Least‐squares reverse time migration based on an efficient pure qP‐wave equation

This paper presents the Lebedev scheme on staggered grids for the numerical simulation of wave propagation in anisotropic elastic media. Primary attention is given to the approximation of the elastic wave equation by the Lebedev scheme. Based on the differential approach, it is shown that the Lebedev scheme approximates a system of equations, which differs from the original equation. It is proved that the approximated system has a set of 24 characteristics, six of them coincide with those of the elastic wave equation and the rest ones are “artifacts.” Requiring the artificial solutions to be equal to zero and the true ones to coincide with those of the elastic wave equation, one comes to the classical definition of the approximation of the initial system on a sufficiently smooth solution. The results obtained and the knowledge of the complete set of characteristics are important for constructing reflectionless boundary conditions during approximation of point sources, etc.

Since a good many of the papers published regularly in Radio Science deal with wave problems in magnetoplasmas, especially those of the ionospheric variety, the readership has constant access to information and techniques pertaining to electromagnetic wave propagation in anisotropic media. Nevertheless, it is appropriate at times to draw attention to work in a given subject area by a concentrated grouping of papers dealing with various facets. The present spedal issue on Electromagnetic Wave Propagation in Anisotropic Media should be regarded in this light.

Advances in computational capabilities as well as ongoing improvements in storage strategies have made reverse time migration (RTM) a feasible method for capturing images of complex structures. However, large storage requirements still restrict RTM applications, especially in anisotropic media. Utilising a first-order quasi-P-wave equation in vertically transversely isotropic (VTI) media, we investigate anisotropy and deduce an RTM equation for a staggered-grid high-order finite difference (FD) scheme incorporating a perfectly matched layer (PML) boundary in this study. We also develop an improved source wavefield storage strategy via a PML boundary method for VTI medium RTM using graphic processing unit (GPU) accelerated computation. Our proposed method significantly reduces the total volume of data storage required for conventional RTM while increasing calculation time by just a small amount. Checkpoints can be set based on GPU memory size, leading to the generation of high precision and high efficiency subsurface images. We carried out a series of numerical tests on simple anisotropic media and complex Hess 2D VTI models to verify the effectiveness of our proposed method.

A three-dimensional high-order reverse-time migration (3-D HO-RTM) method is proposed to perform subsurface electromagnetic imaging with an ultra-wideband radar (UWBR) system consisting of a multi-input and multi-output antenna array. By using a UWBR system to collect temporal scattering signals, subsurface targets can be detected, and the image of targets can be obtained by imaging methods such as the back-propagation method, frequency-wavenumber migration technique, time-reversal mirror, and reverse-time migration method. The proposed HO-RTM method is based on the high-order finite-difference time-domain (HO-FDTD) method to significantly reduce the computational cost in the conventional RTM method. The measured data from an experimental lunar exploration system Chang'E-5 have been collected on a 7 m × 2.5 m × 2.5 m laboratory model with volcanic ash and validated by the 3-D HO-RTM method. Results show that all buried objects can be effectively identified by the HO-RTM, and its computer memory and CPU time are only 3.87% and 0.7128% of the conventional RTM method, respectively.

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common‐midpoint (CMP) spreads is usually well described by the normal‐moveout (NMO) velocity defined in the zero‐offset limit. In their recent work, Grechka and Tsvankin showed that the azimuthal variation of NMO velocity around a fixed CMP location generally has an elliptical form (i.e. plotting the NMO velocity in each azimuthal direction produces an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for the NMO velocity in anisotropic media of arbitrary symmetry.For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The contribution of anisotropy to NMO velocity is contained in the slowness components of the zero‐offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) — quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix‐type average of the matrices responsible for the ‘interval’ NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi‐axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions.Modelling of normal moveout in general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero‐offset) ray. Remarkably, the expressions for geometrical spreading along the zero‐offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi‐azimuth, multi‐offset ray tracing and, therefore, can be used efficiently in traveltime inversion and in devising fast dip‐moveout (DMO) processing algorithms for anisotropic media. This technique becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces.The high accuracy of our NMO expressions is illustrated by comparison with ray‐traced reflection traveltimes in piecewise‐homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P‐wave moveout can be used to find the depth‐dependent fracture orientation and to evaluate the magnitude of azimuthal anisotropy.

The mathematical description of phenomena related to wave propagation in anisotropic media is considerably more complicated than that for isotropic media. This complexity stems from many physical properties distinguishing the anisotropic from isotropic medium. Therefore, calculating the angles of reflection and transmission for a ray impining on a boundary between two anisotropic media become more complicated in mathematical implication. Snell law in isotropic medium has not been applicable in anisotropic medium, which must be modified. This article presents a new method of calculating the angles of reflection and transmission on a boundary between two anisotropic media, called as the modified Snell law.

Studying the anisotropy of seismic wave velocity can provide a theoretical basis for understanding the characteristics of seismic wave propagation in anisotropic media, such as forward modeling of seismic wavefield and simultaneous inversion of anisotropic parameters. It is an important work to calculate the group velocities of three kinds of seismic waves (qP, qS1, and qS2) and the analytical expressions of the partial derivatives of the group velocities with respect to 21 elastic parameters in general anisotropic media. Therefore, a new relationship between the phase and group velocity in the general anisotropic medium is introduced first, and then the analytical expressions of the group velocity of three kinds of waves are deduced. Then, the analytical formulas of partial derivatives of the group (or phase) velocity with respect to 21 elastic parameters are derived. Finally, the distribution of partial derivatives of group slowness with respect to 21 elastic parameters with varied ray angles is analyzed and discussed. The final analytical expressions of group velocity and its partial derivatives are relatively concise and can deal with the singular points and multivalued problems of qS waves. By drawing comparison to the numerical solution of the eigenvector method, the accuracy of the analytical formula of group velocity and its partial derivatives with respect to elastic parameters are verified. The numerical simulation results indicate that the algorithm can accurately obtain the partial derivatives of group slowness with respect to 21 elastic parameters and would provide a theoretical basis for seismic ray tracing and traveltime inversion in general anisotropic media.

Reverse time migration: A prospect of seismic imaging methodology

I derive analytical expressions giving traveltime‐offset curves for multilayered weakly azimuthally isotropic and anisotropic media on terms of the elastic properties of each layer. This method is based upon an approximate skewed hyperbolic moveout formula involving three measured bulk velocities for each reflector. The primary benefits of this technique are: (1) it allows for fast traveltime computation; (2) it makes possible an extremely rapid estimation of the interval elastic parameters; and (3) it provides physical insight into the wave propagation in anisotropic media. I establish a traveltime inversion algorithm which allows the estimation of the five elastic constants together with the orientation of the axis of symmetry for each layer. In the isotropic limit, this algorithm reduces to the conventional one used for determining the interval velocities from stacking velocity measurements in isotropic media. The inversion technique is applied to surface seismic measurements and VSP field surveys. In both cases, the method provides very good estimates of these six parameters, even in highly anisotropic materials. The results also show that the orientation of the horizontal axis of symmetry, for a given azimuthally anisotropic layer at depth, can be obtained using only quasi‐P‐wave information. This technique can be easily incorporated into conventional velocity analysis algorithms. Interval anisotropic properties can then be estimated with little additional effort.

The increasing use of anisotropic elements in many applications ranging from microwaves to optical frequencies has prompted the study of e. m. wave propagation in anisotropic media. In the last decade there has been renewed interest in e. m. waves in anisotropic media and particularly in bounded gyrotropic plasma in connection with magnetized-plasma column, sustained by surface wave [1,2], different applications in microphysics [4,5], high power generation [6,7], energy transform in space [8], etc. The motivation has been strengthened by the successful application of the hydrodynamic model to magnetoplasma excitations in parabolic quantum wells [9]. The new character of the e. m. waves is the base of a big variety of applications.

The paper deals with finite–difference (f‐d) approach to simulation of elastic waves' propagation in anisotropic elastic media with general symmetry. Any implementation of this approach claims resolution of two key problems: construction of an effective f‐d scheme itself; we propose to use the Lebedev's scheme (LS) being a natural generalization of Virieux staggered grid scheme (VS) widely used for isotropy; we prove that LS possesses better dispersion properties in comparison with well known Rotated Staggered Grids Scheme (RSGS). stable domain distension. The Perfectly Matched Layers(PML) useful for isotropic problems can be unstable in the case of anisotropy. Lebedev scheme allows one to use Optimal Grids (OG) which gives a possibility to implement efficient and low cost domain distension. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Along with a wide application of nondestructive evaluation methods by ultrasonic techniques, Rayleigh surface waves are being studied for their applications in material characterization. Because surface waves can offer some sensitive measurement features of wave propagation, it is suggested that surface waves be conveniently used as an experimental technique for the solution of the inverse problem of determining elastic constants and/or other characteristics in materials [1–3]. The most common direct problem is to obtain wave propagation features by theoretical analysis, experimental measurement, or numerical calculation. A desired problem in material evaluation however, is to solve an inverse problem to find material characteristics from a set of field measurement data. In surface wave problems, the closed form solutions may not even exist for some direct problems. Moreover, often material constants collectively influence the ultrasonic wave propagation in anisotropic medium, and we can not decouple them and evaluate them individually by each single ultrasonic measurement. Therefore, a numerical computational procedure is proposed.KeywordsWave PropagationSurface WaveElastic ConstantMaterial ConstantAnisotropic MediumThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

To simulate Electromagnetic wave propagation in anisotropic media, absorbing boundary conditions are needed to truncate the computation domains. Based on the finite difference time domain method in anisotropic medium, the implementation of the modified nearly perfectly matched layer absorbing boundary conditions for truncating anisotropic medium is presented. By using the partial derivatives of space variables stretched-scheme in the coordinate system, the programming complexity is reduced greatly. According to one dimensional numerical simulation analysis, the modified nearly perfectly matched layer absorbing boundary condition is validated.

Wave processes are intensively studied in various fields of physics: electrodynamics, plasma physics, radiophysics, acoustics, hydrodynamics, etc. Along with the study of electromagnetic and elastic wave processes, the research of patterns of wave propagation of various physical nature in the presence of mutual transformation are of particular relevance. Wave processes in coupled fields reflect the mutual influence of elastic, electromagnetic and thermal fields. The coupling of electromagnetic fields to the deformation field takes place in a medium with piezoelectric, piezomagnetic and magnetostrictive properties. In the paper, based on the matrix method, the propagation of coupled elastic and electromagnetic waves in media with different physical and mechanical properties is studied. The paper proposes a generalization of non-classical contact conditions for studying the effect of thin layers with different physical and mechanical properties on wave processes. A system of differential equations of the 1st order with variable coefficients is constructed, which describe the propagation of electroelastic waves in anisotropic media of a rhombic system of class 222. The conditions for nonrigid contact for a thin layer with piezoelectric properties are derived. The possibility of studying layers with δ-shaped properties (δ is the Dirac function) is proved.

Wavefront construction methods allow for efficient simulation of seismic wave propagation in anisotropic media using ray theory. This approach models an individual wavefront as a mesh that is adaptively refined as the wave travels through an earth model. The algorithm, implemented here for quasi-compressional waves, provides an effective tool for modeling and analyzing a large vertical seismic-profile data set from the Gulf of Mexico. Seismic velocities in this region generally increase linearly with depth, but simulations show that an isotropic vertical-gradient model cannot accurately reproduce traveltimes of first arrivals. However, a simple velocity-analysis inversion based on the wavefront-construction method allows a straightforward estimation of improved isotropic and transversely isotropic models that significantly reduce the misfit between computed traveltimes and the observations. The goal of this analysis is to produce a simple model suitable for use as a starting point for migrations and for assessing the importance of anisotropy in the field site. In fact, the final anisotropic model is much more consistent with background geologic information than the best isotropic model, which also has a larger rms traveltime error. The results strongly support the presence of measurable anisotropy.