Forward modeling of ocean-bottom cable data and wave-mode separation in fluid–solid elastic media with irregular seabed

P- and S-wave separated elastic reverse time migration for OBC data from fluid-solid coupled media with irregular seabed interfaces

Summary The marine seismic exploration with the ocean bottom cable (OBC) technology could record the compressional (P-) wave and shear (S-) wave information simultaneously. Elastic full waveform inversion (EFWI) uses both P- and S-waves of OBC data to invert multi-parameters with adequate amplitude information and complete illumination of the subsurface. We use different formats of wave equation in fluid and solid mediums, and the appropriate boundary condition applied enables the energy exchange on the interface, which is more stable and efficient than the traditional integrated simulation scheme. However, if the fluid-solid coupled medium has an extremely irregular seabed interface, the conventional rectangular-grid-based finite difference (RFD) method cannot obtain accurate source and receiver wavefields. We introduce the curvilinear coordinates to overcome the problem of the RFD method on handling the irregular fluid-solid interface. To reduce the crosstalk of inverted P- (Vp) and S-velocities (Vs), we derive the direction formulas of Vp and Vs based on P- and S-wave modes separation in the curvilinear coordinates, and finally propose a Curvilinear-grid-based Fluid-Solid Separated- wavefield EFWI (CFS-SEFWI) method. In the numerical examples, a modified Marmousi II model with irregular fluid-solid interfaces is used to test the proposed CFS-SEFWI method.

A coupled three-dimensional fluid-elastic wave propagation mathematical model has been developed to handle environmental interactions in the ocean between a fluid medium and an elastic bottom. The existing model combines three-dimensional fluid and elastic wave propagation models with the incorporation of a set of horizontal fluid-elastic interface conditions. This paper extends the above model to consider an irregular fluid-elastic interface. The theoretical development of the irregular fluid-elastic interface equations is presented. Comparison of the fluid-elastic irregular interface to the horizontal case is one of the major objectives of this paper. Another major objective of this paper is the construction of a complete set of three-dimensional fluid-elastic wave equations, including the irregular interface, in a form suitable for stable numerical solution.

Coupling of rupture processes in solids with waves also propagating in fluids is a prominent phenomenon arising during tectonic earthquakes. It is executed here in a single `monolithic' model which can asymptotically capture both damageable solids (rocks) and (visco-)elastic fluids (outer core or oceans). Both ruptures on pre-existing lithospheric faults and a birth of new faults in compact rocks are covered by this model, together with emission and propagation of seismic waves, including, e.g., reflection of S-waves and refraction of P-waves on the solid-fluid interfaces. A robust, energy conserving, and convergent staggered FEM discretisation is devised. Using a rather simplified variant of such models for rupture, three computational experiments documenting the applicability of this approach are presented. Some extensions of the model towards more realistic geophysical modelling are outlined, too.

In seismic numerical simulations of wave propagation, it is very important for us to consider surface topography and attenuation, which both have large effects (e.g., wave diffractions, conversion, amplitude/phase change) on seismic imaging and inversion. An irregular free surface provides significant information for interpreting the characteristics of seismic wave propagation in areas with rugged or rapidly varying topography, and viscoelastic media are a better representation of the earth’s properties than acoustic/elastic media. In this study, we develop an approach for seismic wavefield simulation in 2D viscoelastic isotropic media with an irregular free surface. Based on the boundary-conforming grid method, the 2D time-domain second-order viscoelastic isotropic equations and irregular free surface boundary conditions are transferred from a Cartesian coordinate system to a curvilinear coordinate system. Finite difference operators with second-order accuracy are applied to discretize the viscoelastic wave equations and the irregular free surface in the curvilinear coordinate system. In addition, we select the convolutional perfectly matched layer boundary condition in order to effectively suppress artificial reflections from the edges of the model. The snapshot and seismogram results from numerical tests show that our algorithm successfully simulates seismic wavefields (e.g., P-wave, Rayleigh wave and converted waves) in viscoelastic isotropic media with an irregular free surface.

This paper presents a numerical method to simulate the 2D tidal flow and water quality under the orthogonal curvilinear coordinates. In order to overcome the computational difficulties in natural rivers, such as the complicated boundary figures, the great disparity between length and width of computational domain, etc., orthogonal boundary-fitted grid is used, the irregular domain in physical plane is transformed into a rectangular domain in transformed plane, and the depth averaged momentum equations and mass equation are rewritten and discretized based on the alternating direction implicit finite difference scheme in curvilinear coordinates. Practical application of the method is illustrated by an example for the Yangtze River in the vicinity of Nanjing city. A fair agreement between the values measured and computed demonstrates the validity of the method developed.

Seismic wave propagation shows anisotropic characteristics in many sedimentary rocks. Modern seismic exploration in mountainous areas makes it important to calculate P wave travel times in anisotropic media with irregular surfaces. The challenges in this context are mainly from two aspects. First is how to tackle the irregular surface in a Cartesian coordinate system, and the other lies in solving the anisotropic eikonal equation. Since for anisotropic media the ray (group) velocity direction is not the same as the direction of the travel-time gradient, the travel-time gradient no longer serves as an indicator of the group velocity direction in extrapolating the travel-time field. Recently, a topography-dependent eikonal equation formulated in a curvilinear coordinate system has been established, which is effective for calculating first-arrival travel times in an isotropic model with an irregular surface. Here, we extend the above equation from isotropy to transverse isotropy (TI) by formulating a topography-dependent eikonal equation in TI media in the curvilinear coordinate system, and then use a fast sweeping scheme to solve the topography-dependent anisotropic eikonal equation in the curvilinear coordinate system. Numerical experiments demonstrate the feasibility and accuracy of the scheme in calculating P wave travel times in TI models with an irregular surface.

The problem of tsunami warning and the use of the grid-characteristic method developed for studying wave processes in heterogeneous media is addressed. A multilayer geological model with curved boundaries and contrasting elastic parameters is considered. Various seismic survey methods, as well as different geometries of the boundaries, different values of rock densities, and various longitudinal and shear wave velocities, are examined. The boundary value problem is numerically solved jointly for the elastic and acoustic wave equations (seismic wave propagation in rocks and water layer, respectively). We use the numerical grid-characteristic method with combined structured curvilinear and regular computational grids. The wave velocity fields (velocity as derivative of displacement) and stress fields are calculated and analyzed. Synthetic seismograms are calculated for the coastal zone. It is found that some specific features of seismograms and some wave modes can be used for tsunami warning. The optimal parameters of seismic surveying are determined. Numerical simulation of wave propagation effects is used for these purposes. The proposed numerical grid-characteristic method can be used in forward numerical modeling in order to interpret seismograms recorded in the coastal zone.

In this paper, the formulations of the primitive equations for shallow water flow in various horizontal co-ordinate systems and the associated finite difference grid options used in shallow water flow modelling are reviewed. It is observed that horizontal co-ordinate transformations do not affect the chosen co-ordinate system and representation in the vertical, and are the same for the three- and two-dimensional cases. A systematic derivation of the equations in tensor notation is presented, resulting in a unified formulation for the shallow water equations that covers all orthogonal horizontal grid types of practical interest. This includes spherical curvilinear orthogonal co-ordinate systems on the globe. Computational efficiency can be achieved in a single computer code. Furthermore, a single numerical algorithmic code implementation satisfies. All co-ordinate system specific metrics are determined as part of a computer-aided model grid design, which supports all four orthogonal grid types. Existing intuitive grid design and visual interpretation is conserved by appropriate conformal mappings, which conserve spherical orthogonality in planar representation. A spherical curvilinear co-ordinate solution of wind driven steady channel flow applying a strongly distorted grid is shown to give good agreement with a regular spherical co-ordinate model approach and the solution based on a β-plane approximation. Especially designed spherical curvilinear boundary fitted model grids are shown for typhoon surge propagation in the South China Sea and for ocean-driven flows through Malacca Straits. By using spherical curvilinear grids the number of grid points in these single model grid applications is reduced by a factor of 50–100 in comparison with regular spherical grids that have the same horizontal resolution in the area of interest. The spherical curvilinear approach combines the advantages of the various grid approaches, while the overall computational effort remains acceptable for very large model domains.

We have extended a new highly accurate numerical scheme for unstructured 2D and 3D meshes based on the discontinuous Galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid-solid interfaces. Because of the formulation of the wave equations as a unified first-order hyperbolic system in velocity stress, the fluid can be in movement along the interface. The governing equations within the moving fluid are derived from well-known first principles in fluid mechanics. The discontinuous Galerkin approach allows for jumps of the material parameters and the solution across element interfaces, which are handled by Riemann solvers or numerical fluxes. The use of Riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero. No additional compatibility relations, such as vanishing shear stress or continuity of normal stresses, are necessary to couple the solid and fluid along an interface. The Riemann solver automatically recognizes the jump of the material coefficients at the interface and provides the correct numerical fluxes for fluid-solid contacts. Therefore, wave propagation in the entire computational domain containing heterogeneous media, namely moving fluids and elastic solids, can be described by a uniform set of acoustic and elastic wave equations. The accuracy of the proposed scheme is confirmed by comparing numerical results against analytic solutions. The potential of the new method was demonstrated in a 3D model problem typical for marine seismic exploration with a fluid-solid interface determined by a complicated bathymetry.

Tides and their effects on coastal and estuarine water levels are among the most well-known phenomena in coastal research. This study aims to map the tidal constituent features in the near-shore East Sea area, for better understanding of the mechanisms that cause the transport of sediments in the area. Furthermore, the study also points to the potential of tidal energy in the area, clean energy that needs to be studied for exploitation. The authors applied the hydraulic model in the curvilinear coordinates to calculate for 4 main tidal constituents in the near-shore, namely K1, O1, M2, and S2 in the East Sea area, Vietnam. The hydraulic model with two-dimensional orthogonal curvilinear grid has the advantage of increasing the accuracy in the results at the domain boundary, with the applying potential in future small-scale studies in the region. According to this method, the simulation results in areas with complex terrain are better because the velocity field is calculated on a curved grid built by shorelines). The calibration and validation of this model are based on water level data at hydrological stations along the Vietnamese coastline. The result of this model is used to map the harmonic constants and tidal ellipse for four tidal constituents; these help to gain information about the tidal deposition in the East Sea, Viet Nam. The coastal area of Vietnam has potential tidal energy; the largest energy is the tidal constituent O1, followed by the tidal constituents K1, M2, and S2. Tidal ellipses of the residual tidal constituent K1 and O1 are large in the Gulf of Tonkin and the Gulf of Thailand. Meanwhile, residual tidal constituent M2 and S2 are large in the southeast of Vietnam coastal. These results are the primary data for future research in the area.

Steeply dipping structures bring huge challenges in high-resolution imaging. The imaging of the steeply dipping structures can be improved by incorporating the prismatic waves, which has some steeply dipping structure information where primaries cannot contain. In this paper, an elastic reverse time migration of prismatic waves (ERTM-P) is proposed to produce high-resolution multi-component images of steeply dipping structures with complete compressional (P-) and shear (S-) wave illumination of the subsurface. To avoid severe spurious diffraction caused by dramatic surface topography, the velocities are meshed into curvilinear grids and migration is conducted in curvilinear coordinates. In addition, the imaging conditions of prismatic waves in P- and S-components are derived using the decoupled curvilinear-coordinated elastic equations to suppress the imaging artefact from P- and S-wave crosstalk. In numerical examples, we use an elastic horizontal- and vertical-layer model, topographic elastic sag model and salt dome model with surface topography to demonstrate that the proposed method can accurately handle with the dramatic surface topography and obtains high-quality multi-component images by improving the images of steeply dipping structures.

Characteristic equations and their solutions of Rayleigh wa ve in elastic soils,fluid solid medium and saturated soils are presented. Discussion is stressed on wa ve propagation and dispersion in layered soik. Some applications of Rayleigh wa ve dispersion in Civil Engineering are introduced.

In order to describe the reservoirs more accurately, the characteristics of the earth medium must be considered comprehensively, such as the anisotropy, viscoelasticity and pore, etc. Consequently, an anisotropic viscoelastic porous medium model is presented in this paper, and the elastic wave equation in this medium has been derived. The seismic wavefield numerical modeling was simulated by using the pseudo-spectral method. Synthetic seismograms are used to show the characteristics of the seismic wave in complex medium. The research work is useful to understanding seismic wave propagation in earth medium.

The time-reversal imaging method has become a standard technique for seismic source location using both acoustic and elastic wave equations. Although there are many studies on the determination of the relevant parameter for visualization of the time-reversal method, little has been done so far to investigate the accuracy of seismic source location depending on parameters such as the geometry of the seismic network or underestimation of the velocity model. This paper investigates the importance of the accuracy of seismic source location using the time-reversal imaging method of input variables such as seismic network geometry and the assumed geological model. For efficient visualization of seismic wave propagation and interference, peak-to-average power ratio was used. Identification of the importance of variables used in seismic source location was obtained using the Morris elementary effect method, which is a global sensitivity analysis method.