Directional Wave Field Research Articles

Real-time phase-resolved ocean wave prediction in directional wave fields: Second-order Lagrangian wave models

The Improved Choppy Wave Model (ICWM) was established to achieve a second-order Lagrangian expansion of surface waves. The second-order Lagrangian nonlinear interaction terms were found to be negligible and were therefore discarded. However, these interactions in directional wave fields remained unexplored. ICWM was successfully used, especially for floating offshore wind turbines, but it was noted that its accuracy in predicting directional sea states needed improvement. Hence, we formulated the Complementary ICWM (CICWM) with the nonlinear terms for free surface elevation in directional waves. Detailed formulations for data assimilation and wave propagation were provided, enabling real-time wave forecasting for directional seas using a simplified assimilation method. Comparing the model performances against tank-scale experiments showed that CICWM enhances the surface elevation description in directional seas. Ultimately, it was confirmed that the second-order Lagrangian model can reduce the prediction error of the linear wave model by 90% in directional seas for the experimental setups and sea states investigated in this study.

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.

Joint inversion of direct waves, primaries, and multiples for vertical cable seismic data

By anchoring vertical cable in seawater and receiving the data excited by the towing cable seismic source, the reflection data from subsurface structures with a high signal-to-noise ratio, wide frequency band, and wide azimuth angle can be obtained, thereby achieving high-quality imaging of the target area. However, due to drifting receivers driven by ocean currents and the nonuniform coverage caused by the vertical orientation of the receivers, the traditional imaging method cannot be directly used in the vertical cable seismic data. Furthermore, crosstalk between the primaries and multiples commonly contaminates the final imaging. A new inversion method, based on the Bayesian inference and the assumption that all errors are Gaussian distributed, is presented to address these problems. The direct waves are used to invert the source/receiver locations. The primary and multiple wavefields are accurately reconstructed based on the reconstructed direct wavefield, and the corresponding reflectivity updates are obtained. The model-domain Hessian matrices consisting of different data are approximated by their diagonal elements to mitigate the nonuniform coverage issue, and the related covariance matrices and data-domain Hessian matrices are defined and approximated, respectively, to alleviate the calculation problem. In addition, the zero-lag correlated wavefields in the gradient are consistent with the actual wavefield propagation process, e.g., a primary wave is generated when the direct wavefield encounters reflectivity, avoiding the crosstalk. Numerical examples for synthetic models and field data illustrate that our method can accurately recover the source/receiver locations and provide high-resolution balanced-amplitude imaging results without evident crosstalk.

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.

Generative adversarial network-enhanced directional seismic wavefield decomposition and its application in reverse time migration

Reverse time migration (RTM) is an accurate method for imaging complex geologic structures without imposing any dip limitations. However, a large amount of high-amplitude low-frequency noise, which is mainly generated by the crosscorrelation of source and receiver wavefields propagating in the same directions, seriously contaminates the image quality. The causal imaging condition with separated up- and downgoing wavefields is an effective approach to reduce these low-frequency artifacts. Explicit up- and downgoing wavefield decomposition based on the Hilbert transform is computationally expensive due to additional wavefield extrapolation and storage for the imaginary parts. Directionally propagating wavefields have distinctive kinematic patterns such as traveltime and wavefront curvature, which provide us an opportunity to implement the wavefield decomposition using the statistical neural network method. Using extrapolated wavefields as the input and the decomposed up-, down-, left-, and rightgoing wavefields as the labeled data, we train a pair of generative adversarial networks to predict the directional wavefields. The training data sets are generated using seismic full-waveform modeling and explicit wavefield decomposition based on the Hilbert transform. Then, the decomposed directional wavefields are incorporated into a novel imaging condition that depends on the subsurface dip angles to compute the reflectivity perpendicular to the reflectors. Numerical experiments demonstrate that our method can produce accurate directional wavefield decomposition results and high-quality reflectivity images without low-wavenumber artifacts.

A real-time wave prediction in directional wave fields: Strategies for accurate continuous prediction in time

Previous research on real-time deterministic sea wave prediction has generally focused on evaluating the accuracy and efficiency of short-term wave fields within a specific prediction zone. However, for a real-time wave prediction system, it is necessary to provide a continuous description of the ocean wave surface based on short-term prediction segments. In this regard, we have developed algorithms for continuous wave prediction in directional wave fields based on the “practical” prediction zone. The practical prediction zone refers to the time interval available for generating a continuous wave forecast by excluding the reconstructed waves from the prediction zone proposed by Kim et al. (2023). We also introduce and discuss several important time factors, such as the update interval of the spatio-temporal wave dataset, the total computation time, and the length of the practical prediction zone. By gaining a deeper understanding of numerical modeling setups, we have established strategies to reduce computational costs, which are directly related to the accuracy of continuous wave prediction. In particular, the development of these strategies suggests guidance in specifying direction and frequency bandwidths for continuous wave prediction.

A Graphic Processing Unit–High-Order Spectral (GPU-HOS) Numerical Wave Tank for the Simulation of Directional Wave Field Evolution over a Long Time

Existing numerical models for direct simulations of stochastically directional waves are limited by their computational burden, and only a few of them can be applied to modeling directional waves on account of nonlinear evolutions on large time scales. In response to these drawbacks, this paper proposes a modified rectangular numerical wave tank based on the high-order spectral (HOS) method. Three main arrangements are responsible for the model improvement. Firstly, the relaxation technique is applied on the four sides of the tank for generating and attenuating waves, with an enhancement in the total computational efforts required of less than 5% being achieved. Secondly, for this paper, the HOS method for solving the nonlinear evolution of waves was modified to a GPU-speedup version for the first time, and the speedup increases dramatically (up to 16-fold) when the grid number is O(106). Thirdly, the higher-order Runge–Kutta method with order 8 is adopted for the purpose of suppressing cumulative temporal errors. Furthermore, several numerical results are presented for the validation of the model. With our modified numerical wave tank, the nonlinear evolutions of random directional waves can be efficiently studied over a large time and space scale.

Beyond Correlations: Deep Learning for Seismic Interferometry.

Passive seismic interferometry is a vastly generalized blind deconvolution question, where different paths through the Earth correspond to different channels called Green's functions; the sources are completely incoherent and not shared by the channels, and the question is to estimate paths (channels) that are not present in the dataset. SI, turning noise to signal, has numerous applications, from monitoring industrial activities to crustal structure investigation. No standard method of signal processing will solve SI. Instead, domain scientists resort to a simple cross-correlation operation, a.k.a. correlogram, which can retrieve the Green's function directly, but only under restrictive assumptions of ergodicity (energy equipartitioning) of the random process generating the seismic source. However, in practice, correlograms are not equal to the empirical Green's function, because these assumptions are generally far from being satisfied in realistic situations. In the framework of supervised learning, we propose to train deep neural networks (NNs) to overcome two limitations of correlation-based SI: the temporal limitation of passive recordings and the spatial limitation of the random source distribution. Deep NNs are trained to implicitly find the relationship between the empirical Green's functions and the correlograms and then used to extract the correct Green's functions from ambient noise. The input of the network is correlograms (a virtual shot gather), and the desired output is the empirical Green's function (the active shot gather). The NN can often retrieve Green's functions from 5-min passive recordings with acceptable accuracy in our synthetic example. Although an exact estimation of the source locations may not be necessary, a prior knowledge of the source directionality (through a preliminary beamforming step) is helpful when training the NN to mitigate the challenges associated with inhomogeneous source distributions (directional wave fields). In this work, all the numerical examples are based on the retrieval of P-wave reflections in the exploration scale and are conducted on synthetic data. We use a modified ResNet in our numerical experiments.

Real-time phase-resolved ocean wave prediction in directional wave fields: Enhanced algorithm and experimental validation

This paper details a comprehensive study of deterministic real-time wave forecasting in directional seas. By using wave models on the basis of a Lagrangian description, a good balance was achieved between computational efficiency and model accuracy. Nevertheless, due to the highly non-uniform spatial distribution of data and the relatively small size of data in time inherent to remote optical measurements, the initial conditions are determined through an optimization process, which is computationally demanding, especially in multidirectional sea states. Accordingly, in order to offer a real-time system of wave prediction in the case of multidirectional waves, we propose a simplified and succinct assimilation method for the process of wave reconstruction. We also develop a three-dimensional spatio-temporal prediction zone where the future evolution of wave fields can be estimated based on wave measurements. Lastly, we outline a tank-scale experimental campaign conducted to mimic the measurements of a LIDAR in a real configuration. A comparison of model performances with the experimental observations shows that in a multidirectional approach, it is necessary to consider wave components in direction as well as in frequency to achieve nearly the same accuracy as for unidirectional seas.

Freak wave in a two-dimensional directional wavefield with bottom topography change. Part 1. Normal incidence wave

In the propagation and evolution of sea waves, previous studies pointed out that the occurrence of the freak wave height is significantly related to the quasi-resonant four-wave interaction in the modulated waves. From numerical--experimental study over an uneven bottom, the nonlinear effect caused by the bathymetry change also contributes to the occurrence of extreme events in unidirectional waves. To comprehensively analyse the two-dimensional wavefield, this study develops an evolution model for a directional random wavefield based on the depth-modified nonlinear Schrödinger equation, which considers the nonlinear resonant interactions and the wave shoaling the shallow water. Through Monte Carlo simulation, we discuss the directional effect on the four-wave interaction in the wave train and the maximum wave height distribution from deep to shallow water with a slow varying slope. The numerical result indicates that the directional spreading has a dispersion effect on the freak wave height. In a shallow-water environment, this effect becomes weak, and the bottom topography change is the main influencing factor in the wave evolution.

A New Probe Into the Innermost Inner Core Anisotropy via the Global Coda‐Correlation Wavefield

AbstractInvestigations of the Earth's inner core (IC) using seismic body waves are limited by their volumetric sampling due to uneven global distribution of large earthquakes and receivers. The sparse coverage of the IC leads to uncertainties in its anisotropy, the directional dependence of seismic velocity. Yet, detailed constraints on anisotropy, such as its magnitude, and spatial distribution, are required to understand the crystallographic structure of IC's iron and its solidification and deformation processes. Here, we present a new method to investigate the IC's anisotropic properties based on Earth's coda‐correlation wavefield constructed from the late coda of large earthquakes. We perform a comprehensive travel time analysis of I2*, an IC‐sensitive correlation feature identified as a counterpart of the direct seismic wavefield's PKIKPPKIKP waves, yet fundamentally different. Namely, I2* is a mathematical manifestation of similarity among specific seismic phases with the same slowness detected in global correlograms in the short inter‐receiver distance range. Our new spatial sampling of the IC overcomes the shortage of direct seismic wavefield paths sensitive to the IC's central volume, also known as the innermost IC (IMIC). The observed I2*’s travel time variations relative to Earth's rotation axis (ERA) support a model of cylindrical anisotropy with 3.3% strength and a zonal pattern of slow axis oriented 55° from ERA. We thus find compelling evidence for a deep IC structure with distinct anisotropy, although we cannot resolve the depth at which the change occurs. This finding reinforces previous inference on the IMIC, with implications for Earth's evolution.

Monitoring Fracture Saturation With Internal Seismic Sources and Twin Neural Networks

AbstractSeismic coda‐wave analysis is a well‐developed method for detecting subtle physical changes in complex media by measuring arrival times in the late‐arriving energy from multiply scattered or reflected waves. However, a challenge arises when multiply scattered waves are not sufficiently separated in time from the direct arrivals to provide a clear coda wave train. Additional complications for monitoring changes in fracture systems arise when the signals originate from unsynchronized internal sources, such as natural or induced seismicity, from acoustic emission, or from transportable intra‐fracture sources (chattering dust), that generate uncontrolled signals that vary in arrival time, amplitude and frequency content. Here, we use a twin neural network (TNN also known as a Siamese neural network) for dimensionality reduction to analyze signals from chattering dust to classify the fluid saturation state of a synthetic fracture system. The TNN with shared weights generates a low‐dimensional representation of the data input by minimizing contrastive loss, serving as the input to a multiclass classifier that accurately classifies whether multiple fractures in a fracture system are fully saturated or partially saturated, or whether a change in saturation has occurred in different fractures in the system. These results show that information buried in unresolved codas from uncontrolled sources can be extracted using machine learning to monitor the evolution of fracture systems caused by physical and chemical processes even when the scattered and direct wave fields overlap.

An assessment of uncertainties in VS profiles obtained from microtremor observations in the phased 2018 COSMOS blind trials

Site response is a critical consideration when assessing earthquake hazards. Site characterization is key to understanding site effects as influenced by seismic site conditions of the local geology. Thus, a number of geophysical site characterization methods were developed to meet the demand for accurate and cost-effective results. As a consequence, a number of studies have been administered periodically as blind trials to evaluate the state-of-practice on-site characterization. We present results from the Consortium of Organizations for Strong Motion Observation Systems (COSMOS) blind trials, which used data recorded from surface-based microtremor array methods (MAM) at four sites where geomorphic conditions vary from deep alluvial basins to an alpine valley. Thirty-four invited analysts participated. Data were incrementally released to 17 available analysts who participated in all four phases: (1) two-station arrays, (2) sparse triangular arrays, (3) complex nested triangular or circular arrays, and (4) all available geological control site information including drill hole data. Another set of 17 analysts provided results from two sites and two phases only. Although data from one site consisted of recordings from three-component sensors, the other three sites consisted of data recorded only by vertical-component sensors. The sites cover a range of noise source distributions, ranging from one site with a highly directional microtremor wave field to others with omni-directional (azimuthally distributed) wave fields. We review results from different processing techniques (e.g., beam-forming, spatial autocorrelation, cross-correlation, or seismic interferometry) applied by the analysts and compare the effectiveness between the differing wave field distributions. We define the M index as a quality index based on estimates of the time-averaged shear-wave velocity of the upper 10 (VS10), 30 (VS30), 100 (VS100), and 300 (VS300) meters and show its usefulness in quantitative comparisons of VS profiles from multiple analysts. Our findings are expected to aid in building an evidence-based consensus on preferred cost-effective arrays and processing methodology for future studies of seismic site effects.

Day after day variations of arrival angles and polarisation parameters of Q bursts recorded at Antarctic station “Akademik Vernadsky”

We present variations in the source bearings and polarization of the horizontal magnetic field of Q bursts recorded at Antarctic station “Akademik Vernadsky” (65.2458° S, 64.2572° W). We analyzed the parameters of the “polarization ellipse” inscribed in the initial impulse of the Q-burst, which is formed by a direct wave propagating from lightning to the observer along a short path in the Earth-ionosphere cavity. Three days, from March 31 to April 2, 2019, a significant part of the Q burst flux was stably observed in a narrow northerly azimuth sector associated with the direction to lightnings in South America. During this period, regular deviations of the measured azimuths from the northern direction were observed up to 8–10°, to the west when the morning terminator passed and to the east at the evening terminator. Duration of these deviations was about 4 h. The azimuth deviations were accompanied by characteristic oscillations in the “ellipticity” value of the direct wave field, N-shaped during the passage of the evening terminator, and a mirror-symmetric И-shaped oscillation on the morning terminator. We found that, on average, the polarization of the direct wave field is close to linear when the paths from the sources to the observer were on the day side, while at night side its “ellipticity” increases to 0.1. The observed azimuth deviations were qualitatively interpreted using the concept of geometric optics of the reflection of radio waves from the day-night non-uniformity in the Earth-ionosphere cavity. The measurement period was close to the day of the vernal equinox, when the propagation path from the sources and the lines of the solar terminator run almost parallel and radio waves are incident on the day-night interface at “sliding angles”, which enhances their “reflection".

Probabilistic assessment of rogue wave occurrence in directional wave fields

Probabilistic assessment of rogue wave occurrence in directional wave fields

Neural Basis of Numeric Incongruity Processing and Mental Arithmetic in Human Brain

This paper presents a general overview of the practical applicability of non-linear potential flow solvers for water wave propagation. Those numerical models are unable to describe explicitly the wave breaking phenomena, including free surface reconnexion, energy dissipation, etc. This reduces their range of application in real sea waves, which are in the most general context irregular and directional wave fields. This study covers the influence of discretization, water depth, directional spreading and spectral peakedness on the limitations in the use of such non-linear potential models. The approach at use relies on a highly non-linear model based on the High-Order Spectral method.

Vector elastic deconvolution migration with dual wavefield decomposition

Elastic-wave imaging using multicomponent data can provide more useful subsurface information than acoustic-wave imaging, but it is usually algorithmically challenging. We have developed a vector elastic deconvolution migration method for high-resolution imaging of subsurface structures in isotropic and anisotropic elastic media. Our new method uses a vector deconvolution imaging condition based on dual wavefield decomposition, including an explicit directional wavefield separation using the Hilbert transform and a P/S vector wavefield decomposition using the low-rank decomposition method. Using three elastic models, we numerically determine that our new method produces notably higher resolution and more amplitude-balanced elastic images compared with a crosscorrelation-based vector elastic reverse time migration method.

Imaging ahead of and around the bit in a desert environment: DrillCAM field trial with wireless geophones and top-drive sensor

Advanced seismic-while-drilling (SWD) technologies are being utilized to steer drilling operations and provide high-resolution subsurface images around and ahead of the bit. We present a case study of SWD imaging using a recently acquired field data set from a desert environment with a complex near surface. Data acquisition is performed with wireless geophones and top-drive sensors using continuous real-time recording. The drill-bit noise data are analyzed while continuously recording in real time by using a specialized workflow that combines elements of SWD and conventional vertical seismic profiling processing with controlled seismic sources. First, the workflow enhances the direct wavefield to retrieve accurate first-break picks for traveltime tomographic inversion along east–west- and north–south-striking walkaway lines. Then, it extracts and enhances upgoing reflection events, illuminating parts of the subsurface around and ahead of the bit. During the final step, these upgoing reflections are imaged using the inverted velocity model to reconstruct a migrated subsurface image around the well. As is the case for land surface seismic in the presence of a complex near surface, we observe a significant variation of data quality for the orthogonal receiver lines. As a result, each line provides a robust image of a different part of the subsurface. The east–west-striking line's migrated image delineates a major shallow reflector that serves as a marker for predicting the drilling depth of a deeper horizon. Likewise, migrating upgoing reflections from the north–south line accurately maps a deeper target horizon ahead of the bit. The obtained SWD images assist in setting the casing points accurately and provide a more precise ahead-of-the-bit depth for different horizons with significantly less uncertainty than surface seismic.

Extreme wave events and sampling variability

Wave field data are affected not only by the accuracy of instruments recording them but also by sampling variability, an uncertainty due to the limited number of observations. For stationary meteorological conditions, due to the randomness of the sea surface elevation, wave parameters derived from a temporal or spatial wave record will depend on which part of a wave record is used in an analysis as well as on the length of a wave record and/or size of the investigated ocean area. This study demonstrates, using numerical simulations, challenges that sampling variability brings in the interpretation of nonlinear wave characteristics of the surface elevation when single 20- or 30-min field wave records are used in an analysis. As examples, we use sea states in which rogue waves were observed in the North Sea and investigate them using linear, second-, and third-order numerical simulations. The third-order wave data are simulated by a numerical solver based on the higher order spectral method (HOSM) which includes the leading order nonlinear dynamical effects, accounting for the effect of modulational instability. Wave steepness, the maximum wave crest, skewness, and kurtosis are investigated in unidirectional and directional wave fields. The study shows that having a single 20- or 30-min wave record may make it difficult to determine on the degree of wave field nonlinearity and the accuracy of derived wave parameters, as well as to evaluate the validity of wave models. Both single-point temporal and stereo-video camera data are discussed. We demonstrate that numerical simulations represent important supporting tools for wave field measurements.

Stochastic ion and electron heating on drift instabilities at the bow shock

Abstract The analysis of the wave content inside a perpendicular bow shock indicates that heating of ions is related to the lower hybrid drift (LHD) instability, and heating of electrons is related to the electron cyclotron drift (ECD) instability. Both processes represent stochastic acceleration caused by the electric field gradients on the electron gyroradius scales, produced by the two instabilities. Stochastic heating is a single-particle mechanism where large gradients break adiabatic invariants and expose particles to direct acceleration by the direct current and wave fields. The acceleration is controlled by function $\chi = m_iq_i^{-1} B^{-2}$div(E), which represents a general diagnostic tool for processes of energy transfer between electromagnetic fields and particles, and the measure of the local charge non-neutrality. The identification was made with multipoint measurements obtained from the Magnetospheric Multiscale spacecraft. The source for the LHD instability is the diamagnetic drift of ions, and for the ECD instability the source is ExB drift of electrons. The conclusions are supported by laboratory diagnostics of the ECD instability in Hall ion thrusters.