the surface at successive times, with an amplitude calculated using Equation 6.4 and accounting for free-surface doubling. Well-defined Love waves originate from the edge and propagate horizontally, exhibiting phase velocities close to β2 and β1. The group velocity varies between these values, reflecting higher-mode Love wave characteristics. Sharp displacement peaks near the edge are attributed to interference between the Love waves and vertically arriving waves. The simulation results, for a high dimensionless pulse frequency, demonstrate the generation and propagation of Love waves from the layer's edge. The clear presence of Love waves with distinct phase and group velocities and their interference with vertically arriving waves are highlighted as key observations. The Ricker pulse characteristics and the resulting surface displacements are analyzed, showing the importance of the dimensionless frequency in shaping the wave propagation.

The analysis reveals that the surface displacement response is highly sensitive to changes in input frequency, specifically the ratio of wavelength to layer depth. High input frequencies, low shear-wave velocities, and deep layers produce narrow constructive edge amplification, accurately predicted by the theoretical wavefront analysis. Long-period inputs, stiff sediments, and shallow layers result in edge responses more dependent on the interaction between the dispersive Love wave and layer resonance. The results show a relationship between frequency and the propagation velocity, where high-frequency components travel at speeds closer to the layer velocity (β1), while lower frequencies demonstrate higher-mode Love wave velocities. The simulations clearly show that the ground motion amplification pattern and wave velocities are strongly dependent on the input frequency. High frequencies create narrow amplification patterns, while low frequencies show more interaction with layer resonance. The observed velocity patterns reveal how specific frequencies within the Love waves are amplified, even with a smooth distribution of frequencies in the input Ricker wavelet, highlighting the effects of dispersion curves.

This section examines the relationship between phase velocity and group velocity in the generated waves. A line through the centroids of moving peaks defines the group velocity, while lines along peak trajectories represent phase velocity. These velocities are linked to common frequencies, with an example for the Airy phase frequency (η = 1.01) provided. The study observes that group velocities don't follow linear trajectories from a point source, indicating a frequency-dependent generation of waves near the edge. High-frequency components appear closer to the edge, while lower-frequency energy is generated further away. The analysis of phase velocities and group velocities helps to determine the dispersion characteristics of the observed Love waves. The deviation from linearity of group velocities suggests a frequency-dependent generation of wave energy, with higher-frequency components closer to the edge. This detailed analysis of phase and group velocities contributes significantly to understanding the observed amplification patterns.

The primary constructive interference is identified between phases of the edge-generated Love wave and base-refracted pulses. Amplification further from the edge stems from constructive interference between subcritical reflections of the base-refracted wave. These interferences are highly sensitive to the shape and timing of peaks within the Ricker pulse. While undispersed edge-refracted waves show constructive interference, the associated amplification is generally small, except for high-frequency inputs and steep edge slopes. For high-frequency inputs, the phase velocity of the edge-refracted wave is close to the Love wave phase velocity, allowing rough approximation of peak displacement. For low-frequency inputs, the effects are masked by interference between vertically reflecting wavefronts. This section explores the multiple mechanisms contributing to observed ground motion amplification. The dominant mechanism is identified as the constructive interference between Love waves and base-refracted pulses, while other smaller contributions arise from reflections and refractions. The complex interactions highlight the challenges in creating simple predictive models for all scenarios.

The Lower Hutt Valley case study investigates ground motion amplification in a real-world setting. The valley's subsurface geology consists of Holocene sediments (top 20 meters) overlying the Torlesse Complex basement rock. The Holocene sediments comprise loose sands, silts, and peat, with a gravelly colluvial fan deposit beneath. Shear wave velocities of 168-190 m/s have been measured in the lower valley using a seismic cone penetrometer (SCPT) by Stephenson and Barker (1992). The study uses an array of seismographs deployed along the fault-bounded northwestern edge of the valley to record ground motion data from local and distant seismic events during December 1998 to January 1999. The data collection and processing involved careful instrument siting (by the thesis author), data acquisition by Emma Winthrop and Neal Osborne (as part of an honors project supervised by John Taber), and subsequent data analysis. The Lower Hutt Valley's geological composition, particularly the soft Holocene sediments, plays a crucial role in the study. The location of the seismograph array, along the valley's edge, allows for a targeted investigation of the basin edge effect. The specific characteristics of the Holocene sediments (shear wave velocity, thickness) and their contrast with the underlying basement rock are highlighted. The data-collection methodology is outlined to emphasize the rigor and practical aspects of the study.

The recorded seismic data underwent a rigorous processing pipeline. A 20-second window encompassing the shear wave arrival was selected for each event. Data from station E04, identified as faulty, was visually corrected. Displacement traces were obtained by integrating velocity records using the trapezoidal method, followed by baseline error correction. To minimize contamination from site effects at the reference station E01 (rock site), Fourier amplitude spectra were smoothed using a moving triangular window (1 Hz for E01 and 0.2 Hz for others), combining horizontal components via root-mean-squared (RMS) averaging before calculating the Fourier spectral ratio (FSR). This data processing was crucial for obtaining reliable ground motion amplification estimates. The use of smoothing techniques to minimize the influence of site effects at the reference station emphasizes the importance of data quality control in seismic studies. The detailed description of the data processing steps ensures transparency and allows for better understanding of the data analysis methodology. The method aimed to minimize the contamination of the FSR by site effects at E01 by combining horizontal components using an RMS average and smoothing with a moving triangular window, this is crucial in obtaining reliable results.

Analysis of the time-domain ground motion records shows significant amplitude variations between stations. The peak amplitude generally occurs within the first few pulses of the shear wave arrival (around 2.0-2.2 Hz). The initial arrival of the Ricker pulse through the rock is compared to the arrival through the sediments. The peak amplitude is shown to be influenced by several factors, including time of arrival, phase, and frequency content of pulses. Three distinct amplification zones are identified: two near shallow-dipping bedrock edges and one further into the valley. The largest displacement occurs due to constructive interference between the initial Ricker pulse and an edge-generated phase. Apparent horizontal velocities of edge-generated pulses vary depending on valley-edge geometry (250 m/s in Lower Hutt Central; 730 m/s in Petone). The analysis of the time-domain ground motion reveals significant amplification patterns in the Lower Hutt Valley. Three distinct regions of amplification are identified, with the largest amplification near the fault-bounded edge. This is attributed to constructive interference involving the edge-generated phases, highlighting the importance of edge effects in shaping the observed amplification. The variation in the apparent velocity of the edge-generated pulses based on the valley geometry is discussed.

The study attempts to identify edge-generated surface waves in the recorded data, although this proves challenging compared to numerical modeling. The strong, variable amplification in the valley strongly suggests the presence of edge-generated surface waves. While directly observing surface wave propagation paths is difficult, narrow band-pass filtering helps analyze frequency-dependent wave packet propagation (group velocity). Estimating phase velocities, however, is hindered by the complex mixture of vertically and horizontally traveling waves. Time-variant cross-multiplication is used to estimate Love wave phase velocities, but this method proves unsuccessful due to the dominance of body wave motion and strong resonance in the Holocene sediments. The study explores the methods for estimating the phase velocity and group velocity of surface waves in the Lower Hutt Valley. While the direct identification of surface waves from the recorded data is difficult, the observed amplification patterns strongly suggest their presence. The analysis emphasizes the inherent challenges of interpreting real-world seismic data, where complex wave interactions and site effects can mask subtle features such as surface wave propagation. The study focuses on group velocity estimation, providing insights into the frequency-dependent propagation of wave packets and identifying a possible Airy phase amplification at 2.0-2.5Hz. This is consistent with the theoretical framework developed in earlier sections.