Wavenumber Integration Method Research Articles

An Inversion Method for Geoacoustic Parameters in Shallow Water Based on Bottom Reflection Signals

The inversion method based on the reflection loss-grazing angle curve is an effective tool to obtain local underwater acoustic parameters. Because geoacoustic parameters vary in sensitivity to grazing angle, it is difficult to get accurate results in geoacoustic parameter inversion based on small-grazing-angle data in shallow water. In addition, the normal-mode model commonly used in geoacoustic parameter inversion fails to meet the needs of accurate local sound field simulation as the influence of the secant integral is ignored. To solve these problems, an acoustic data acquisition scheme was rationally designed based on a sparker source, a fixed vertical array, and ship drifting with the swell, which could balance the trade-off among signal transmission efficiency and signal stability, and the actual local acoustic data at low-to-mid frequencies were acquired at wide grazing angles in the South Yellow Sea area. Furthermore, the bottom reflection coefficients (bottom reflection losses) corresponding to different grazing angles were calculated based on the wavenumber integration method. The local seafloor sediment parameters were then estimated using the genetic algorithm and the bottom reflection loss curve with wide grazing angles, obtaining more accurate local acoustic information. The seafloor acoustic velocity inverted is cp=1659 m/s and the sound attenuation is αp=0.656 dB/λ in the South Yellow Sea. Relevant experimental results indicate that the method described in this study is feasible for local inversion of geoacoustic parameters for seafloor sediments. Compared with conventional large-scale inversion methods, in areas where there are significant changes in the seabed sediment level, this method can obtain more accurate local acoustic features within small-scale areas.

A Vector Wavenumber Integration Model of Underwater Acoustic Propagation Based on the Matched Interface and Boundary Method

Acoustic particle velocities can provide additional energy flow information of the sound field; thus, the vector acoustic model is attracting increasing attention. In the current study, a vector wavenumber integration (VWI) model was established to provide benchmark solutions of ocean acoustic propagation. The depth-separated wave equation was solved using finite difference (FD) methods with second- and fourth-order accuracy, and the sound source singularity in this equation was treated using the matched interface and boundary method. Moreover, the particle velocity was calculated using the wavenumber integration method, consistent with the calculation of the sound pressure. Furthermore, the VWI model was verified using acoustic test cases of the free acoustic field, the ideal fluid waveguide, the Bucker waveguide, and the Munk waveguide by comparing the solutions of the VWI model, the analytical formula, and the image method. In the free acoustic field case, the errors of the second- and fourth-order FD schemes for solving the depth-separated equation were calculated, and the actual orders of accuracy of the FD schemes were tested. Moreover, the time-averaged sound intensity (TASI) was calculated using the pressure and particle velocity, and the TASI streamlines were traced to visualize the time-independent energy flow in the acoustic field and better understand the distribution of the acoustic transmission loss.

What Is Fling Step? Its Theory, Simulation Method, and Applications to Strong Ground Motion near Surface Fault Ruptures

ABSTRACT We present the theory of the fling step and a theoretical method for simulating accurately the near-fault strong motions, and apply it to reproduce various strong-motion records near surface faults. Theoretically, the fling step is the contribution of the static Green’s function in the representation theorem (Hisada and Bielak, 2003), and we show that this theory holds for any seismic velocity structure. We first demonstrate the validity of this theory using theoretical solutions of a circular fault model in a homogeneous full-space. Next, we apply the theory to layered half-spaces, present a theoretical method based on the wavenumber integration method, and introduce various techniques to simulate the near-fault ground motions including fling steps with high accuracy. Finally, we demonstrate the effectiveness of the method by reproducing various strong-motion records near surface fault ruptures and discuss the characteristics of near-fault strong motions including the fling step and the forward directivity pulse. We made all of the software and data used in this article available on the internet.

A Green’s Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method

The Green’s function (GF) directly eases the efficient computation for acoustic radiation problems in shallow water with the use of the Helmholtz integral equation. The difficulty in solving the GF in shallow water lies in the need to consider the boundary effects. In this paper, a rigorous theoretical model of interactions between the spherical wave and the liquid boundary is established by Fourier transform. The accurate and adaptive GF for the acoustic problems in the Pekeris waveguide with lossy seabed is derived, which is based on the image source method (ISM) and wave acoustics. First, the spherical wave is decomposed into plane waves in different incident angles. Second, each plane wave is multiplied by the corresponding reflection coefficient to obtain the reflected sound field, and the field is superposed to obtain the reflected sound field of the spherical wave. Then, the sound field of all image sources and the physical source are summed to obtain the GF in the Pekeris waveguide. The results computed by this method are compared with the standard wavenumber integration method, which verifies the accuracy of the GF for the near- and far-field acoustic problems. The influence of seabed attenuation on modal interference patterns is analyzed.

Sound-Source Localization in Range-Dependent Shallow-Water Environments Using a Four-Layer Model

Sound-source localization in shallow water is a difficult task due to the complicated environment, e.g., complex sound-speed profile and irregular water bottom reflections. Full-wave numerical techniques are currently able to accurately simulate the propagation of sound waves in such complex environments. However, the source localization problem, which generally involves a large number of sound propagation calculations, still requires a fast computation of the wave equation, and thus a simplified model is well advised. In this paper, a four-layer model is considered, which is able to approximate a wide range of shallow-water environments, particularly those in summer conditions. More specifically, the medium is assumed to be horizontally stratified and vertically divided into four layers, and the sound speed in each layer is assumed to be constant or varying linearly. Under this assumption, the wave propagation can be rapidly computed via a classical wave number integration method. The main contribution of this paper is to show the suitability of the four-layer model in terms of source localization in a complex (range-dependent) environment. The sound-speed profile is assumed to be vertically irregular and horizontally slowly varying and the bottom is nonflat. In the forward problem, sound propagation in complex underwater environments is simulated via a time-domain full-wave simulation approach called the spectral-element method. The source localization error due to model imprecision is analyzed.

The 2011 Mw 5.2 Lorca earthquake as a case study to investigate the ground motion variability related to the source model

Near-field recordings are very sensitive to the spatiotemporal details of the rupture process while far-field signals show the signature of the overall “point-source” earthquake mechanism. Near- and far-field recording ranges are dependent on the event magnitude and modulate the variability of the ground motion. This study investigates the ground motion and the source-related near-field variability for the 2011 Lorca earthquake, a moderate seismic event (Mw = 5.2) that caused significant localized damage in the Region of Murcia, Spain. The low-frequency content (up to 1 Hz) is simulated by the wavenumber integration method assuming four different source models obtained by inversion of geodetic or seismological data. As a first result, we estimate the variability of the ground motion. We observe that the dispersion in the peak and spectral parameters is larger at LOR, the closest station to the source, and decreases as the source distance increases (more than 50 km far from the source) where the finite-fault effects become negligible. The variability of the pseudo spectral velocity at 2 s is within the ground motion prediction equation ±1 σ, apart from the very near-source station and those stations affected by forward directivity effects. These effects are also found in high-frequency seismograms obtained by the empirical Green’s functions approach.

Modeling and experimental validation of an evanescent wavefield using a wavenumber integration method

When sound is incident on the boundary between two media, where the first medium has a lower sound speed than the second (such as sound in water incident on some sediments), a critical angle exists. If the incident angle is below this critical angle an evanescent wavefield is formed in the second medium. Experimental work measured the structure of such a wavefield at an oil-water interface insonified by an appropriately oriented transducer, where the region of interest is not in the transducer far-field [Osterhoudt et al., IEEE J. Ocean. Eng. 33, 397–404 (2008)]. Several features of note within the wavefield were observed. This work attempts to model those observations using wavenumber integration techniques. In the incident plane wave approximation, the evanescent wavefield is vertically phase-locked and the characteristic exponential spatial decay is constant across the wavefield. However, the contribution of many wavenumber components across the region of interest is key to understanding the wavefield structure seen in experiments. This model will be compared to the experimental results, and the details of phase, amplitude, and the overall wavefield structure discussed. An improved approximation of the transducer orientation used in experiments may also be inferred from these results. [Work supported by ONR.]

Analytical solution based on the wavenumber integration method for the acoustic field in a Pekeris waveguide**Project supported by the National Natural Science Foundation of China (Grant No. 11125420), the Knowledge Innovation Program of the Chinese Academy of Sciences, the China Postdoctoral Science Foundation (Grant No. 2014M561882), and the Doctoral Fund of Shandong Province, China (Grant No. BS2012HZ015).

An exact solution based on the wavenumber integration method is proposed and implemented in a numerical model for the acoustic field in a Pekeris waveguide excited by either a point source in cylindrical geometry or a line source in plane geometry. Besides, an unconditionally stable numerical solution is also presented, which entirely resolves the stability problem in previous methods. Generally the branch line integral contributes to the total field only at short ranges, and hence is usually ignored in traditional normal mode models. However, for the special case where a mode lies near the branch cut, the branch line integral can contribute to the total field significantly at all ranges. The wavenumber integration method is well-suited for such problems. Numerical results are also provided, which show that the present model can serve as a benchmark for sound propagation in a Pekeris waveguide.

SH‐wave propagation in a continuously inhomogeneous half‐plane with free‐surface relief by BIEM

The anti‐plane strain elastodynamic problem for a continuously inhomogeneous half‐plane with free‐surface relief subjected to time‐harmonic SH‐wave is studied. The computational tool is a boundary integral equation method (BIEM) based on analytically derived Green's function for a quadratically inhomogeneous in depth half‐plane. To show the versatility of the proposed BIE method, it is considered SH‐wave propagation in an inhomogeneous half‐plane with free surface relief presented by a semi‐circle, semi‐elliptic and triangle canyon. The inhomogeneous in depth half‐plane is modeled in two different ways: (i) the material properties vary continuously in depth and BIEM based on Green's function is used; (ii) the material properties vary in a discrete way and the half‐plane is presented by a set of homogeneous layers with horizontal interfaces and a hybrid technique based on wave number integration method (WNIM) and BIEM is applied. The equivalence of these two different models is shown. The simulations reveal a marked dependence of the wave field on the material inhomogeneity and the potential of the BIEM based on the Green's function for half‐plane to produce highly accurate results by using strongly reduced discretization mesh in comparison with the conventional boundary element technique using fundamental solution for the full plane.

強震動予測手法に関するベンチマークテスト：理論的手法の場合（その３）

We conducted a benchmark tests for the strong motion simulation methods using various theoretical methods (the wavenumber integration method, the discrete wavenumber method, and the thin layer method). For steps 5 and 6 in 2011, we conducted benchmark tests for actual sources in the Kanto sedimentary basin, such as the 1923 Kanto earthquake (M7.9). All the results generally show good agreements among simulations, but the surface waves seen in later part of observed ground motions were not reproduced well due to the limitation of use of flat-layered half-space models.

Probabilistic inversion of Rayleigh-wave dispersion data: an application to Mt. Etna, Italy

We present a methodology for determining the elastic properties of the shallow crust from inversion of surface wave dispersion characteristics through a fully nonlinear procedure. Using volcanic tremor data recorded by a small-aperture seismic array on Mount Etna, we measured the surface waves dispersion curves with the multiple signal classification technique. The large number of measurements allows the determination of an a priori probability density function without the need of making any assumption about the uncertainties on the observations. Using this information, we successively conducted the inversion of phase velocities using a probabilistic approach. Using a wave-number integration method, we calculated the predicted dispersion function for thousands of 1-D models through a systematic grid search investigation of shear-wave velocities in individual layers. We joined this set of theoretical dispersion curves to the experimental probability density function (PDF), thus obtaining the desired structural model in terms of an a posteriori PDF of model parameters. This process allowed the representation of the objective function, showing the non-uniqueness of the solutions and providing a quantitative view of the uncertainties associated with the estimation of each parameter. We then compared the solution with the surface wave group velocities derived from diffuse noise Green’s functions calculated at pairs of widely spaced (~5–10 km) stations. In their gross features, results from the two different approaches are comparable, and are in turn consistent with the models presented in several earlier studies.

Elastic wave scattering and stress concentration effects in non-homogeneous poroelastic geological media with discontinuities

In this work, we investigate elastic wave propagation in complex geological media that exhibit material inhomogeneity in the form of depth-dependent material parameters and heterogeneity in the lateral direction. In addition, we consider a layered structure with non-parallel interfaces, free-surface relief, plus the presence of different type of discontinuities such as interface and/or internal cracks. The seismic load comprises incident, time-harmonic pressure (P) and vertically polarized shear (SV) waves, thus covering the plane strain case. Next, the computational technique developed herein is a hybridization of the semi-analytical wave number integration method (WNIM) with the boundary element method (BEM) cast in the frequency domain for solving 2D problems of elastodynamics. Transient response to this problem is recovered through standard Fourier synthesis of the frequency domain results. In essence, the present modeling effort interfaces continuum mechanics with linear fracture mechanics, since it focuses on wave scattering by cracks in complex geological regions. Material behavior is reproduced by the viscoelastic equivalent to Biot's poroelasticity. More specifically, by assuming saturated geomaterials, Bardet's model is introduced in the analysis as the computationally efficient viscoelastic isomorphism to Biot´s equations of dynamic poroelasticity, thus replacing the original two-phase material by a single phase one. Finally, the role of stationary cracks in poroelastic materials is investigated through an extensive series of parametric studies, whereupon these discontinuities act as both wave scatterers and stress concentrators. In sum, our simulations serve to quantify the sensitivity of the near field stress intensity factors (SIF) and of the far field free surface motions to incoming wave characteristics, surface and underground topography, and mechanical properties of the geological deposits.

BENCHMARK TESTS FOR STRONG GROUND MOTION PREDICTION METHODS: CASE FOR THEORETICAL METHODS (PART 2)

Following the 2009 benchmark test, we conducted a subsequent test for the strong motion simulation methods in more complex media, using various theoretical methods (the wavenumber integration method, the discrete wavenumber method, and the thin layer method). All the results show generally good agreements, but the results for damping media show discrepancies, especially in the fling step at very close stations to a surface fault. This is because the frequency-dependent quality factors violate severely the causality condition in the waveforms, especially at lower frequencies.

Full-waveform based complete moment tensor inversion and source parameter estimation from downhole microseismic data for hydrofracture monitoring

Downhole microseismic monitoring is a valuable tool in understanding the efficacy of hydraulic fracturing. Inverting for the moment tensor has gained increasing popularity in recent years as a way to understand the fracturing process. Previous studies utilize only part of the information in the waveforms, such as direct P- and S-wave amplitudes, and make far-field assumptions to determine the source mechanisms. The method is hindered in downhole monitoring, when only limited azimuthal coverage is available. In this study, we develop an approach to invert for complete moment tensor using full-waveform data recorded at a vertical borehole. We use the discrete wavenumber integration method to calculate full wavefields in the layered medium. By using synthetic data, we find that, at the near-field range, a stable, complete moment tensor can be retrieved by matching the waveforms without additional constraints. At the far-field range, we discover that the off-plane moment tensor component is poorly constrained by waveforms recorded at one well. Therefore, additional constraints must be introduced to retrieve the complete moment tensor. We study the inversion with three different types of constraints. For each constraint, we investigate the influence of velocity model errors, event mislocations, and data noise on the extracted source parameters by a Monte Carlo study. We test our method using a single well microseismic data set obtained during the hydraulic fracturing of the Bonner sands in East Texas. By imposing constraints on the fracture strike and dip range, we are able to retrieve the complete moment tensor for events in the far-field. Field results suggest that most events have a dominant double-couple component. The results also indicate the existence of a volumetric component in the moment tensor. The derived fracture plane orientation generally agrees with that derived from the multiple event location.

Wavefield simulation and data-acquisition-scheme analysis for LWD acoustic tools in very slow formations

Propagating wavefields from monopole, dipole, and quadrupole acoustic logging-while-drilling (LWD) tools in very slow formations have been studied using the discrete wavenumber integration method. These studies examine the responses of monopole and dipole systems at different source frequencies in a very slow surrounding formation, and the responses of a quadrupole system operating at a low source frequency in a slow formation with different S-wave velocities. Analyses are conducted of coherence-velocity/slowness relationships (semblance spectra) in the time domain and of the dispersion characteristics of these waveform signals from acoustic LWD array receivers. These analyses demonstrate that, if the acoustic LWD tool is centralized properly and is operating at low frequencies (below 3 kHz), a monopole system can measure P-wave velocity by means of a “leaky” P-wave for very slow formations. Also, for very slow formations a dipole system can measure the P-wave velocity via a leaky P-wave and can measure the S-wave velocity from a formation flexural wave. With a quadrupole system, however, the lower frequency limit (cutoff frequency) of the drill-collar interference wave would decrease to 5 kHz and might no longer be neglected if the surrounding formation becomes a very slow formation, with S-wave velocities at approximately 500 m/s.

BENCHMARK TESTS FOR STRONG GROUND MOTION PREDICTION METHODS: CASE FOR THEORETICAL METHODS (PART 1)

We carried out benchmark tests for the strong motion simulation methods using various theoretical methods (the wavenumber integration method, the discrete wavenumber method, and the thin layer method). All the results generally show good agreements, but the results for damping media show large discrepancies, especially in the later phases at very far stations. This is because the most methods used the quality factors of the complex parts of the medium velocities, whereas one method used the factors not only in the complex parts, but also in the real parts, in order to satisfy the causality condition in the waveforms.

A relationship between the waveguide invariant and wavenumber integration

The waveguide invariant is typically defined using of normal modes or ray theory, but it can also be related to the wavenumber-integration method for calculating the acoustic field in a waveguide. In this letter, the Wiener-Khinchin Theorem is used to show that the autocorrelation of the wavenumber-integration kernel, when plotted versus wavenumber difference and frequency, contains striations that can be described by the waveguide invariant.

The influence of shallow atmospheric structure on tropospheric ducted infrasound from the Buncefield oil depot explosion

The vapour cloud explosion which destroyed the Buncefield oil depot, UK, on 11th December 2005, has proven to be a benchmark ground truth event for infrasonic studies. The regional infrasonic returns, those that propagated in the stratosphere and thermosphere, have been analysed in detail elsewhere. Here, we present the results of a study into infrasound ducted in the troposphere, recorded within 250 km of the source as air-to-ground coupled waves by a dense seismometer network. These tropospheric arrivals exhibit large waveform differences across the UK, both in amplitude and waveform shape. We numerically model these infrasound arrivals using a wave number integration method, incorporating a velocity profile derived from the UK Met Office numerical weather prediction model. Although some of the waveform variability is due to ground conditions at the recording site, we show that consistent changes in waveform shape across a 200 km swath of stations are correlated with a change in the wind vector in the lowermost 2 km of the atmosphere. Also, small amplitude, high-frequency precursors to the dominant acoustic signal, which might be misinterpreted as evidence of a small initial explosion, are shown to be consistent with the dispersion expected from a thin, shallow wind jet.

Numerical Modeling of the 13 December 1990 M 5.8 East Sicily Earthquake at the Catania Accelerometric Station

The record of Catania enea–enel accelerometric station, with its anomalously high amplitude, represents an outlier for the dataset of seismograms recorded by the Italian accelerometric network during the 13 December 1990 M 5.8 East Sicily earthquake. Using numerical modeling tools, we corroborate the interpretation of this anomaly as an effect of the crustal structure or site response, rather than as a finite source effect. In the first part of this study, we compare the recorded seismograms to those computed by two numerical methods—the 2D Spectral Element Method (spem), a technique which solves the 2D full-wave propagation through a complex geological structure; and the Wavenumber Integration Method (wim), which solves the 3D full-wave propagation in a horizontally-layered structure. The comparison shows that the recorded waveforms are reproduced accurately using a laterally heterogeneous structure, whereas none of the representations in terms of horizontal plane layers, which simplify the same structure, provides satisfactory results. Furthermore, we compare the horizontal-to-vertical spectral ratios (hvsr) obtained from (1) the earthquake record, (2) the seismogram simulated by the spem, and (3) the seismic noise recorded at the same site. Again, the overall agreement is very good. This study has several outcomes. Firstly, that the numerical modeling approach based on the spem, and used in a previous study to simulate ground shaking for a destructive scenario earthquake, provides reliable results. Secondly, that the 2D model used to represent the crustal structure beneath this area is realistic. Indeed, simplified 1D models may not be an adequate means to reproduce realistic seismograms and predict the ground motion in the frequency band of interest for seismic hazard (i.e., 0.5–10 Hz). Finally, the high amplitude displayed by the Catania station during the 1990 earthquake is explained as a combined effect of site and structure-path, while finite source models appear unnecessary. In general, this study emphasizes the importance of methods that accurately model the wavefield propagation through realistic geologic structures for predicting ground motion.

Mode count and modal density of structural systems: relationships with boundary conditions

The effects of boundary conditions on the mode count and modal density of one- and two-dimensional structural systems, beams and plates, respectively, are investigated by using the wavenumber integration method. Bending vibrations are examined first for a single beam. In this case it is demonstrated that the average mode count is reduced by between 0 and 1 for each boundary constraint, depending on the type of boundary conditions. For more generalised mass and stiffness constraints a frequency-dependent coefficient, which need not lie between 0 and 1, is obtained. The effects of line constraints on the mode count of two-dimensional systems are similar to the equivalent one-dimensional constraints but they are always frequency dependent. Then the mode count of systems of multiple collinear beams and coplanar plates is studied. It is found that an intermediate constraint has the same effect on the average mode count as the same type of constraint applied at an end of the system. The modal density is largely independent of boundary conditions for one-dimensional systems although there are exceptions, while it is dependent on boundary conditions for two-dimensional systems. The results are compared with those from previously published formulae for natural frequencies and with results from finite element method (FEM) analysis. Inclusion of the effect of the boundary conditions in statistical energy analysis (SEA) estimations will result in improved agreements with both analytical and numerical results.