Isotropic Wave Field Research Articles

Statistics of rogue waves in isotropic wave fields

We investigate the statistics of rogue waves occurring in the inverse cascade of surface gravity wave turbulence. In such statistically homogeneous, stationary and isotropic wave fields, low-frequency waves are generated by nonlinear interactions rather than directly forced by a wave maker. This provides a laboratory realization of arguably the simplest nonlinear sea state, in which long-time acquisitions are performed and compared with theoretical models. The analysis of thousands of rogue waves reveals that some of their properties crucially depend on four-wave resonant interactions, large crests being for instance more likely than predicted by second-order models.

Microtremor surveys based on rotational seismology: theoretical analysis with focus on separation of Rayleigh and Love waves in general wavefield of microtremors

SUMMARYThe applicability of rotational seismology to the general wavefield of microtremors is theoretically demonstrated based on a random process model of a 2-D wavefield. We show the effectiveness of taking the rotations (i.e. spatial differentiation) of microtremor waveforms in separating the Rayleigh and Love waves in a wavefield where waves are simultaneously arriving from various directions with different intensities. This means that a method based on rotational seismology (a rotational method) is capable of separating Rayleigh and Love waves without adopting a specific array geometry or imposing a specific assumption on the microtremor wavefield. This is an important feature of a rotational method because the spatial autocorrelation (SPAC) method, a conventional approach for determining phase velocities in microtremor array surveys, requires either the use of a circular array or the assumption of an isotropic wavefield (i.e. azimuthal averaging of correlations is required). Derivatives of the SPAC method additionally require the assumption that Rayleigh and Love waves are uncorrelated. We also show that it is possible to apply a rotational method to determine the characteristics of Love waves based on a simple three-point microtremor array that consists of translational (i.e. ordinary) three-component sensors. In later sections, we assume realistic data processing for microtremor arrays with translational sensors to construct a theoretical model to evaluate the effects of approximating spatial differentiation via finite differencing (i.e. array-derived rotation, ADR) and the effects of incoherent noise on analysis results. Using this model, it is shown that in a short-wavelength range compared to the distance for finite differencing (e.g. $\lambda < 3h$, where $\lambda $ and $h$ are the wavelength and distance for finite differencing, respectively), the leakage of unwanted wave components can determine the analysis limit. It is also shown that in a long-wavelength range (e.g. $\lambda > 3h$), the signal intensity gradually decreases, and thus the effects of incoherent noise increase (i.e. the signal-to-noise ratio, SNR decreases) and determine the analysis limit. We derive the relation between the SNR and wavelength. Although the analysis results quantitatively depend on the array geometry used for finite differencing, the qualitative understanding supported by mathematical expressions with a physically clear meaning can serve as a guideline for the treatment of data obtained from ADR.

Application of in-situ anisotropic parameters in 3D seismic velocity analysis for improved pre-drill pore pressure prediction in the onshore Niger Delta basin, Nigeria

The inaccurate prediction and characterization of anisotropic properties of hydrocarbon reservoirs during seismic processing have been a major obstacle to optimal time-to-depth conversion, well-to-seismic ties, and a robust fit-for-purpose velocity model for accurate seismic pore pressure (SPP) imaging. Though some levels of exploration successes have been achieved using isotropic velocity models in parts of the Niger Delta Basin, serious challenges however abound at the deeper shale-dominated intervals of the basin because of the intrinsic anisotropic properties of shale. The application of in situ anisotropic parameters in 3D seismic velocity analysis for improved pore pressure prediction in the study area was carried out with the objectives of applying the appropriate anisotropic parameters for optimal processing of seismic velocity field for overpressure studies. Anisotropic parameters for vertical transverse isotropic (VTI) wave-field were estimated from available wire-line logs using standard empirical relationships and were used to correct for shale anisotropy on a 6.0-km offset seismic data. The migrated velocities were further converted to geological depth and calibrated to wells. Isotropic and anisotropic pre-stack depth migrated (preSDM) velocities were inverted to seismic pore pressure prediction. Results of the study revealed that the anisotropic preSDM velocities yielded higher resolution in the vertical and lateral positioning of geological features when compared with the isotropic velocity model. Also, the seismic velocity fields generated from the anisotropic preSDM seismic velocities were observed to optimally tie to the well and accurately imaged the reservoir pore pressures at deeper intervals. These findings, therefore, revealed the appropriateness of the applied anisotropy parameters (−0.05 ≤ δ ≤ 0.05 and 0.08 ≤ η ≤ 0.48) in predicting elasto-thermal properties of the area. These parameters can therefore be incorporated into anisotropic velocity modeling and processing of long offset seismic data for accurate prediction of pore pressure, lithological, and fluid properties of reservoirs in the area.

2-D WFS Approach Through Field Transformation—Application to Performance Prediction in Standard Antenna Simulators

This paper presents a 2-D wavefield synthesis approach directly implementable in standard antenna simulation software. The approach uses a field transformation matrix and two array sets to design vertically and horizontally polarized wavefields, implemented with thin electric dipoles for their simple geometry. The synthesized wavefields positively correlate with numerical methods (FIT, FEM) obtained from full-wave simulations over various array sets (size, number, polarization, frequency). Synthesizing 2-D isotropic wavefields in both polarizations under an error vector magnitude (EVM) equal to or smaller than -30 dB is demonstrated over the largest test zone radius of λ/10 with just three dipoles located at a minimum distance of 5λ far from the phase center. Application to a test antenna (folded dipole) anticipates the received power and MEG directly inside the antenna simulator in receiving mode. This valuable design option, not handled by most antenna simulators, helps optimize the antenna with the propagation medium characteristics during the design stage.

Diffusion of inertia-gravity waves by geostrophic turbulence

The scattering of inertia-gravity waves by large-scale geostrophic turbulence in a rapidly rotating, strongly stratified fluid leads to the diffusion of wave energy on the constant-frequency cone in wavenumber space. We derive the corresponding diffusion equation and relate its diffusivity to the wave characteristics and the energy spectrum of the turbulent flow. We check the predictions of this equation against numerical simulations of the three-dimensional Boussinesq equations in initial-value and forced scenarios with horizontally isotropic wave and flow fields. In the forced case, wavenumber diffusion results in a $k^{-2}$ wave energy spectrum consistent with as-yet-unexplained features of observed atmospheric and oceanic spectra.

Scattering of internal tides by barotropic quasigeostrophic flows

Oceanic internal tides and other inertia–gravity waves propagate in an energetic turbulent flow whose length scales are similar to the wavelengths. Advection and refraction by this flow cause the scattering of the waves, redistributing their energy in wavevector space. As a result, initially plane waves radiated from a source such as a topographic ridge become spatially incoherent away from the source. To examine this process, we derive a kinetic equation which describes the statistics of the scattering under the assumptions that the flow is quasigeostrophic, barotropic and well represented by a stationary homogeneous random field. Energy transfers are quantified by computing a scattering cross-section and shown to be restricted to waves with the same frequency and identical vertical structure, hence the same horizontal wavelength. For isotropic flows, scattering leads to an isotropic wave field. We estimate the characteristic time and length scales of this isotropisation, and study their dependence on parameters including the energy spectrum of the flow. Simulations of internal tides generated by a planar wavemaker carried out for the linearised shallow-water model confirm the pertinence of these scales. A comparison with the numerical solution of the kinetic equation demonstrates the validity of the latter and illustrates how the interplay between wave scattering and transport shapes the wave statistics.

Nearly cloaking for the elasticity system with residual stress

The nearly cloaking via transformation optics approach for the isotropic elastic wave fields is considered. This work extends the study of the nearly cloaking scheme to the elasticity system with residual stress in \mathbb{R}^{N} for N=2,3, which is an anisotropic elasticity system. It is worth to mention that there are no minor symmetric properties for the elastic tensor with residual stress and this system is invariant under a coordinate transformation. Therefore, we think the elasticity system residual stress model is more natural than the isotropic elasticity system in designing the elastic cloaking medium in the physical sense. In addition, the main difficulty of treating this problem lies in the fact that there are no layer potential theory for the residual stress system. Instead, we will derive suitable elliptic estimates for the elasticity system residual stress by comparing with the Lam\'e system to achieve desired results.

Elastic wavefield tomography with physical model constraints

Data-domain elastic wavefield tomography is an effective method for updating multiparameter elastic models that exploits much of the information provided by observed multicomponent data. However, poor illumination of the subsurface by P- and S-waves often prevents reliable updates of the model parameters. Moreover, differences in illumination, amplitude, and wavelength between P- and S-waves can distort the intrinsic physical relationships between the reconstructed model parameters. We have developed a method for elastic isotropic wavefield tomography that explicitly constrains the relationship between P- and S-wave velocities. By incorporating a model constraint term in the objective function, we confine P- and S-velocity updates to a physical area defined by prior information, for example, by laboratory measurements or well logs. We have determined that this physical constraint yields models that are more physically plausible, compared with models obtained using only the data misfit objective function.

Geophysics Bright Spots

An explosion of Bright Spots illuminate the July–August issue of Geophysics. In this issue, we learn an efficient technique for up/down and P/S elastic wavefield separation in reverse time migration (RTM) that yields more accurate Poynting vectors while suppressing RTM artifacts. We see the derivation of an efficient formula for correlative least squares in RTM. We learn about a Q-compensated RTM with one-step wave extrapolation. We are introduced to a new approach to isotropic elastic wavefield imaging using an energy norm. We also see the derivation of a seismoelectric Green's function for homogeneous media. We learn a new method by which P- and S-wave velocities are extracted via surface-wave multimodal inversion. We also learn about a new method for finding the relative location of microseismic events with multiple master events, and we see how to apply frequency cycling for the compensation for undesired off-resonance effects in surface nuclear magnetic resonance (NMR).

An efficient wave extrapolation method for anisotropic media with tilt

ABSTRACTWavefield extrapolation operators for elliptically anisotropic media offer significant cost reduction compared with that for the transversely isotropic case, particularly when the axis of symmetry exhibits tilt (from the vertical). However, elliptical anisotropy does not provide accurate wavefield representation or imaging for transversely isotropic media. Therefore, we propose effective elliptically anisotropic models that correctly capture the kinematic behaviour of wavefields for transversely isotropic media. Specifically, we compute source‐dependent effective velocities for the elliptic medium using kinematic high‐frequency representation of the transversely isotropic wavefield. The effective model allows us to use cheaper elliptic wave extrapolation operators. Despite the fact that the effective models are obtained by matching kinematics using high‐frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The methodology developed here offers a much better cost versus accuracy trade‐off for wavefield computations in transversely isotropic media, particularly for media of low to moderate complexity. In addition, the wavefield solution is free from shear‐wave artefacts as opposed to the conventional finite‐difference‐based transversely isotropic wave extrapolation scheme. We demonstrate these assertions through numerical tests on synthetic tilted transversely isotropic models.

Elliptical dip moveout (EMO) for 3D seismic imaging in the presence of azimuthal anisotropy

Seismic images of the earth?s interior can be significantly distorted by complex wave propagation effects arising from 3D structural velocity variations, combined with the presence of azimuthal velocity anisotropy within some of the rock layers. Most image processing techniques attempt to separate and compensate for both of these phenomena sequentially; they rarely address both simultaneously. These approaches implicitly assume that the effects of 3D structural velocity and azimuthal anisotropy are separable, when in fact both effects are coupled together in the seismic data. The presence of strong azimuthal velocity anisotropy can lead to significant errors in estimated seismic velocity and degraded quality of subsurface images, especially for large source-receiver offsets, wide azimuths and steep geologic dips. Such imaging errors can greatly increase the uncertainty associated with exploring, characterizing, developing and monitoring subsurface geologic features for hydrocarbons, geothermal energy, CO2 sequestration, and other important geophysical imaging applications. Our approach is to simultaneously address velocity structure and azimuthal anisotropy by development of an elliptic dip moveout (DMO) operator. We combine the structural-velocity insensitivity of isotropic DMO with elliptic moveout representative of azimuthal velocity anisotropy. Forward and adjoint elliptical DMO operators are then cascaded together to form a single elliptical moveout (EMO) operation, which has a skewed saddle-like impulse response that resembles an isotropic azimuthal moveout operator. The EMO operator can be used as a prestack data conditioner, to estimate azimuthal anisotropy in a domain that is relatively insensitive to 3D velocity structure, or to compensate and map the data back to its original prestack domain in its approximately equivalent isotropic wavefield form. We show that EMO can reduce structural dip image errors of 10-20o or more for realistic azimuthal anisotropy values at far offsets.

Wave number filtering on a finite periodically supported plate: Implications on the vibration field, radiated power, and validity of SEA

Preston Smith published and presented a number of papers on wave propagation and sound radiation from periodically supported plates. An excellent reference is, “Radiation from Periodically Supported Fluid-Loaded Plates”, BBN Report No. 3999, January 1979. In these papers he identifies the physics of Bloch wave radiation, coherent scattering from ribs and spatial attenuation. As an extension to Preston’s work, the problem of propagation of a local isotropic wave field on a finite periodically supported plate is considered. The specific question to be addressed is: do the rib supports provide wave number filtering of the isotropic field as it propagates throughout the plate-stiffener system? This question is answered via a series of analytical and finite element models. The implications on radiated power and validity of the SEA isotropic vibration field assumption will be discussed.

Analytical linear theory for the interaction of a planar shock wave with a two- or three-dimensional random isotropic acoustic wave field

We present an analytical model that describes the linear interaction of a planar shock wave with an isotropic random sonic field. First, we study the interaction with a single-mode acoustic field. We present the exact evolution for the pressure, velocity, vorticity, and density field generated behind the shock wave, and we also calculate exact and closed analytical expressions for the asymptotic behavior of these modes. Applying superposition, we use the results obtained from the single-mode analysis in order to compute the interaction with 2D/3D isotropic random acoustic fields. We present analytical expressions for the average turbulent kinetic energy generated behind the shock, as well as the averaged vorticity and the density perturbations as a function of the shock strength M(1) and the gas compressibility γ. We also study the acoustic energy flux emitted by the shock front. Exact asymptotic analytical scaling laws are given for all the 3D averages downstream. A detailed comparison with previous works is shown.

Elliptical dip moveout for 3D seismic imaging in the presence of azimuthal anisotropy

Seismic images of the earth’s interior can be significantly distorted by complex wave propagation effects arising from 3D structural velocity variations, combined with the presence of azimuthal velocity anisotropy within some of the rock layers. Most image-processing techniques attempt to separate and compensate for both of these phenomena sequentially; they rarely address both simultaneously. These approaches implicitly assume that the effects of 3D structural velocity and azimuthal anisotropy are separable, whereas in fact, both effects are coupled together in the seismic data. In the presence of strong azimuthal velocity anisotropy, this can lead to significant errors in seismic velocity estimation and degraded quality of subsurface images, especially for large source-receiver offsets, wide azimuths, and steep geologic dips. Such imaging errors can greatly increase the uncertainty associated with exploring, characterizing, developing and monitoring subsurface geologic features for hydrocarbons, geothermal energy, [Formula: see text] sequestration, and other important geophysical imaging applications. Our approach simultaneously addressed velocity structure and azimuthal anisotropy by development of an elliptic dip moveout (DMO) operator. We combined the structural-velocity insensitivity of isotropic DMO with elliptic moveout representative of azimuthal velocity anisotropy. Forward and adjoint elliptical DMO operators were then cascaded together to form a single elliptical moveout (EMO) operation, which had a skewed saddle-like impulse response that resembles an isotropic azimuthal moveout operator. The EMO operator can be used as a prestack data conditioner, to estimate azimuthal anisotropy in a domain that is relatively insensitive to 3D velocity structure, or to compensate and map the data back to its original prestack domain in its approximately equivalent isotropic wavefield form. We demonstrated that EMO can reduce structural dip image errors of 10°–20° or more for realistic azimuthal velocity anisotropy values at far offsets.

2D and 3D elastic wavefield vector decomposition in the wavenumber domain for VTI media

A pragmatic decomposition of a vector wavefield into P- and S-waves is based on the Helmholtz theory and the Christoffel equation. It is applicable to VTI media when the plane-wave polarization is continuous in the vicinity of a given wavenumber and is uniquely defined by that wavenumber, except for the kiss singularities on the VTI symmetry axis. Unlike divergence and curl, which separate the wavefield into a scalar and a vector field, the decomposed P- and S-wavefields are both vector fields, with correct amplitude, phase, and physical units. If the vector components of decomposed wavefields are added, they reconstruct those of the original input wavefield. Wavefield propagation in any portions of a VTI medium that have the same polarization distribution (i.e., the same eigenvector) in the wavenumber domain have the same decomposition operators and can be recon-structed with a single 3D Fourier transform for each operator (e.g., one for P-waves and one for S-waves).This applies to isotropic wavefields and to VTI anisotropic wavefields, if the polarization distribution is constant, regardless of changes in the velocity. Because the anisotropic phase polarization is local, not global, the wavefield decomposition for inhomogeneous anisotropic media needs to be done separately for each region that has a different polarization distribution. The complete decomposed vector wavefield is constructed by combining the P-, SV-, and SH-wavefields in each region into the corresponding composite P-, SV-, and SH-wavefields that span the model. Potential practical applications include extraction of separate images for different wave types in prestack reverse time migration, inversion, or migration velocity analysis, and calculation of wave-propagation directions for common-angle gathers.

Passive Imaging in Nondiffuse Acoustic Wavefields

A main property of diffuse acoustic wavefields is that, taken any two points, each of them can be seen as the source of waves and the other as the recording station. This property is shown to follow simply from array azimuthal selectivity and Huygens principle in a locally isotropic wavefield. Without time reversal, this property holds approximately also in anisotropic azimuthally uniform wavefields, implying much looser constraints for undistorted passive imaging than those required by a diffuse field. A notable example is the seismic noise field, which is generally nondiffuse, but is found to be compatible with a finite aperture anisotropic uniform wavefield. The theoretical predictions were confirmed by an experiment on seismic noise in the mainland of Venice, Italy.

On bias and noise in passive seismic data from finite circular array data processed using SPAC methods

The finite nature of typical small seismic arrays used in conjunction with spatial autocorrelation (SPAC) processing for observing the microtremor wavefield causes predictable perturbations of the SPAC spectrum when sources of seismic noise are confined to a restricted range of azimuths. Such perturbations are especially evident at higher frequencies where wavelengths are on the order of the array radius. The effects are readily modeled and show that the triangular array geometries commonly used for microtremor studies require azimuthal distributions of wave energy on the order of [Formula: see text] or greater to have a high probability of being free of such perturbations. The imaginary component of the SPAC spectrum, which is ideally zero for a sufficiently dense circular array and/or a sufficiently isotropic wavefield, is in practice often nonzero and provides three quality-control indicators: (1) an indication of insufficient spatial averaging, (2) an empirical measure of the level of statistical uncertainty in SPAC spectral estimates, and (3) an indication of departures from plane-wave stationarity of the seismic noise wavefield.

Correlation in ambient seismic noise and the reconstruction of Green function

Cross-correlations between long continuous records of ambient seismic noise at distant stations are investigated. The dominant part of the Green function, namely Rayleigh waves, are reconstructed in a broad period range. This property reminds of the fluctuation-dissipation theorem that relates the random fluctuations of a linear system and the system’s response to an external force. Ambient seismic noise is indeed not a thermal noise but it can be considered as a random and isotropic wave field both because the distribution of the ambient sources responsible for the noise randomizes when averaged over long periods and because of scattering from heterogeneities that occur within the Earth. The dispersion curves of Rayleigh waves for the paths between the stations are measured from the correlations. On paths where direct measurements between earthquake and station are available, we show that they are in good agreement with those deduced from noise correlation. The measurement of correlation along paths crossing different geological structures allows to differentiate them, opening the way for a passive imaging of the Earth structure. The dispersion measurements are applied to seismic tomography at the regional scale. They make it possible to image crustal structures with a resolution higher than conventional techniques.

Interferometric coherency determination of wavelength or what are broadband ELF waves?

To determine the wavelength of waves within a random, isotropic wave field, we introduce the observable of wave coherency measured with plasma wave interferometers. We show generally that within a random direction wave field, wavelengths large compared to the interferometer length produce large coherency (nearly 1), but wavelengths the order of a few times the interferometer length, or smaller, produce small coherency (close to zero). We apply this principle first to examining auroral hiss and lower hybrid waves measured by the Physics of Auroral Zone Electrons (PHAZE) 2 and Topside Probe of the Auroral Zone (TOPAZ) 3 experiments and show that the implied wavelengths are consistent with the expected dispersion relations and with other, different estimates of wavelength for these modes. Next, we apply the principle to broadband extra low frequency (BB‐ELF) electric fields observed in both experiments and conclude that the wavelengths are small. In one case we calculate the coherency of BB‐ELF electric fields, using an ensemble average of 7889 data samples, and demonstrate that the coherency near the oxygen gyrofrequency is very small (≅0.15), corresponding to wavelengths of 10 m and the order of the ion gyroradius. We conclude that because of the short wavelengths, previous satellite measurements of BB‐ELF electric fields may have underestimated the electric field amplitudes, unless ion gyroradii are substantially larger than the case for these rocket measurements. Although the wavelengths and frequencies of BB‐ELF electric fields are now known, we are unable to assign the wave to a known, normal mode of homogeneous plasmas. This suggests that inhomogeneities may be essential for describing BB‐ELF electric fields.

Level-crossing densities in random wave fields

Level-crossing densities along two orthogonal directions in an isotropic two-dimensional Gaussian random wave field are discussed for the real and the imaginary parts of the wave function, for the intensity, for the phase, and for all the first- and second-order spatial derivatives of these functions. Analytical expressions are given for most crossing densities and are supplemented by numerical densities obtained from multidimensional Monte Carlo evaluations in cases in which analysis proved intractable. The analytical results and the Monte Carlo evaluations are generally found to be in good agreement with densities derived from a computer simulation that yields an accurate numerical representation of the wave function.