Background Medium Research Articles

3-D scattering of elastic waves by small-scale heterogeneities in the Earth’s mantle

SUMMARY Small-scale heterogeneities in the Earth’s mantle, the origin of which is likely compositional anomalies, can provide critical clues on the evolution of mantle convection. Seismological investigation of such small-scale heterogeneities can be facilitated by forward modelling of elastic wave scattering at high frequencies, but doing so with conventional 3-D numerical methods has been computationally prohibitive. We develop an efficient approach for computing high-frequency synthetic wavefields originating from small-scale mantle heterogeneities. Our approach delivers the exact elastodynamic wavefield and does not restrict the geometry or physical properties of the local heterogeneity and the background medium. It combines the technique of wavefield injection and a numerical method called AxiSEM3D. Wavefield injection can decompose the total wavefield into an incident and a scattered part. Both these two parts naturally have low azimuthal complexity and can thus be solved efficiently using AxiSEM3D under two different coordinate systems. With modern high-performance computing (on an order of magnitude of 105 CPU-hr), we have achieved a 1 Hz dominant frequency for global-scale problems with strong deep Earth scattering. Compared with previous global injection approaches, ours allows for a 3-D background medium and yields the exact solution without ignoring any higher-order scattering by the background medium. Technically, we develop a traction-free scheme for realizing wavefield injection in a spectral element method, which brings in several flexibilities and simplifies the implementation by avoiding stress or traction computation on the injection boundary. For a spherical heterogeneity in the mid-lower mantle, we compare the 3-D full-wave solution with two approximate ones obtained, respectively, by the perturbation theory and in-plane (axisymmetric) modelling. As a comprehensive application, we study S-wave scattering by a 3-D ultra-low velocity zone, incorporating 3-D crustal structures on the receiver side as part of the background model.

Orthotropic anisotropy: on contributions of elasticity parameters to a difference in quasi‐P‐wave‐squared velocities resulted from propagation in two orthogonal symmetry planes

ABSTRACTWe investigate the dependence of quasi P‐wave phase velocity propagating in orthotropic media on particular elasticity parameters. Specifically, due to mathematical facilitation, we consider the squared‐velocity difference, , resulted from propagation in two mutually perpendicular symmetry planes. In the context of the effective medium theory, may be viewed as a parameter evaluating the influence of cracks – embedded in the background medium – parallel to one or both aforementioned planes. Our investigation is both theoretical and numerical. Based on Christoffel's equations, we propose two accurate approximations of . Due to them, we interpret the aforementioned squared‐velocity difference as being twice more dependent on , than on . To describe the magnitude of the dependence, we consider the proportions between the partial derivatives of . Further, it occurs that is influenced by the ratio of vertically propagating quasi P‐wave to vertically propagating quasi S‐wave. Anomalously high might be caused by the low P/S ratio, which in turn can be an indicator of the presence of gas in natural fractures or aligned porosity. Also, we carry out numerical sensitivity study, according to which is approximately twice more dependent on than on , twice more sensitive to than to , and equally dependent on as on . The dependence on and can be neglected, especially for small phase angles. We verify the approximations and perform the sensitivity study, using eight examples of the elasticity tensors.

Quality Enhancement in Holographic Imaging by Background Property Estimation

In this communication, we propose a technique to enhance the quality of images obtained from holographic microwave imaging. It is based on estimating the electrical properties of the background medium. To accomplish that, background properties to be estimated are assumed to be within a preknown range. Then, two sets of frequencies are employed in wideband holographic imaging and images are reconstructed from each set and for the assumed property values within the predetermined range. An error is calculated according to the differences between the two sets of images. The error is expected to be minimum at the true values of the background medium's properties. The estimated properties, in turn, are used to reconstruct images with the best quality. The validity of the proposed technique is demonstrated via simulation and experimental results.

Analysis of wave dispersion and attenuation effects on seismic low-frequency reflections of a poroelastic layer

We investigated reflection dispersion characteristics of a poroelastic layer with patchy saturations in seismic low-frequency range. Frequency-dependent attenuation and dispersion of phase velocity were determined with varying fluid saturation, porosity, permeability and heterogeneity size, using analytical solutions of a generalised 1D White’s model proposed here, obtained by incorporating capillary forces into the patchy saturation model. Numerical predictions were employed in combinations with an approximation for the complex and frequency-dependent reflection coefficient from an attenuating thin layer embedded in a dissipative background medium, so as to give a deep insight into the impacts of velocity dispersion and tuning, as well as acoustic impedance contrast, on the seismic low-frequency reflections. Results from this study suggest that wave-induced fluid flow in poroelastic media might account for low-frequency reflection signature anomalies that are associated with fluid saturation in the pore space. We also observed that seismic reflection coefficients of a partially saturated layer can vary strongly versus frequencies, especially in low-frequency range where high attenuation and dispersion can take place due to fluid partial saturation. Analysis of the crossplots of the magnitude for reflection coefficients and dispersion showed that attenuation, dispersion and tuning are more sensitive to gas saturation, porosity, and heterogeneity size. Thus the crossplot provides a useful means for detecting changes in fluid saturation and rock properties.

Hybrid frequency-domain full-waveform inversion using ray+Born sensitivity kernels

Full-waveform inversion (FWI) using the scattering integral (SI) approach is an explicit formulation of the inversion optimization problem. The inversion procedure is straightforward, and the dependence of the data residuals on the model parameters is clear. However, the biggest limitation associated with this approach is the huge computational cost in conventional exploration seismology applications. Modeling from each of the source and receiver locations is required to compute the update at every iteration, and that is prohibitively expensive, especially for 3D problems. To deal with this issue, we have developed a hybrid implementation of frequency-domain FWI, in which forward modeling is combined with ray tracing to compute the update. We use the sensitivity kernels computed from dynamic ray tracing to build the gradient. The data residual is still computed using finite-difference wavefield modeling. With ray theory, the Green’s function can be approximated using a coarser grid compared to wave-equation modeling. Therefore, the memory requirements, as well as the computational cost, are reduced significantly. Considering that in transmission FWI long-to-intermediate wavelengths are updated during the early iterations, we obtain accurate inverted models. The inversion scheme captured the anomaly embedded in the homogeneous background medium. For more complex models, the hybrid inversion method helps in improving the initial model with little cost compared with conventional SI inversion approaches. The accuracy of the inversion results shows the effectiveness of the hybrid approach for 3D realistic problems.

3D Printed Meta‐Helmet for Wide‐Angle Thermal Camouflages

AbstractThermal camouflage utilizes the manipulation of heat fluxes to conceal an arbitrary object in various environments from being detected via thermography. In the past decade, the field of thermal metamaterials and the technique of 3D printing have been rapidly developed, which makes nonintuitive heat flux manipulation feasible. However, when thermal metamaterials are applied to the thermal camouflaging, their conductivities are dependent on the properties of background, leading to the damage of background integrality. Moreover, previous thermal camouflaging schemes have mostly worked in the 2D regime, largely restricting their functional angles and application scenarios, especially in complex environments. Here, wide‐angle radiative thermal camouflaging is realized by using a 3D‐printed meta‐helmet of extremely anisotropic thermal conductivities. Based on 3D coordinate transformation, this meta‐helmet directly maps temperature distributions from the background to the metamaterial surface without damaging background integrity. The non‐invasive device is efficient in wide‐angle thermal camouflage by rendering the same emissivity to the background medium and can self‐adjust to various even unknown background thermal fields, which is demonstrated in numerical simulations and experiments. This work opens a door to the 3D transformation‐thermotics‐based devices for versatile practical applications in thermal infrared stealth of macro‐sized objects and others.

Gap localization of multiple TE‐Modes by arbitrarily weak defects

This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. Using the Floquet–Bloch theory in negative-order Sobolev spaces, we characterize the precise number of eigenvalues created by the line defect in terms of the band functions of the original periodic background medium for arbitrarily weak defects.

Enhanced toroidal localized spoof surface plasmons in homolateral double-split ring resonators.

In this paper, toroidal localized spoof surface plasmons (LSSPs) based on homolateral double-split ring resonators is proposed and experimentally demonstrated at microwave frequencies. By introducing a new split in the conventional single-split ring resonator, the magnetic field in resonator is locally modified. The double-split ring resonator can create the mixed coupling in the structure, leading to the enhancement of magnetic field. Both numerical simulations and experiments are in good agreement. Compared with traditional toroidal LSSPs based on the single-split ring resonators, the imperfection of toroidal LSSPs is resolved, the intensity of toroidal resonance and the figure of merit (FoM) are significantly enhanced. To understand and clarify the enhanced magnetic field phenomena, we analyze the role of the double-split ring resonator. The effect of location of source and spacing between two splits on the resonance intensity are also discussed. A higher intensity of toroidal LSSPs resonance could be achieved by changing the spacing between two splits. Additionally, it is experimentally demonstrated that the enhanced toroidal LSSPs resonance is sensitivity to the background medium. The results of our research provide a new idea for exciting the enhanced toroidal dipole.

Fluorescence Spectroscopy Study of Protoporphyrin IX in Optical Tissue Simulating Liquid Phantoms.

Fluorescence spectroscopy has been extensively investigated for disease diagnosis. In this framework, optical tissue phantoms are widely used for validating the biomedical device system in a laboratory environment outside of clinical procedures. Moreover, it is fundamental to consider that there are several scattering components and chromophores inside biological tissues and the interplay between scattering and absorption may result in a distortion of the emitted fluorescent signal. In this work, the photophysical behaviour of a set of liquid, tissue-like phantoms containing different compositions was analysed: phosphate buffer saline (PBS) was used as the background medium, low fat milk as a scatterer, Indian ink as an absorber and protoporphyrin IX (PpIX) dissolved in dimethyl formamide (DMF) as a fluorophore. We examined the collected data in terms of the impact of surfactant Tween-20 on the background medium, scattering effects and combination of scattering and absorption within a luminescent body on PpIX. The results indicated that the intrinsic emission peaks are red shifted by the scattering particles or surfactant, whilst the scattering agent and the absorbent can alter the emission intensity substantially. We corroborated that phantoms containing higher surfactant content (>0.5% Tween 20) are essential to prepare stable aqueous phantoms.

Time-domain simulation of wave propagation across resonant meta-interfaces

Meta-interface models stem from the homogenization, in a low-frequency dynamic regime, of thin heterogeneous layers that are structured to achieve uncommon properties at the macroscopic level. When the layer is composed of a thin periodic array of highly-contrasted inclusions embedded within a homogeneous background medium then the corresponding effective interface model is characterized by jump conditions that, in the harmonic regime, involve some singular frequency-dependent terms. In this context, the article is concerned with the simulation of transient waves across such resonant meta-interfaces and a numerical method is proposed to handle the associated resonant jump conditions. To do so, a set of auxiliary variables is introduced locally along the interface and an augmented system of first-order equations in time accompanied with local-in-time jump conditions is derived. This system is then discretized on a Cartesian grid and solved using a high-order finite-difference scheme while the complexity associated with the geometry of the interface and the jump conditions is handled using an immersed interface method. A set of numerical examples in 1D and 2D is proposed to illustrate and validate the overall numerical approach, and quantitative comparisons with semi-analytical solutions are also provided.

Exact and approximate reflection/transmission responses from a layer containing vertical fractures

SUMMARYAnalyses of vertical fractures are of great interest in characterizing the fluid flow and minimum in situ stress direction in reservoirs. Long-wavelength equivalent orthorhombic (ORT) media typically characterize the anisotropy induced by a set of vertical parallel fractures or two sets of vertical and mutually orthogonal fractures embedded into a transversely isotropic medium with a vertical symmetry axis (VTI). Reflection and transmission (R/T) responses quantify wave amplitude variations in 1-D media and help to reveal the model property enclosing the heterogeneity. Conventionally, the R/T responses are analysed for an interface bounded by two half-spaces. However, for a plane wave travelling through a subsurface layer, the wave scattering effects at the top and bottom of the layer interact with each other. For a continuous infinite ORT space cut in two halves along the horizontal symmetry plane, we focus on the plane wave R/T responses from an ORT layer that is placed between the two halves, where the azimuths of the vertical symmetry plane in the layer and in the upper and lower half-spaces are identical. The R/T coefficient modelling method can be found in many publications for the ORT layer with an arbitrary finite thickness. We decompose the exact R/T coefficients into series expansions that correspond to different orders of intrabed multiples in the ORT layer. Under the weak-contrast assumption for the ORT half-spaces and the ORT layer, we use the anisotropic background medium to obtain the first-order R/T coefficient approximations and second-order reflectivity approximations. There is no constraint for the middle layer thickness in the obtained first-order reflectivity approximations. In the proposed first-order transmissivity approximations and second-order reflectivity approximations, the layer thickness is assumed to be thin to obtain appropriate approximations for a few wave modes. The isotropic background medium is also considered for weakly anisotropic models to obtain simpler approximations that facilitate parametric analyses. For the ORT layer with its thickness much smaller than the propagating wave's wavelength, the influences of the layer thickness on R/T coefficients can be inspected conveniently from the derived approximations. Particularly, the R/T coefficients are analysed for the model which would be a homogeneous VTI medium, if the vertical parallel fractures were absent from the middle layer. Numerical tests demonstrate that the proposed R/T coefficient approximations perform well for the thin ORT layer. The approximation accuracy decreases when the thin layer thickness increases.

Negative apparent resistivities in surface resistivity measurements

Apparent resistivity is traditionally defined as the resistivity of a hypothetical homogeneous half-space that would yield the same response as that measured over the real earth with the same acquisition parameters. Negative apparent resistivity contradicts this definition of apparent resistivity as we cannot physically attain a homogeneous medium with negative resistivity. However, we often encounter negative apparent resistivity in field data acquisition; it is regarded as the effect of measurement error or noise. In this paper, we re-investigated the idea that negative apparent resistivity, especially in the dipole-dipole measurement, can be caused by a natural geological structure even when the earth is isotropic and flat, and the survey line is straight. The results from 3D numerical modelling revealed that negative apparent resistivity in the dipole-dipole measurement can be encountered particularly when the resistivity contrast between the conductive body and background medium is extraordinarily high. Moreover, negative apparent resistivity can occur only when the current source is very close to one side of the conductive body, like in the mise-ả-la-masse case, and the potential dipole is placed at the opposite side of the body.

Numerical solutions of the forward and inverse problems arising in diffuse optical tomography

Optical tomography is a typical non-invasive medical imaging technique, aiming to reconstruct the optical properties of tissues from boundary measurements by passing near infrared light through the tissues. We are concerned with the forward and inverse problems for a time-dependent diffusion equation model arising in diffuse optical tomography. To solve the forward problem, we develop a boundary integral equation method, and then derive a discretization scheme for the resulting system of boundary integral equations. As for the inverse problem, rather than the precise values of the optical parameters, we recover geometric information on unknown inclusions as anomalies where the optical properties are significantly different from the background medium. A noniterative reconstruction method with its rigorous justification is established. Finally, we present some numerical results that illustrate the efficiency and effectiveness of the proposed methods for both the forward and inverse problems.

Scattering Matrix for the Interaction between Solar Acoustic Waves and Sunspots. II. Theory

Abstract The interaction between solar acoustic waves and an isolated sunspot is a scattering problem. A common tool to study scattering problems is the scattering matrix, which is the amplitude for the transition from one mode to another after the interaction. In the previous study (Part I), the scattering matrix elements were determined with the measured scattered wavefunction of the sunspot. In this study (Part II), we obtain an analytical relation between the scattering matrix elements and the perturbed quantities of the background medium of the sunspot region. The sunspot is considered a perturbed region relative to the quiet Sun. The perturbation of the background medium includes the magnetic field, the flow velocity, and perturbed thermodynamics quantities, such as the density and pressure perturbations. Inferring these perturbed quantities from measured quantities is one of the goals of helioseismology. Here, with the help of Green’s functions, the scattering matrix elements are expressed as a spatial integral, which contains these unknown perturbed quantities. This integral equation, together with the measured scattering matrix elements, could be used to infer the perturbed quantities with the forward and inversion methods. Besides the typical approximations for solar acoustic waves, two additional assumptions are made here: one is the Born approximation, and the other is that the background medium of the sunspot region does not change with time.

Physical Realization of Elastic Cloaking with a Polar Material.

An elastic cloak is a coating material that can be applied to an arbitrary inclusion to make it indistinguishable from the background medium. Cloaking against elastic disturbances, in particular, has been demonstrated using several designs and gauges. None, however, tolerate the coexistence of normal and shear stresses due to a shortage of physical realization of transformation-invariant elastic materials. Here, we overcome this limitation to design and fabricate a new class of polar materials with a distribution of body torque that exhibits asymmetric stresses. A static cloak for full two-dimensional elasticity is thus constructed based on the transformation method. The proposed cloak is made of a functionally graded multilayered lattice embedded in an isotropic continuum background. While one layer is tailored to produce a target elastic behavior, the other layers impose a set of kinematic constraints equivalent to a distribution of body torque that breaks the stress symmetry. Experimental testing under static compressive and shear loads demonstrates encouraging cloaking performance in good agreement with our theoretical prediction. The work sets a precedent in the field of transformation elasticity and should find applications in mechanical stress shielding and stealth technologies.

Spin decoherence in inhomogeneous media

A Green’s function formalism for describing the decoherence of a ‘central spin’ in inhomogeneous media is developed. By embedding the ‘central spin’ in a background medium and performing real-cavity, local-field corrections on the macroscopic fields at the location of the ‘central spin’ one can show that the Green’s function splits up into two main contributions, a contribution that is related to the bulk properties of the background medium and a contribution that is related inhomogeneities within the background medium. As an example, the coherence time of a shallow NV center in diamond close to a planar interface, both in the absence and presence of surface spins, is computed. It is found that the coherence time of the NV center increases as it moves away from the interface and, at distances greater than ≈1 nm, the interaction with the interface is negligible with the main source of decoherence coming from the interaction with the surface spins. Above ≈50 nm the interaction with the surface spins is also negligible and one recovers the bulk coherence time.

Crack‐Induced Shear‐Wave Orthorhombic Anisotropy: Modeling and Inversion of Shear‐Wave Borehole Measurements

AbstractRocks in horizontally layered sediments often exhibit transversely isotropic (TI) characteristics and contain aligned fractures/cracks due to tectonic movement and stress. We model the effective elastic properties of the TI medium containing vertically aligned cracks using the sphere‐equivalency of the effective scattering method. Results show that the corresponding anisotropy of the VTI background medium containing vertically fluid‐filled ellipsoidal inclusions can be characterized by the orthorhombic anisotropy model. The shear‐wave anisotropy of the model is governed by three anisotropy parameters, corresponding to the horizontal plane, and the two vertical planes that are normal and parallel to the crack plane, respectively. Effects on anisotropy parameters due to cracks are examined in detail, showing that one can use shear‐wave measurements to evaluate both the TI and crack parameters of the cracked TI‐rock. Based on the theoretical analysis, we develop an inversion method to estimate shear anisotropy parameters from the four‐component shear‐wave logging measurement in a vertical borehole penetrating the orthorhombic rock. The inversion method is validated with synthetic data and applied to interpret acoustic anisotropy from borehole measurements acquired in a fractured shale formation, where theoretical predictions and measurements are found in good agreement.

Acoustics and dynamic materials

The governing equations for small-amplitude acoustic disturbances are derived by combining an admissible background flow or medium, balance laws, an equation of state and perturbation theory. This standard approach is used here when the background medium can vary in both space and time, thus defining a dynamic material. When the background velocity is zero, the background density cannot vary with time, assuming that the usual balance laws (such as conservation of mass) are obeyed. When mass conservation is discarded, perhaps replaced by a growth model, a time-dependent background density is permitted, leading to new equations governing acoustic disturbances in certain dynamic materials.

Reflection and transmission approximations for weak contrast orthorhombic media

Subsurface media are in general anisotropic, and this fact should be taken into account for analyzing reflection and transmission (R/T) coefficients. Orthorhombic (ORT) media are commonly regarded as a practical symmetry system to account for polar anisotropy and azimuthal anisotropy. We have focused on the model made up of two welded ORT half-spaces to analyze the R/T coefficients normalized by vertical energy flux. The two half-spaces have azimuthally nonaligned vertical symmetry planes and are parameterized in two local 3D Cartesian coordinate frames, respectively. The vertical coordinate planes in each local frame coincide with the vertical symmetry planes for the corresponding ORT half-spaces. Under the weak contrast assumption for the two half-spaces, this model is taken as the perturbed model in R/T approximations with the perturbation theory. The unperturbed model is also composed of two unperturbed ORT half-spaces with their different vertical symmetry plane orientations inheriting the counterparts for the perturbed half-spaces above and below the interface, respectively. The unperturbed ORT half-spaces above and below the interface have the same model parameters defined in the two local coordinate frames, respectively. With the perturbations respectively evaluated in the local coordinate frames above and below the interface, the azimuth angle that indicates the local frames’ azimuthal difference is decoupled from the model parameter contrasts. Compared with the traditional approximation method with the perturbation theory in a global coordinate frame, the proposed R/T approximations depend on fewer model parameter discontinuities. We also consider the isotropic background medium under the weak anisotropy assumption. Influences of S-wave singularity points are mitigated by introducing pseudowaves for approximations. Numerical tests are implemented to demonstrate the accuracy.

Small pertubations of an interface for Stokes system

AbstractWe consider the Stokes system for a viscous medium consisting of an impurely (inclusion) merged in a consistent background medium. Appointed on both field expansion and layer potential methods, we strictly derive the asymptotic expansion of the perturbed velocity field because of small perturbations in the interface of the inclusion. We use these techniques to determine a relationship between Stokes solutions measurements and the shape of the object. Moreover, we may prove an asymptotic expansion for the perturbation in the viscosity moment tensors (VMTs) caused by the presence of small changes in the interface of the impurity.