Reflected Wave Field Research Articles

Directional Spectrum Estimation for Sea States Generated by the Single Summation Method

The influence of directional spreading of waves is significant for wave-induced loads, wave breaking and nonlinearity of the waves. For physical model testing performed at test facilities such as the Ocean and Coastal Engineering Laboratory at Aalborg University, it is crucial to validate if the test conditions match the target sea states by measurement and analysis of the generated directional wave field. Most of the existing methods assumes a double summation sea state to be present which is valid in the prototype. However, waves in the laboratory are usually generated by single summation. The current paper presents a method to analyse short-crested waves generated by the single summation method. Compared to similar methods oblique reflections are considered instead of only in-line reflections. The results show that the method successfully decomposes the incident and reflected wave fields in the time domain. Thus, for example the incident wave height distribution may be obtained. The sensitivity of the new method to additional reflective directions, noise, calibration errors and positional errors of the wave gauges was found small.

A microlocal and visual comparison of 2D Kirchhoff migration formulas in seismic imaging*

Abstract The term Kirchhoff migration refers to a collection of approximate linearized inversion formulas for solving the inverse problem of seismic tomography which entails reconstructing the Earth’s subsurface from reflected wave fields. A number of such formulas exists, the first dating from the 1950 s. As far as we know, these formulas have not yet been mathematically compared with respect to their imaging properties. This shortcoming is to be alleviated by the present work: we systematically discuss the advantages and disadvantages of the formulas in 2D from a microlocal point of view. To this end we consider the corresponding imaging operators in an unified framework as pseudodifferential or Fourier integral operators. Numerical examples illustrate the theoretical insights and allow a visual comparison of the different formulas.

Characteristic reflection wavefield-based wave equation traveltime inversion

Introducing a high-precision deep-depth velocity model is essential for accurately imaging seismic data in complex areas such as deepwater and deep depth in oil and gas exploration. Conventional methods, such as reflection full-waveform inversion (RFWI), face challenges, notably the cycle-skipping issue, especially when the initial velocity model is far away from the true model. To address this, we develop a novel wave equation traveltime inversion (WETI) using the characteristic reflection wavefield (CRW) to update the low-wavenumber components of the velocity model. The CRWs are primary reflections in seismic data that contribute significantly to velocity model updates. We implement a multistep approach, including migration, characteristic reflector structure (CRS) picking, and demigration to extract the CRW. This extraction, constrained by CRS, ensures accurate matching between the observed and the simulated CRWs, even with an inaccurate initial velocity model, thus enabling an accurate traveltime shift estimation during inversion. Unlike conventional methods that use all reflection data simultaneously, CRW-WETI strategically selects fewer CRWs, enhancing inversion convexity and avoiding local minima from the cycle-skipping problem. This selective inversion begins with a limited number of CRWs to ensure stability and progressively incorporates more as the WETI process advances, eventually transitioning into conventional RFWI. Moreover, with the CRS constraint, CRW can be selected based on the target region. The target-oriented velocity inversion can also be implemented easily to update a local anomalous velocity model. Numerical examples of synthetic and field data demonstrate the effectiveness and validity of our method.

Directional Spectrum Estimation For Sea States Generated By The Single Summation Method

The influence of directional spreading of waves is significant for wave-induced loads, wave breaking and nonlinearity of the waves. For physical model testing performed at test facilities such as the Ocean and Coastal Engineering Laboratory at Aalborg University, it is crucial to validate if the test conditions match the target sea states by measurement and analysis of the generated directional wave field. Most of the existing methods assumes a double summation sea state to be present which is valid in the prototype. However, waves in the laboratory are usually generated by single summation. The current paper presents a method to analyse short-crested waves generated by the single summation method. Compared to similar methods oblique reflections are considered instead of only in-line reflections. The results show that the method successfully decomposes the incident and reflected wave fields in the time domain. Thus, for example the incident wave height distribution may be obtained. The sensitivity of the new method to additional reflective directions, noise, calibration errors and positional errors of the wave gauges was found small.

Wave power extraction from slanted oscillating water columns in the presence of an array of arbitrary trenches

This paper investigates the impact of arbitrary bottom topography on the efficiency of a slanted oscillating water column device located in near-shore areas. The device incorporates a turbine positioned at its opening, which harnesses the oscillation of the fluid at the free surface, created by recurrent incident waves, to propel an air volume in a uni-direction. To analyze the wave power absorption capability of the system, an eigenfunction expansion and a boundary element model based on linear water wave theory are employed. The notion is to use the anticipated amplification of scattered and reflected wave fields resulting from the existence of a sloping wall to improve the wave power absorption caused by the walls' wave reflections. The accuracy of the current study's numerical results is ratified by conducting a comparative study using the available data given by other researchers. The study demonstrates that multiple efficiencies occur in the case of the long-wave region as compared to the short-wave region. Additionally, both the shape and magnitude of the efficiency curves are influenced by the slanted design of the device's front wall and the number of trenches.

Wave power extraction by a dual OWC chambers over an undulated bottom

The impacts of sloping bottom and slanted impenetrable wall on dual oscillating water columns (OWCs) with a slanted front wall are examined. To tackle the physical model, a coupled eigenfunction expansion — boundary element system is employed. The concept is based on employing the anticipated amplification of scattered and reflected wave fields generated by the existence of sloping walls to enhance the absorption of wave power driven by the walls’ wave reflections. The study concludes that 100% efficiency occurs for smaller incident wave angles. It is also observed that compared with short waves, the efficiency of the device is more in the long-wave region. OWCs are observed to be more effective over the undulating bottom and sloping impermeable wall with an increased spacing between the water columns and the second OWC to the impermeable wall. Additionally, the chambers’ width and the sloping bottom angle are vital parameters for higher power extraction from the OWCs. Furthermore, the optimal values of slope bottom and wall inclination may be employed to build and implement an efficient OWC device. Moreover, the proposed model can also be used as a breakwater to protect the coastline.

Flow Characteristics of the Rip Current System near a Shore-Normal Structure with Regular Waves

Rip currents are the strong narrow seaward current produced by waves breaking on the coast. Of the many types of rip currents, the present study investigates, experimentally and numerically, the rip current system produced by the wave reflection on a finite-length shore-normal structure (corresponding to groin, jetty and headland in a real-word environment) with strong wave reflection strength. The incident wave condition of an obliquely propagating monochromatic surface gravity wave is considered for the bottom topographies of planar and barred beaches without or with a rip channel. The corresponding laboratory experiment was conducted and used to validate the numerical model. A set of higher order Boussinesq equations is used to reproduce the experimental observation and produce the results for the related prototype cases, with the latter considering more factors which may influence the formation and evolution of rip current flow patterns. The resulting rip current system contains two parts: the node rips formed due to the presence of the longshore standing wave in the wave reflection field and the deflection rip formed due to the interaction between the incoming longshore current and the reflection wave field mentioned above. A new formation mechanism of the deflection rip has been proposed, i.e., the deflection rip is formed by the meeting and then deflection of the longshore current and lateral flow. The effects of different wave reflection strengths of the structure, wave incident angles and bathymetric conditions on the pattern and flow volume of the deflection rip are also studied.

Boundless Metamaterial Experimentation: Physical Realization of a Unidirectional Virtual Periodic Boundary Condition

We experimentally implement a one-directional virtual geometric periodicity in an elastic metamaterial. First, unwanted boundary reflections at the domain ends are cancelled through the iterative injection of the polarity-reversed reflected wave field. The resulting boundless experimental state allows for a much better analysis of the influence of metamaterials on the propagating wave field. Subsequently, the propagating wave field exiting on one end of the structure is reintroduced at the opposite end, creating a virtual geometric periodicity in one propagation direction. We find that the experimentally observed band gap converges to the analytical solution through the introduction of the virtual periodicity. The established work flow introduces an approach to the experimental investigation and validation of metamaterial prototypes in the presence of strongly dispersive wave propagation and internal scattering. The fully data-driven ad hoc treatment of boundary conditions in metamaterial experimentation with arbitrary mechanical properties enables reflection suppression, virtual periodicity, and the introduction of more general fictitious boundaries.

A simple method for fast simulation of total reflection waves of single-well imaging with eccentric tools

Abstract In the exploration of unconventional oil and gas reservoirs, single-well acoustic imaging has become an important technique to detect geophysical structures around the borehole. In this study, we propose a simple method to calculate the total reflection wavefields of pressure and displacement for single-well imaging (SWI) with eccentric monopole or dipoles. To calculate the pressure at the eccentric site for incident waves with different angles of incidence and azimuth, reciprocity is applied to the radiation from the eccentric monopole source and the far-field forces generating the waves corresponding to the incident waves. Displacement solutions are derived by calculating the gradient of the pressure fields. When considering eccentric monopole or dipole sources for an inclined reflector outside the borehole, reciprocity is again used to derive the total reflection waves, including the same type of waves (P-P/SV-SV/SH- SH) and the converted waves (P- SV/SV-P). The analytical results agree well with the reference solutions of the 3D finite-difference time-domain method, illustrating the validation of the method. The analysis shows that the pressure amplitude in the eccentric position is greater than that in the borehole axis, proving that the eccentric monopole can increase the weak amplitude of reflection waves. We also simulate the wavefields for the acoustic imaging model for eccentric monopole and dipoles. The results show that the pressure and displacement wavefields in the eccentric position are sensitive to the azimuth angle of the reflector, proving that SWI with eccentric logging tools is capable of identifying the azimuth angle.

Tunable reflection and broadband absorption of flexural waves by adaptive elastic metasurface with piezoelectric shunting circuits

Elastic metasurfaces have attracted lots of attention due to their extraordinary ability in manipulating elastic waves. Among various elastic metasurfaces, the adaptive elastic metasurface (AEM) has more flexibility because of the tunability in function and working frequency band without changing the geometrical configuration. In this paper, we propose an AEM composed of sandwiched plates with mass blocks at their free ends to realize tunable reflection and high-efficiency absorption of flexural waves in broadband. The upper and lower parts of the sandwiched plate are piezoelectric patches individually shunted with a hybrid circuit in series of a resistance and negative capacitance. We solve the reflection coefficient/phase shift of the subunit and the full reflected wave field of the AEM by using the transfer matrix method and coupled-mode theory, respectively. The modulation mechanisms of the phase shift and reflection coefficient are revealed. Especially, the influence of negative effective rigidity on the phase shift is investigated. The moment of inertia generated by the mass block plays a key role in reducing the sensitivity of the phase shift to negative capacitance. Based on the theoretical analyses, the AEMs are designed to realize tunable reflection, switchable asymmetric reflection and high-efficiency absorption. The results obtained from analytical solutions and finite element simulations are consistent with each other. The proposed AEM may have potential applications in vibration control and noise reduction.

Offshore subsurface characterization enabled by fiber-optic distributed acoustic sensing (DAS): An East China Sea 3D VSP survey example

Recent advances in distributed acoustic sensing (DAS) technology have allowed more intense measurements of subsurface and environment events, providing improved geohazard monitoring and subsurface characterization. This study discussed the subsurface evaluation enabled by the DAS-vertical seismic profiling (DAS-VSP) system in the East China Sea, China. Taking advantage of a continuous recording of the vibrational wavefields through the strain deformation of the fiber deployed along the borehole, DAS-VSP is considered an emerging and promising alternative borehole acquisition method. It provides a wider-spectrum range of recordings in a cheaper, denser, and more continuous tense compared to conventional geophones. We explored the 3D DAS-VSP signal processing and imaging strategy. Based on a set of 3D DAS-VSP data of exceptional quality, this study reviewed the common processing challenges and practical solutions for de-noise, de-ghosting, de-multiple, and wavefield separation arising from the DAS acquisition mechanism in the offshore VSP scenario. High-quality down-going multiples were separated and imaged using the pre-stack Gaussian beam depth migration in addition to the primary reflection wavefield, providing significant additional illumination to support the subsurface evaluation. The current results validated the efficiency of the DAS-VSP survey and encouraged better geological interpretation.

Joint passive seismic imaging based on surface wave inversion and reflection wavefield retrieval: A case study in the sedimentary basin areas adjacent to Well Songke-2

Owing to their cost-effectiveness, environmental friendliness, and high-efficiency, passive seismic methods are widely used in geophysical exploration. However, field passive seismic data suffer from low signal-to-noise ratio, as well as the fact that different types of seismic waves are naturally mixed together, which can affect the accuracy of subsurface imaging results and the subsequent geological interpretation. In this paper, we demonstrate that these problems can be addressed and present a case study on passive seismic detection in sedimentary basin areas adjacent to Well Songke-2 in Songliao Basin. To obtain accurate and reliable imaging results of the sedimentary strata, a detailed practical scheme is proposed, where fundamental and higher mode surface wave dispersion curves are inverted for obtaining the near-surface S-wave velocity profile, and the body wave component of ambient noise is utilized to retrieve the reflection wavefield information. The obtained profiles from surface wave inversion and reflection wavefield retrieval illustrate similar underground structures. The marker boundaries T2 (1.5 s) and T4 (2.1 s) are well demonstrated, and a low velocity stratum (0.4 s) is detected at a shallow depth of around 400–600 m. Further, the results are highly consistent with the data obtained from borehole logging of Well Songke-2 and the deep reflection seismic profile adjacent to this area, which indicates that the surface wave and body wave in passive seismic data can be utilized together to contribute to a detailed and accurate subsurface imaging and interpretation. Overall, this study investigated and validated the reliability and accuracy of the combination use of passive seismic methods for geological structure exploration, which can further boost their applications for geological interpretation in sedimentary basin areas.

The influence of curvilinear form of boundaries on reflected wave field

The paper is devoted to the influence of curvilinear reflection boundaries on kinematic and dynamic 1haracteristics of reflected seismic waves. First of all, the paper describes the major reasons of low geological efficiency of CDP seismic survey in the areas of Azerbaijan, characterized by complex surface and deep seismic and geological environmental features and in this aspect, the special attention is paid to curvilinear boundaries. The paper displays statistical data for Shamakhy-Gobustan province, where of 132 anticlinal folds only a few are under production currently. It is considered, how the spatial curvilinear nature of reflection surfaces influence reflection waves propagation. In addition, by use of 3D seismic data the formula is given for breakdown of reflection waves along propagation planes in space. As the example, VSP data from a well in Absheron peninsula are displayed and on the seismic records, it is possible to clearly observe the out of plane waves, which do not propagate in the section plane (in the vertical plane). Velocity analysis is based on evaluation of coherency measure, which allow to assess the correlation level of reference point of seismic traces in the given time window along the hyperbolic trajectory of curvature depending on time and velocity. It is known that in case of curvilinear reflection boundaries the observed travel time graphs are far from hyperbolic by their shape, and due to this, in the process of velocity analysis we derive both optimal stacks, which are the subject to the rule of velocity change versus time, common to this area, and also quite different stacks, which we name as “anomalous”. The great importance is to make clear the nature of these “anomalous” stacks at the initial stage of study as they keep some hidden information about the geology of the area. Analysis display that in addition to folded reflections, the cause of these anomalous stacks may be the curvilinear boundaries, horizontal gradient of the velocity. Curvilinear boundaries influence also the dynamic parameters, wave amplitudes in particular, as in the case of curvilinear boundaries the seismic rays are focusing due to the concaved shape of boundary. It has been noted that if the curvature of a syncline form boundary is not constant, then at the defined reception points the reflections arrive from several parts (points) of syncline and the reflected energy will be recorded as several branches (most frequently as three). As the light is focused when it passes through the lens, the seismic waves are similarly focused by curvilinear boundaries and due to the refraction the seismic rays are bended and we observe intensive reflections on seismic sections.

Acoustic vortex beam reflection, refraction, and orbital Hall effect

Some of our recent work concerned the interactions of twisted wave fronts of acoustic vortex beams with boundaries and inhomogeneities of media. Experimental measurements and numerical simulations were conducted to investigate vortex beam reflection from a water-air interface and vortex beam refraction when travelling in an inhomogeneous medium or through an acoustic metasurface. The results reveal spatial evolution of the wave fields and the carried angular momenta. In particular, the investigation observed a reversal of orbital angular momentum in the reflected wave field [Zou et al., Phys. Rev. Lett. 125(7), 074301 (2020)], and a shift of beam center and an asymmetric vortex pattern in the refracted wave field in the direction transverse to the plane of incidence [Fan et al., Phys. Rev. Res. 1(3), 032014 (2019); 3(1), 013251 (2021)]. The shift relates to conservation of angular momenta and coupling of the twisted wave fronts with the refraction and is termed as acoustic orbital Hall effect.

The Inverse Fresnel Beam XSP-CDP Stack Imaging in Crosswell Seismic

In order to overcome the shortcomings of serious arc drawing and low computational efficiency in the crosswell seismic migration method and the problems of the inaccurate velocity model and sparse distribution of reflection points in the traditional stack imaging method, the article proposes an inverse Fresnel beam XSP-CDP stack imaging method based on first-arrival wave velocity tomography combined with the characteristics of crosswell seismic wave field. Firstly, an accurate crosswell velocity model is established by the first-arrival wave tomography inversion method based on the characteristics of high energy and easy pick-up of the first-arrival wave in crosswell seismic. Secondly, the velocity model is optimized, and the energy contribution weights of effective rays to the receiver point are calculated through the crosswell seismic Fresnel beam wave field forward numerical simulation method. Then, the reflected wave field is dynamically migrated to the reflection points within the first Fresnel zone according to the weight function, and the intensive common reflection point (CRP) gather after normal moveout (NMO) correction is generated. Finally, an appropriate bin is selected for stacking. In this article, the inverse Fresnel beam method is used to decompose the single-channel seismic wave field into the effective reflection points in the Fresnel zone, which makes the fold of the reflection point more uniform and improves the imaging accuracy. The model test and actual data processing results proved the validity and robustness of this method.

Effective diffraction separation using the improved optimal rank-reduction method

Seismic diffractions encoding subsurface small-scale geologic structures have great potential for high-resolution imaging of subwavelength information. Diffraction separation from the dominant reflected wavefields still plays a vital role because of the weak energy characteristics of the diffractions. Traditional rank-reduction (RR) methods based on the low-rank assumption of reflection events have been commonly used for diffraction separation. However, these methods using truncated singular-value decomposition (TSVD) suffer from the problem of reflection-rank selection by singular-value spectrum analysis, especially for complicated seismic data. In addition, the separation problem for the tangent wavefields of reflections and diffractions is challenging. To alleviate these limitations, we have developed an effective diffraction separation strategy using an improved optimal RR (ORR) method to remove the dependence on the reflection rank and improve the quality of separation results. The improved RR method adaptively determines the optimal singular values from the input signals by directly solving an optimization problem that minimizes the Frobenius-norm difference between the estimated and exact reflections instead of the TSVD operation. This improved method can effectively overcome the problem of reflection-rank estimation in the global and local RR methods and adjusts to the diversity and complexity of seismic data. The adaptive data-driven algorithms indicate good performance in terms of the trade-off between high-quality diffraction separation and reflection suppression for the ORR operation. Applications of our strategy to synthetic and field examples demonstrate the superiority of diffraction separation in detecting and revealing subsurface small-scale geologic discontinuities and inhomogeneities.

Detecting small-scale geological structures using diffraction attributes

We developed a method that uses diffracted wave data to address the issue of precisely characterizing small geological structures in a reservoir and demonstrated its effectiveness using model calculations. The method was implemented to characterize a fractured reservoir and igneous rock intruded in a carbonate reservoir. Two diffraction attributes were analyzed to precisely characterize the fine, small-scale geological structures therein. Principal component analysis was performed to extract diffraction data from seismic wave fields, and the reflection and diffraction wave fields were separated based on the kinematics and dynamics of the diffracted waves. The resulting diffracted wave fields provided an excellent basis upon which to assess the distribution of small geological structures in the study area. This proved, experimentally, that our diffraction-based method has the potential to characterize the distribution of small geological structures precisely and holistically.

Tangential cone condition and Lipschitz stability for the full waveform forward operator in the acoustic regime

Time-domain full waveform inversion (FWI) in the acoustic regime comprises a parameter identification problem for the acoustic wave equation: pressure waves are initiated by sources, get scattered by the earth’s inner structure, and their reflected parts are picked up by receivers located on the surface. From these reflected wave fields the two parameters, density and sound speed, have to be reconstructed. Mathematically, FWI reduces to the solution of a nonlinear and ill-posed operator equation involving the parameter-to-solution map of the wave equation. Newton-like iterative regularization schemes are well suited and well analyzed to tackle this inverse problem. Their convergence results are often based on an assumption about the nonlinear map known as tangential cone condition. In this paper we verify this assumption for a semi-discrete version of FWI where the searched-for parameters are restricted to a finite dimensional space. As a byproduct we establish that the semi-discrete seismic inverse problem is locally Lipschitz stable, in particular, it is conditionally well-posed.

Hydrodynamic Efficiency of a Wave Energy Converter in Front of an Orthogonal Breakwater

In the present study, the hydrodynamic efficiency of a cylindrical wave energy converter (WEC) of vertical symmetry axis and arranged in front of a reflecting orthogonal breakwater is explored. The idea is based on exploiting the anticipated amplification of the scattered and the reflected wave fields originating from the presence of the vertical walls, towards increasing the WEC’s wave power absorption due to the walls’ wave reflections. Two types of converters are examined, namely the heaving device and the oscillating water column (OWC) device, assuming linear potential theory. The associated diffraction-, motion-, and pressure-radiation problems are solved using axisymmetric eigenfunction expansions for the velocity potential around the WECs by properly accounting for the wave field’s modification due to the walls’ presence. To this end, a theoretical formulation dealing with the evaluation of the converter’s performance is presented accounting for the coupling between the WEC and the reflecting vertical walls. The results depict that the amount of the harvested wave power by the WEC in front of an orthogonal wall is amplified compared to the absorbed wave power by the same WEC in the open sea.

Efficient reflection waveform inversion using a locally normalized zero‐lag correlative objective function

ABSTRACTFull‐waveform inversion is characterized by cycle‐skipping when the starting background model differs significantly from the true model and low‐frequency data are unavailable. To mitigate this problem, reflection waveform inversion is applied to provide a background velocity model for full‐waveform inversion. This technique attempts to extract background velocity updates along the reflection wavepath by matching the reflection waveforms. However, two issues arise during the implementation of reflection waveform inversion: amplitude and efficiency. The amplitude is always underestimated due to the complex subsurface parameter (i.e. the source signature, density, attenuation etc.). This makes it unreasonable to match the reflection amplitude involved in waveforms, especially in the filed data cases. In addition, generating the background velocity gradient requires the simulation of the reflection wavefield. However, simulating the reflection wavefield is time‐consuming. To address the former, we introduced a locally normalized objective function, while for the latter, we used an efficient strategy by avoiding the explicit generation of the reflection wavefield. Results show that applying the proposed method to both synthetic and field data can provide a good background velocity model for full‐waveform inversion with high efficiency.