A hybrid method for calculating seismic wave first-arrival traveltimes in two-dimensional models with an irregular surface

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.

A fast marching algorithm for the factored eikonal equation

A stable moving-particle semi-implicit method for free surface flows

To simulate heterogeneous bubbly flows, a hybrid method has been developed in several studies by combining the two-fluid model and the interface tracking method. Since this original hybrid method is based on the one-pressure modeling, which is a common approach for the two-fluid model, unphysical flow would occur in the flow where the surface tension is dominant. To resolve this inherent weakness of one-pressure modeling, we proposed a new formulation of the hybrid method in this study. The proposed hybrid method includes no pressure term in the dispersed phase because we regarded the dispersed phase as a group of discrete particles. The phase-field method is employed as the interface-tracking method. The proposed hybrid method was validated in several numerical simulations. Using the static bubble problem, we confirmed that in the proposed hybrid method simulation, there is no unphysical flow that is observed in the original hybrid method. In the bubble plume problem, we compared the bubble plume behavior and the time-averaged void fraction of the simulation results with those of previously reported experiments and found good agreement between them. From these validations, we confirmed that the proposed hybrid method was applicable to heterogeneous bubbly flow simulation, and it avoided the weakness which the original hybrid method has.

A hybrid method combining tabu search (TS) and Simplex NeIder-Mead (NM) has been proposed for solving global continuous optimization problems. Similar to enhanced continuous tabu search (ECTS) method, the proposed method works in three sequential phases: diversification, semi-intensification and intensification. TS strategy just works in the diversification. NM strategy works not only in semi-intensification but also in intensification which is different from other hybrid methods where NM strategy is just applied in the third phase. In the first phase, we proposed four methods to generate neighbors. The first numerical experiment comparing the results of four methods of neighbor generation is carried out and three important conclusions are given. First, the proposed hybrid method has a good performance in solving the global continuous optimization problems. Second, the right direction of neighbors is more important than its right hierarchy for most functions. Third, the spiral method to generate neighbors has the moderate performance both in the robustness and in the calculation speed. To compare the proposed hybrid method to the best and the most recent other published hybrid methods, the second numerical experiment with spiral method to generate neighbors is carried out. The comparison shows that our hybrid method is better than other methods when the dimension of the functions is no more than 4. Moreover, although the dimension is more than 4, it is not worse than many of them.

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.

In the projection system, the projected image should be able to meet the normal visual needs of observer, so that the viewer can get the correct projection. When the projection screen surface is the irregular free surface, the projection geometric correction of the projected image becomes the research focus. For the question of irregular projection screen surface, this paper presents effective geometric correction method. The geometric correction method based on the geometry of irregular projection screen surface and ideal location of viewport to calculate the points’ three-dimensional coordinate values. In this paper, the projection geometric correction method is widely applicable to various projection screen surface environments.

This article proposes a novel hybrid finite-difference time-domain (FDTD) method to calculate the transient responses of the transmission line network based on the leapfrog alternating direction implicit FDTD (leapfrog ADI-FDTD) algorithm and the traditional FDTD method. The proposed hybrid method can be implemented by dividing the transmission line network into two parts that are the interconnects part in a printed circuit board solved by the leapfrog ADI-FDTD and the cable part solved by the traditional FDTD with different mesh sizes and the same time step, respectively. In addition, some boundary conditions should be introduced based on the modified nodal approach and Kirchhoff's law at the terminals of the transmission line network. The numerical results show that the proposed hybrid method is in good agreement with HSPICE. Especially, it is more efficient and applicative for the multiscale transmission line network problems due to the use of the implicit FDTD solver with unconditional stability compared with the traditional FDTD method.

A hybrid method is proposed for synthesizing sparse concentric ring arrays (CRAs) to get circularly symmetric and invariant three-dimensional (3-D) patterns over all-azimuth directions. The hybrid method involves a convex optimization approach and a determined approach together. Based on the Bessel function approximation of the circular array pattern and the compressed sensing (CS) theory, the synthesis of a sparse concentric current ring array can be formulated as a convex problem with the minimization of l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm, and the corresponding array parameters including the number of concentric rings, the ring radii, and the current ring excitations can be determined simultaneously to satisfy the requirements on radiation pattern. In order to reconstruct the 3-D pattern accurately with discrete antenna elements in place of the current rings, a determined approach with or without radial amplitude weighting is proposed to define the number of elements equally distributed on each ring. The proposed hybrid method can be applied to synthesizing CRAs with 3-D pattern either matching a desired pattern or satisfying a prescribed power mask. Moreover, the method is very easy to implement, while has good numerical stability and computational efficiency. Numerical experiments are conducted to validate the flexibility and effectiveness of the proposed method.

Modeling of seismic wave propagation near the earth's surface

We describe recent work on striped patterns in a system of block copolymers. A by-product of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation, the unknown is a field of projection matrices of the form P=e⊗e, where e is a unit vector field. We describe how this formulation is better adapted to the description of striped patterns than the classical eikonal equation, and illustrate this with examples.

This work presents a new hybrid method to compute boundaries of power flow feasible region, which is pertinent to static voltage stability analysis. The hybrid method is derived from a 'predictor-corrector' framework. It converts the problem into an optimization problem, whose objective is to determine the minimum distance from an external point to the boundary of the feasible region. The proposed hybrid method can reduce the computation dimension in comparison with using 'predictor-corrector' method. In addition to visualizing boundaries in two dimension space, the method can be utilized as a powerful tool to compute the closest boundary point associated with a specific operating point. It also can be used to compute the operating limit of facilities. In order to handle concave topology on the boundary, a modification of hybrid method is also proposed. Numerical simulations on two systems illustrate that the hybrid method is reliable and effective in terms of computing the boundaries of power flow feasible region.

In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, J. Comput. Phys., 227 (2008) pp. 8191–8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the $L^1$ norm and a very fast convergence speed, but only first order accuracy in the $L^\infty$ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the $L^\infty$ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples.

Efficient and accurate traveltime calculations of seismic waves have important applications in traveltime tomography for initial velocity model building, Kirchhoff depth migration, earthquake location, etc. Because anisotropy significantly affects the traveltimes of seismic waves, neglecting it would result in inaccurate imaging and inversion of the underground structure. Therefore, it is necessary to consider anisotropy in the traveltime calculations. The fast sweeping method (FSM) has important applications in computing the anisotropic first-arrival traveltime. The conventional method, which solves a transformed traveltime quartic equation combined with the FSM, is well suited for general anisotropic media and does not rely on approximations to Christoffel’s equation. However, this method has the following problems: (1) the form of the traveltime quartic equation is highly complex and exhibits multiple solutions, requiring additional effort to find solution intervals and determine which solution meets the criteria, and (2) Solving high-order polynomial equations may encounter numerical instability, especially when coefficients undergo minor variations or rounding errors occur for 3D problems. To address the stability and efficiency issues in anisotropic traveltime calculation, we analyze the characteristics of the constructed triangular-pyramid local solver and the quartic coupled slowness equation for the qP and qSV waves in tilted transversely isotropic (TTI) media. We observe that the decomposed slowness equation yields only one or no solution in the triangular-pyramid local solver and satisfies monotonicity conditions. Therefore, we develop the use of the Newton method to solve the factorized slowness equation efficiently, thus addressing stability and computational efficiency concerns inherent in the conventional approach. For the qSH wave, its slowness equation is quadratic and simple to solve. Our method provides an efficient and stable procedure for the traveltime calculations of qP, qSV, and qSH waves in 3D general TTI media. Numerical examples verify the efficiency and accuracy of our method.

Improving forecasting accuracy of time series data using a new ARIMA-ANN hybrid method and empirical mode decomposition