Secondary Wave Field Research Articles

Analytical seismic model for a tunnel with segmental liner buried in the half-space soil

In this paper, by using the wave function expansion (WFE) and the expansion of the cylindrical wave into plane wave (ECPW) methods, an analytical method for the dynamic response of a circular tunnel with segmental liner buried in the half-space soil to harmonic elastic waves is developed. The liner of the tunnel is supposed to consist of several segments and joints. Both the segments and joints are treated as open cylindrical shells, and the segment and joint shells together thus form an equivalent continuous shell (ECS) liner, and the thin shell theory is utilized to describe its vibration. The scattered wave field due to the presence of the tunnel and boundary of the half-space soil is divided into two parts, namely, the direct scattered wave field and secondary scattered wave field. The expressions for the direct scattered cylindrical waves in the soil is determined via the WFE method, while the secondary scattered waves form the boundary of the half-space soil are obtained by the ECPW method together with the application of the boundary condition along the soil surface. Applying the cylindrical shell theory on the ECS liner and using the Fourier series expansions for the variables and parameters of the ECS liner along the azimuthal direction together with introducing the Fourier component constitutive relation for the ECS liner, a system of equations for the Fourier component ECS displacements are derived with the differential equations for the ECS liner displacements. By using the system of equations, the expressions for the free and scattered wave fields and the continuity conditions at the interface between the liner and soil, the system of equations for the ECS liner coupled with the soil are derived. By using the developed analytical method for the tunnel, some results of the ECS tunnel under different incident waves are given.

Finite-element modelling of seismoelectric and electroseismic waves in frequency domain: 2-D SHTE mode

SUMMARYWe propose a frequency-domain finite-element (FDFE) method to simulate the 2-D SHTE mode seismoelectric and electroseismic waves. By neglecting the secondary weak wavefield feedbacks, the SH and TE waves are solved, separately. In a finite plane region, propagations of both SH and TE waves can be described as the Helmholtz equation with boundary conditions, which is proved to be equivalent to the extremum of functional by conducting calculus of variation. The computation region is partitioned into structured rectangular elements with the bilinear interpolation. The proposed FDFE algorithm solves the wavefield in frequency domain and avoids adopting the quasi-static approximation. One advantage of the proposed algorithm is its ability to accurately simulate the seismoelectric and electroseismic responses generated from the free surface. We verify the proposed algorithm based on a layered model beneath a free surface by comparing the waveforms calculated using the FDFE algorithm with those calculated using analytically-based method. The proposed algorithm is applied in feasibility studies of interface seismoelectric and electroseismic responses in exploring the hydrocarbon reservoir and monitoring the time-lapse pollutant within a sand channel.

Predicting ship waves in sheltered waterways – An application of XBeach to the Stockholm Archipelago, Sweden

Predicting hydrodynamic forces from ship waves along a waterway is essential for managing a sustainable shoreline in fairways. In this context, a numerical model can be a useful tool for calculating ship waves in fairways with complex geometry. XBeach is a numerical model offering the potential to be a one-tool solution for simulating ship waves from the ship to the shore. However, XBeach has not been evaluated against field measurements of ship waves. Thus, the objective of this paper is (1) to compare XBeach against high-quality field measurements of ship waves and (2) to assess its general applicability to ship wave simulations through an extensive sensitivity analysis with regard to the main input parameters. Water level measurements from 42 passages of the M/S Baltic Princess in the Furusund navigational fairway in the Stockholm Archipelago were chosen to test and evaluate XBeach. The results showed that XBeach was able to generate and propagate the primary and secondary waves. Primary waves were reproduced well with a mean absolute error (MAE) of 3 cm for the drawdown height. The overall secondary wave field pattern was represented by XBeach; however, for the largest, single divergent wave comparison for height, period, and timing showed deviation from the observations. Overall, XBeach was found to be a robust model for ship wave applications, only requiring input data typically available. Sensitivity analysis for a generic hull, derived from the type of vessel studied and with the corresponding dimensions as the real hull, showed similar results enabling robust and reliable simulations without access to detailed technical drawings.

Broadband global acoustic cloaking experiments

Rendering objects invisible to impinging acoustic waves (i.e., acoustic cloaking), is a mature topic in the acoustic community and has diverse applications. Nevertheless, existing approaches and physical demonstrations still exhibit a large range of assumptions and limitations. Passive approaches, which surround the scattering object with a suitable (meta-)material, are usually only effective in a narrow frequency-band and cannot adapt to changing incident fields. Active approaches, which rely on the emission of a secondary wavefield that destructively interferes with the scattered field, have the potential to respond dynamically and be broad band. However, without knowledge of the primary or scattered field, the signals for the secondary sources need to be estimated from wavefield measurements in real-time. To our knowledge, this was never demonstrated beyond 1D acoustic experiments to achieve global broadband control of forward- and backscattered waves. Here, we present experimental results of active, broadband 2D acoustic cloaking without any assumptions of the primary or scattered wavefields. Using a low-latency control system and a dense array of control microphones, the signals for 20 control sources surrounding the scatterer are estimated and applied in real-time to achieve fully deterministic, global acoustic cloaking.

In the eye of the storm

AbstractThe airborne NOAA Wide Swath Radar Altimeter (WSRA) is a 16 GHz digital beamforming radar altimeter that produces a topographic map of the waves as the aircraft advances. The wave topography is transformed by a two-dimensional FFT into directional wave spectra. The WSRA operates unattended on the aircraft and provides continuous real-time reporting of several data products: (1) significant wave height, (2) directional ocean wave spectra, (3) the wave height, wavelength, and direction of propagation of the primary and secondary wave fields, (4) rainfall rate and (5) sea surface mean square slope (mss). During hurricane flights the data products are transmitted in real-time from the NOAA WP-3D aircraft through a satellite data link to a ground station and on to the National Hurricane Center (NHC) for use by the forecasters for intensity projections and incorporation in hurricane wave models. The WSRA is the only instrument that can quickly provide high-density measurements of the complex wave topography over a large area surrounding the eye of the storm.

Quest CCS Facility: Microseismic System Monitoring and Observations

In August 2015, CO2 injection commenced at the Quest Carbon Capture and Storage (CCS) Facility located near Fort Saskatchewan, Alberta, Canada. Quest is a fully integrated CCS facility with a capture target of just over one million metric tonnes of CO2 per year. CO2 is injected into a deep saline aquifer, the Basal Cambrian Sandstone (BCS), at a depth of about two kilometers below ground. In order to demonstrate containment and conformance of the injected CO2, a Measurement, Monitoring and Verification (MMV) plan has been implemented. Although Quest is in an extremely quiet tectonic location, induced seismicity has been recognized as a potential risk for all large-scale injection sites. As a result, microseismic monitoring is a key component of the Quest MMV Plan to ensure the continued assessment of that risk and to provide early notice of any changes. The aim of this presentation is to detail why we monitor for microseismic activity and how we do this efficiently using proactive technical monitoring (PTM) and exception based surveillance (EBS), to report on six years of microseismic monitoring observations through the pre- and post-start of injection periods and to describe the empirical analysis that has allowed the Quest CCS Facility to de-risk microseismicity as an active threat to containment. Microseismic data is continuously recorded at the Quest CCS storage site using a commercial downhole geophone array installed in a deep monitoring well (DMW). The data are processed to produce trigger files using a common algorithm for event detection and triggering, based on the ratio of short-time-average to long-time-average (STA/LTA) amplitudes. The triggered events are analyzed, classified and reported daily. Categories for various trigger types are defined based on the characteristics of each event. Operationally, the status of the microseismic array is continually assessed by the presence of trigger events. A trigger event is defined to be locatable when it shows clear primary (P) and secondary (S) wave field arrivals. The locatable events are processed and located using an anisotropic velocity model. The p-wave velocities were derived from a vertical seismic profile survey, and the s-wave velocities were obtained from sonic well logs. Pre-CO2 injection recording of microseismic data began in November 2014. Most of the trigger files generated were related to surface, automatic and noise triggers. These events were used to assess the health of the system. No locatable events were detected during the pre-injection monitoring phase. The first locatable event was recorded 10 months after start of CO2 injection in July 2016 with a magnitude of -1.3. Since January 2017, sustained low level, small magnitude microseismic activity have been located within the Quest microseismic area of review (AOR) within the Precambrian basement. The AOR as defined in the 2017 Quest MMV plan, extends 10 km radially outwards from each active injection well (IW). As of December 31st, 2020, 486 locatable events have been detected, with an average magnitude of -0.7, a maximum magnitude of 0.8 and a typical occurrence rate of 1-2 events per week. Using the current methodology, all locatable events have been located within the Precambrian basement, below the injection zone. None of this microseismic activity has represented a risk to containment.

Full-waveform inversion of intensity-focused seismic data

Full-waveform inversion (FWI) is a promising tool for the comprehensive analysis of seismic data because it involves modeling the complete elastodynamic phenomena in 3D for the estimation of medium parameters. However, the application for reservoir characterization is still limited in achieving the inverse problem solution using reasonable computational resources and the required model resolution. Following a data-driven approach, we have implemented the isotropic elastic FWI with sensitivity emphasis on a prescribed target zone of the propagation volume, showing that the method is useful in improving the estimation over the target zone compared with nonlocalized methods. Our method uses the superposition principle to construct multipoint sources that focus the primary wave intensity (seismic illumination) over the specific target area. Thus, the secondary wavefield scattered from the target area to the receivers increases, providing improved information on the model parameters that control the property heterogeneity within the target zone. We design two types of focused macrosource (FMS) configurations that accomplish this objective: the convergent FMS and the combined beam FMS. Although multipoint seismic sources are not implemented in the field for practical reasons, they can be synthetized computationally using the superposition principle, interpolation, and the current seismic survey data (point-source) to compose the FMS observed gathers, which are then used as input data for the FWI. We perform elastic FWI synthetic tests in a 2D modified Marmousi model to compare the inversion performance with focused and nonfocused data, under the same computational conditions, with and without the presence of noise. Our tests show faster convergence and improved estimation in the target zone with the use of the intensity-focused seismic data for FWI; improved focusing effects are expected in 3D. Being target-oriented, the method is suitable for reducing computational requirements in elastic FWI application for reservoir description.

Physical implementation of immersive boundary conditions in acoustic waveguides

The physical implementation of so-called immersive boundary conditions (IBCs) allows the construction of anechoic chambers, where wavefield reflections from the boundary of a physical domain, such as a wave propagation laboratory, are actively suppressed by emitting a secondary wavefield at the domain boundary that destructively interferes with the reflected waves. Moreover, IBCs enable immersive wave propagation experimentation by linking the wave propagation in the physical domain with the propagation in a numerical domain enclosing the physical domain. In this case, IBCs correctly account for all wavefield interactions between both domains, including higher-order scattering. The physical implementation of IBCs is achieved by densely populating the boundary surrounding the physical domain with transducers that enforce the necessary boundary conditions. The signals required at the injection boundary are predicted with the help of a secondary surface of transducers that record the wavefield on a surface slightly inside the physical domain. The recorded wavefield is extrapolated to the boundary by evaluating a Kirchhoff-Helmholtz integral in real-time using an FPGA-enabled data acquisition, computation and control system. Here, we present results of the active suppression of broadband boundary reflections in 1D and 2D acoustic waveguides.

Method for determining size of inhomogeneity localization region based on analysis of secondary wave field of second harmonic

The article describes the research of the method of localization and determining the size of heterogeneity in biological tissues. The equation for the acoustic harmonic wave, which propagates in the positive direction, is taken as the main one. A three-dimensional expression that describes the field of secondary sources at the observation point is obtained. The simulation of the change of the amplitude values of the vibrational velocity of the second harmonic of the acoustic wave at different coordinates of the inhomogeneity location in three-dimensional space is carried out. For the convenience of mathematical calculations, the area of heterogeneity is reduced to a point.

Non-collinear interaction of guided elastic waves in an isotropic plate

The nonlinear wave propagation in a homogeneous and isotropic elastic plate is analyzed theoretically to investigate the non-collinear interaction of plate wave modes. In the presence of two primary plate waves (Rayleigh-Lamb or shear horizontal modes) propagating in arbitrary directions, an explicit expression for the modal amplitude of nonlinearly generated wave fields with the sum or difference frequency of the primary modes is derived by using the perturbation analysis. The modal amplitude is shown to grow in proportion with the propagation distance when the resonance condition is satisfied, i.e., when the wavevector of secondary wave coincides with the sum or difference of those of primary modes. Furthermore, the non-collinear interaction of two symmetric or two antisymmetric modes is shown to produce the secondary wave fields consisting only of the symmetric modes, while a pair of symmetric and antisymmetric primary modes is shown to produce only the antisymmetric modes. The influence of the intersection angle, the primary frequencies, and the mode combinations on the modal amplitude of secondary wave is examined for a low-frequency range where the lowest-order symmetric and antisymmetric Rayleigh-Lamb waves and the lowest-order symmetric shear horizontal wave are the only propagating modes.

High Frequency Field Measurements of an Undular Bore Using a 2D LiDAR Scanner

The secondary wave field associated with undular tidal bores (known as whelps) has been barely studied in field conditions: the wave field can be strongly non-hydrostatic, and the turbidity is generally high. In situ measurements based on pressure or acoustic signals can therefore be limited or inadequate. The intermittent nature of this process in the field and the complications encountered in the downscaling to laboratory conditions also render its study difficult. Here, we present a new methodology based on LiDAR technology to provide high spatial and temporal resolution measurements of the free surface of an undular tidal bore. A wave-by-wave analysis is performed on the whelps, and comparisons between LiDAR, acoustic and pressure-derived measurements are used to quantify the non-hydrostatic nature of this phenomenon. A correction based on linear wave theory applied on individual wave properties improves the results from the pressure transducer (Root mean square error, R M S E of 0 . 19 m against 0 . 38 m); however, more robust data is obtained from an upwards-looking acoustic sensor despite high turbidity during the passage of the whelps ( R M S E of 0 . 05 m). Finally, the LiDAR scanner provides the unique possibility to study the wave geometry: the distribution of measured wave height, period, celerity, steepness and wavelength are presented. It is found that the highest wave from the whelps can be steeper than the bore front, explaining why breaking events are sometimes observed in the secondary wave field of undular tidal bores.

Tidal bore dynamics in funnel‐shaped estuaries

AbstractThe formation and dynamics of tidal bores in funnel‐shaped estuaries is investigated from both a global tidal wave scaling analysis and new quantitative field observations. We show that tidal bore occurrence in convergent estuaries can be estimated from a dimensionless scaling parameter characterizing the relative intensity of nonlinear friction versus local inertia in the momentum equation. A detailed analysis of tidal bore formation and secondary wave structure is presented from a unique long‐term database (observations of more than 200 tides) acquired during four campaigns in the two main French tidal‐bore estuaries: the Seine and Gironde/Garonne estuaries. We describe the effect of freshwater discharge on the global tidal wave transformation at the estuarine scale and on local tidal bore occurrence in the upper estuary. Our field data suggest that the tidal bore intensity is mainly governed by the dimensionless tidal range, which characterizes the local tidal wave nonlinearity. We also show that the secondary wavefield associated with tidal bore propagating in natural estuaries differs significantly from those associated to undular bores in rectangular channels. In particular, we observe an abrupt decrease of the whelp steepness when the Froude number goes below 1.1. This secondary field transition can explain why tidal bore occurrence in worldwide estuaries is certainly underestimated.

Higher order interaction of elastic waves in weakly nonlinear hollow circular cylinders. I. Analytical foundation

A mathematical framework to analyze the cumulative behavior of higher order harmonic generation due to the interaction of two collimated waves in a weakly nonlinear hollow circular cylinder is formulated in this article. A total number of (N + 1)(N + 2)/2 − 3 nonlinear boundary problems are formulated due to the Nth order mode interactions in a cylinder with Kth order nonlinearity (N ≤ K). The cumulative criteria for the second order harmonics (second harmonics, sum and difference harmonics) due to the quadratic interactions of two waves are examined based on the nonlinear forcing terms in curvilinear coordinates. These criteria are formulated by a synchronism condition, the circumferential orders of the primary modes, as well as the nature of the primary and the secondary wave fields, i.e., torsional or longitudinal. A generalized analysis that provides insight into the cumulative nature of the Nth order harmonics by Nth order interaction of two collimated waves is conducted by considering a cylinder with strain energy function written as Murnaghan's power series. The nature of the cumulative Nth order harmonics can be determined by the parity of the number of times the primary waves interact, and their circumferential orders.

On selection of primary modes for generation of strong internally resonant second harmonics in plate

The selection of primary shear-horizontal (SH) and Rayleigh–Lamb (RL) ultrasonic wave modes that generate cumulative second harmonics in homogeneous isotropic plates is analyzed by theoretical modeling. Selection criteria include: internal resonance (synchronism and nonzero power flux), group velocity matching, and excitability/receivability. The power flux, group velocity matching, and excitability are tabulated for the SH and RL internal resonance points. The analysis indicates that SH waves can generate cumulative symmetric RL secondary wave fields. Laboratory experiments on aluminum plates demonstrate that excitation of the SH3 primary mode generates the s4 secondary RL mode and that the secondary wave field amplitude increases linearly with propagation distance. Simple magnetostrictive transducers were used to excite the primary SH wave and to receive the SH and RL wave signals. Reception of these wave modes having orthogonal polarizations was achieved by simply reorienting the electrical coil. The experiment was complicated by the presence of a nonplanar primary wavefront, however finite element simulations were able to clarify the experimental results.

Interaction of torsional and longitudinal guided waves in weakly nonlinear circular cylinders

The nonlinear forcing terms for the wave equation in general curvilinear coordinates are derived based on an isotropic homogeneous weakly nonlinear elastic material. The expressions for the nonlinear part of the first Piola-Kirchhoff stress are specialized for axisymmetric torsional and longitudinal fundamental waves in a circular cylinder. The matrix characteristics of the nonlinear forcing terms and secondary mode wave structures are manipulated to analyze the higher harmonic generation due to the guided wave mode self-interactions and mutual interactions. It is proved that both torsional and longitudinal secondary wave fields can be cumulative by a specific type of guided wave mode interactions. A method for the selection of preferred fundamental excitations that generate strong cumulative higher harmonics is formulated, and described in detail for second harmonic generation. Nonlinear finite element simulations demonstrate second harmonic generation by T(0,3) and L(0,4) modes at the internal resonance points. A linear increase of the normalized modal amplitude ratio A2/A1(2) over the propagation distance is observed for both cases, which indicates that mode L(0,5) is effectively generated as a cumulative second harmonic. Counter numerical examples demonstrate that synchronism and sufficient power flux from the fundamental mode to the secondary mode must occur for the secondary wave field to be strongly cumulative.

Forward modelling the multiply scattered 2.5-D teleseismicPwaves accelerated by the multilevel fast multipole method

SUMMARY The present study introduces an algorithm to compute the forward modelling of multiple scattering. The propagating medium is defined, in general, by 3-D heterogeneities embedded in a 1-D layered structure, assumed as the reference structure. The 3-D formulation is consequently modified to suit 2.5-D scattering modelling. The basic formulation relies on the Green's function method and employs (1) the reflectivity method (RM) to acquire the 1-D multilayered, half-space Green's function, (2) the multilevel fast multipole method (MLFMM) to gain speed and (3) the generalized minimal residual method (GMRES) to iteratively solve the linear system in a memory efficient manner. The heterogeneities are modelled as point scatterers, creating a secondary wavefield that superimposes on the waves in the reference structure. The algorithm is efficient since the CPU time increases logarithmically with the size of the model. The test cases included various anomalous lithosphere structures, ranging from a simple point scatterer to a suture zone.

Variation of teleseismic short-period waveforms at Matsushiro Seismic Array System

AbstractWe investigate the origin of variation of teleseismic P phase waveforms recorded at Matsushiro Seismic Array System. Short-period, vertical component waveforms from three source regions were analyzed separately in the frequency domain. Observed variation is mainly attributed to the site effects, which are frequency dependent and strongly depend on the incoming azimuth. Such variation is not seen for the pre-event noise spectra. The secondary wavefield that is excited by the incident P phase is the likely cause of the observed waveform variation.

Characterization of secondary ultrasonic waves radiated by two oscillating bubbles.

A bubble oscillating under an ultrasonic wave produces an interference sound field with the incident ultrasonic wave, forming fringe patterns of trapped microbubbles. We discuss secondary ultrasonic waves radiated by two oscillating bubbles in the incident ultrasonic wave in terms of both theory and experimental conditions. Two oscillating bubbles produce an inherent secondary wave field that varies with the distance between the two bubbles and the phases of these waves. Here we discuss the Bjerknes force derived for two bubbles and microbubble trapping around the bubbles. The experiments were carried out using microbubbles produced with an ultrasonic contrast agent (Levovist). The microbubble-trapping pattern observed around the bubbles in the experiment closely agrees with the numerical simulation.

Phase‐bunching effects in triggered VLF emissions: Antenna effect

We consider the effect of initial phase bunching of energetic electrons on the generation of triggered ELF/VLF emissions in the magnetosphere. The electrons interacted with the primary whistler wave packet form a phase‐bunched beam in the velocity space, which serves as a traveling‐wave antenna emitting secondary waves. We calculate this antenna field and study its spectral and amplitude characteristics. It is shown that under the typical magnetospheric conditions, the secondary wave field is by about an order of magnitude smaller than the primary wave field. Therefore this antenna field is significant as the seed wave which is further amplified by the same beam via the cyclotron resonant mechanism. The frequency band of the antenna effect is of the order of 100 Hz near the primary‐wave frequency. This is consistent with the observed frequency shift of triggered emissions.

SELF-ACTION OF A SUPERSHORT LASER PULSE IN A MEDIUM WITH NORMAL GROUP-VELOCITY DISPERSION

We present the results of numerical and analytical analysis of solutions of the three-dimensional (3D) nonlinear Schroodinger equation with hyperbolic spatial operator. Evolution of the system is considered in separate for two types of the initial field: a Gaussian distribution and a hollow-type (tubular or horseshoe) distribution. The effect of the nonlinear dispersion on wave-packet splitting during self-compression toward the system axis is studied. It is shown that additional focusing of Gaussian wave packets takes place in a wide range of the nonlinear-dispersion parameter. This effect results in a noticeable amplitude growth of one of the two secondary pulses formed as a result of the splitting. For hollow-type distributions, we note the formation of moving inhomogeneities and the excitation of secondary wave fields typical of the hyperbolic system.