Seismic modeling by combining the finite-difference scheme with the numerical dispersion suppression neural network

Seismic finite-difference(FD) modeling suffers from numerical dispersion including both the temporal and spatial dispersion, which can decrease the accuracy of the numerical modeling. To improve the accuracy and efficiency of the conventional numerical modeling, I develop a new seismic modeling method by combining the FD scheme with the numerical dispersion suppression neural network(NDSNN). This method involves the following steps. First, a training data set composed of a small number of wavefield snapshots is generated. The wavefield snapshots with the low-accuracy wavefield data and the high-accuracy wavefield data are paired, and the low-accuracy wavefield snapshots involve the obvious numerical dispersion including both the temporal and spatial dispersion. Second, the NDSNN is trained until the network converges to simultaneously suppress the temporal and spatial dispersion.Third, the entire set of low-accuracy wavefield data is computed quickly using FD modeling with the large time step and the coarse grid. Fourth, the NDSNN is applied to the entire set of low-accuracy wavefield data to suppress the numerical dispersion including the temporal and spatial dispersion.Numerical modeling examples verify the effectiveness of my proposed method in improving the computational accuracy and efficiency.

The Precise Finite Difference Method for Seismic Modeling

D seismic modeling can be used to study the propagation of seismic wave exactly and it is also a tool of 3-D seismic data processing and interpretation. In this paper the arbitrary difference and precise integration are used to solve seismic wave equation, which means difference scheme for space domain and analytic integration for time domain. Both the principle and algorithm of this method are introduced in the paper. Based on the theory, the numerical examples prove that this hybrid method can lead to higher accuracy than the traditional finite difference method and the solution is very close to the exact one. Also the seismic modeling examples show the good performance of this method even in the case of complex surface conditions and complicated structures.

Irregular surface seismic forward modeling by a body-fitted rotated–staggered-grid finite-difference method

Finite-difference(FD) methods are widely used in seismic forward modeling owing to their computational efficiency but are not readily applicable to irregular topographies. Thus, several FD methods based on the transformation to curvilinear coordinates using body-fitted grids have been proposed, e.g., stand staggered grid(SSG) with interpolation, nonstaggered grid, rotated staggered grid(RSG), and fully staggered. The FD based on the RSG is somewhat superior to others because it satisfies the spatial distribution of the wave equation without additional memory and computational requirements; furthermore, it is simpler to implement. We use the RSG FD method to transform the firstorder stress–velocity equation in the curvilinear coordinates system and introduce the highprecision adaptive, unilateral mimetic finite-difference(UMFD) method to process the freeboundary conditions of an irregular surface. The numerical results suggest that the precision of the solution is higher than that of the vacuum formalism. When the minimum wavelength is low, UMFD avoids the surface wave dispersion. We compare FD methods based on RSG, SEM, and nonstaggered grid and infer that all simulation results are consistent but the computational efficiency of the RSG FD method is higher than the rest.

Numerical modeling of wave equation by a truncated high-order finite-difference method

Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with increased order of accuracy. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note that there exist some very small coefficients. With the order increasing, the number of these small coefficients increases, however, the values decrease sharply. An error analysis demonstrates that omitting these small coefficients not only maintain approximately the same level of accuracy of finite difference but also reduce computational cost significantly. Moreover, it is easier to truncate for the high-order finite-difference formulas than for the pseudospectral for- mulas. Thus this study proposes a truncated high-order finite-difference method, and then demonstrates the efficiency and applicability of the method with some numerical examples.

Flow difference effect in the lattice hydrodynamic model

In this paper, a new lattice hydrodynamic model based on Nagatani＇s model INagatani T 1998 Physica A 261 5991 is presented by introducing the flow difference effect. The stability condition for the new model is obtained by using the linear stability theory. The result shows that considering the flow difference effect leads to stabilization of the system compared with the original lattice hydrodynamic model. The jamming transitions among the freely moving phase, the coexisting phase, and the uniform congested phase are studied by nonlinear analysis. The modified KdV equation near the critical point is derived to describe the traffic jam, and kink -antikink soliton solutions related to the traffic density waves are obtained. The simulation results are consistent with the theoretical analysis for the new model.

Second-order difference scheme for a nonlinear model of wood drying process

A numerical simulation for a model of wood drying process is considered. The model is given by a couple of nonlinear differential equations. One is a nonlinear parabolic equation and the other one is a nonlinear ordinary equation. A difference scheme is derived by the method of reduction of order. First, a new variable is introduced and the original problem is rewritten into a system of the first-order differential equations. Secondly, a difference scheme is constructed for the later problem. The solvability, stability and convergence of the difference scheme are proved by the energy method. The convergence order of the difference scheme is secondorder both in time and in space. A prior error estimate is put forward. The new variable is put aside to reduce the computational cost. A numerical example testifies the theoretical result.

Effect of strain rate difference between inside and outside groove in M-K model on prediction of forming limit curve of Ti6Al4V at elevated temperatures

The influence of initial groove angle on strain rate inside and outside groove of Ti6Al4V alloy was investigated.Based on the evolution of strain rate inside and outside groove,the effect of strain rate difference on the evolution of normal stress and effective stress inside and outside groove was also analyzed.The results show that when linear loading path changes from uniaxial tension to equi-biaxial tension,the initial groove angle plays a weaker role in the evolution of strain rate in the M-K model.Due to the constraint of force equilibrium between inside and outside groove,the strain rate difference makes the normal stress inside groove firstly decrease and then increase during calculation,which makes the prediction algorithm of forming limit convergent at elevated temperature.The decrease of normal stress inside groove is mainly caused by high temperature softening effect and the rotation of groove,while the increase of normal stress inside groove is mainly due to strain rate hardening effect.

A geometrically and locally adaptive remeshing method for finite difference modeling of mining-induced surface subsidence

Surface subsidence induced by underground mining is a typical serious geohazard.Numerical approaches such as the discrete element method(DEM)and finite difference method(FDM)have been widely used to model and analyze mining-induced surface subsidence.However,the DEM is typically computationally expensive,and is not capable of analyzing large-scale problems,while the mesh distortion may occur in the FDM modeling of largely deformed surface subsidence.To address the above problems,this paper presents a geometrically and locally adaptive remeshing method for the FDM modeling of largely deformed surface subsidence induced by underground mining.The essential ideas behind the proposed method are as follows:(i)Geometrical features of elements(i.e.the mesh quality),rather than the calculation errors,are employed as the indicator for determining whether to conduct the remeshing;and(ii)Distorted meshes with multiple attributes,rather than those with only a single attribute,are locally regenerated.In the proposed method,the distorted meshes are first adaptively determined based on the mesh quality,and then removed from the original mesh model.The tetrahedral mesh in the distorted area is first regenerated,and then the physical field variables of old mesh are transferred to the new mesh.The numerical calculation process recovers when finishing the regeneration and transformation.To verify the effectiveness of the proposed method,the surface deformation of the Yanqianshan iron mine,Liaoning Province,China,is numerically investigated by utilizing the proposed method,and compared with the numerical results of the DEM modeling.Moreover,the proposed method is applied to predicting the surface subsidence in Anjialing No.1 Underground Mine,Shanxi Province,China.

Finite-difference calculation of traveltimes based on rectangular grid

To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, to improve calculating efficiency and adaptability, the calculation method of first-arrival traveltime of finite-difference is de- rived based on any rectangular grid and a local plane wavefront approximation. In addition, head waves and scat- tering waves are properly treated and shadow and caustic zones cannot be encountered, which appear in traditional ray-tracing. The testes of two simple models and the complex Marmousi model show that the method has higher accuracy and adaptability to complex structure with strong vertical and lateral velocity variation, and Kirchhoff prestack depth migration based on this method can basically achieve the position imaging effects of wave equation prestack depth migration in major structures and targets. Because of not taking account of the later arrivals energy, the effect of its amplitude preservation is worse than that by wave equation method, but its computing efficiency is higher than that by total Green′s function method and wave equation method.

Poroelastic finite-difference modeling for ultrasonic waves in digital porous cores

Scattering attenuation in short wavelengths has long been interesting to geophysicists. Ultrasonic coda waves, observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements, are important for such studies where ultrasonic waves interact with smallscale random heterogeneities on a scale of micrometers, but often ignored as noises because of the contamination of boundary reflections from the side ends of a sample core. Numerical simulations with accurate absorbing boundary can provide insight into the effect of boundary reflections on coda waves in laboratory experiments. The simulation of wave propagation in digital and heterogeneous porous cores really challenges numerical techniques by digital image of poroelastic properties, numerical dispersion at high frequency and strong heterogeneity, and accurate absorbing boundary schemes at grazing incidence. To overcome these difficulties, we present a staggered-grid high-order finite-difference （FD） method of Biot＇s poroelastic equations, with an arbitrary even-order （2L） accuracy to simulate ultrasonic wave propagation in digital porous cores with strong heterogeneity. An unsplit convolutional perfectly matched layer （CPML） absorbing boundary, which improves conventional PML methods at grazing incidence with less memory and better computational efficiency, is employed in the simulation to investigate the influence of boundary reflections on ultra- sonic coda waves. Numerical experiments with saturated poroelastic media demonstrate that the 2L FD scheme with the CPML for ultrasonic wave propagation significantly improves stability conditions at strong heterogeneity and absorbing performance at grazing incidence. The boundary reflections from the artificial boundary surrounding the digital core decay fast with the increase of CPML thick- nesses, almost disappearing at the CPML thickness of 15 grids. Comparisons of the resulting ultrasonic coda Qsc values between the numerical and experimental ultrasonic S waveforms for a cylindrical rock sample demonstrate that the boundary reflection may contribute around one-third of the ultrasonic coda attenuation observed in laboratory experiments.

Finite-difference model of land subsidence caused by cluster loads in Zhengzhou,China

Groundwater exploitation has been regarded as the main reason for land subsidence in China and thus receives considerable attention from the government and the academic community.Recently,building loads have been identified as another important factor of land subsidence,but researches in this sector have lagged.The effect of a single building load on land subsidence was neglected in many cases owing to the narrow scope and the limited depth of the additional stress in stratum.However,due to the superposition of stresses between buildings,the additional stress of cluster loads is greater than that of a single building load under the same condition,so that the land subsidence caused by cluster loads cannot be neglected.Taking Shamen village in the north of Zhengzhou,China,as an example,a finite-difference model based on the Biot consolidation theory to calculate the land subsidence caused by cluster loads was established in this paper.Cluster loads present the characteristics of large-area loads,and the land subsidence caused by cluster loads can have multiple primary consolidation processes due to the stress superposition of different buildings was shown by the simulation results.Pore water migration distances are longer when the cluster loads with high plot ratio are imposed,so that consolidation takes longer time.The higher the plot ratio is,the deeper the effective deformation is,and thus the greater the land subsidence is.A higher plot ratio also increases the contribution that the deeper stratigraphic layers make to land subsidence.Contrary to the calculated results of land subsidence caused by cluster loads and groundwater recession,the percentage of settlement caused by cluster loads in the total settlement was 49.43%and 55.06%at two simulated monitoring points,respectively.These data suggest that the cluster loads can be one of the main causes of land subsidence.

ON LINEARIZED FINITE DIFFERENCE SIMULATION FOR THE MODEL OF NUCLEAR REACTOR DYNAMICS

The model of nuclear reactor dynamics is an initial-boundary value problems of a cou- pled nonlinear integrodifferential equation system of one ordinary differential equation and one par-- tial differential equation. In this this,paper,a linearized difference scheme is derived by the method of reduction of order.It is proved that the scheme is uniquely solvable and unconditionally convergent with the convergence rate of order two both in discrete H1norm and in discrete maxinum narm,and one needs only to solve a tridiagonal system of linear algebraic equations at each time lev- el.The method of reduction of order is an indirect constructing-difference-scheme method,which aim is for the analysis of solvablity and convergence of the constructed difference scheme.

The time-dependent Ginzburg-Landau equation for the two-velocity difference model

A thermodynamic theory is formulated to describe the phase transition and critical phenomenon in traffic flow. Based on the two-velocity difference model, the time-dependent Ginzburg-Landau （TDGL） equation under certain condition is derived to describe the traffic flow near the critical point through the nonlinear analytical method. The corresponding two solutions, the uniform and the kink solutions, are given. The coexisting curve, spinodal line and critical point are obtained by the first and second derivatives of the thermodynamic potential. The modified Korteweg- de Vries （mKdV） equation around the critical point is derived by using the reductive perturbation method and its kink antikink solution is also obtained. The relation between the TDGL equation and the mKdV equation is shown. The simulation result is consistent with the nonlinear analytical result.

Multiple flux difference effect in the lattice hydrodynamic model

Considering the effect of multiple flux difference, an extended lattice model is proposed to improve the stability of traffic flow. The stability condition of the new model is obtained by using linear stability theory. The theoretical analysis result shows that considering the flux difference effect ahead can stabilize traffic flow. The nonlinear analysis is also conducted by using a reduetive perturbation method. The modified KdV （mKdV） equation near the critical point is derived and the kink antikink solution is obtained from the mKdV equation. Numerical simulation results show that the multiple flux difference effect can suppress the traffic jam considerably, which is in line with the analytical result.

A new control method based on the lattice hydrodynamic model considering the double flux difference

A new feedback control method is derived based on the lattice hydrodynamic model in a single lane. A signal based on the double flux difference is designed in the lattice hydrodynamic model to suppress the traffic jam. The stability of the model is analyzed by using the new control method. The advantage of the new model with and without the effect of double flux difference is explored by the numerical simulation. The numerical simulations demonstrate that the traffic jam can be alleviated by the control signal.

Flow difference effect in the two-lane lattice hydrodynamic model

By introducing a flow difference effect, a modified lattice two-lane traffic flow model is proposed, which is proved to be capable of improving the stability of traffic flow. Both the linear stability condition and the kink-antikink solution derived from the modified Korteweg-de Vries （mKdV） equation are analyzed. Numerical simulations verify the theoretical analysis. Futhermore, the evolution laws under different disturbances in the metastable region are studied.

Finite Difference Method of Modelling Groundwater Flow

In this study, finite difference method is used to solve the equations that govern groundwater flow to obtain flow rates, flow direction and hydraulic heads through an aquifer. The aim therefore is to discuss the principles of Finite Difference Method and its applications in groundwater modelling. To achieve this, a rectangular grid is overlain an aquifer in order to obtain an exact solution. Initial and boundary conditions are then determined. By discretizing the system into grids and cells that are small compared to the entire aquifer, exact solutions are obtained. A flow chart of the computational algorithm for particle tracking is also developed. Results show that under a steady-state flow with no recharge, pathlines coincide with streamlines. It is also found that the accuracy of the numerical solution by Finite Difference Method is largely dependent on initial particle distribution and number of particles assigned to a cell. It is therefore concluded that Finite Difference Method can be used to predict the future direction of flow and particle location within a simulation domain.

Modeling Dynamic Systems by Using the Nonlinear Difference Equations Based on Genetic Programming

When acquaintances of a model are little or the model is too complicate to build by using traditional time series methods, it is convenient for us to take advantage of genetic programming (GP) to build the model. Considering the complexity of nonlinear dynamic systems, this paper proposes modeling dynamic systems by using the nonlinear difference e-quation based on GP technique. First it gives the method, criteria and evaluation of modeling. Then it describes the modeling algorithm using GP. Finally two typical examples of time series are used to perform the numerical experiments. The result shows that this algorithm can successfully establish the difference equation model of dynamic systems and its predictive result is also satisfactory.

Identification of Water Quality Model Parameter Based on Finite Difference and Monte Carlo

Identification results of water quality model parameter directly affect the accuracy of water quality numerical simulation. To overcome the difficulty of parameter identification caused by the measurement’s uncertainty, a new method which is the coupling of Finite Difference Method and Markov Chain Monte Carlo is developed to identify the parameters of water quality model in this paper. Taking a certain long distance open channel as an example, the effects to the results of parameters identification with different noise are discussed under steady and un-steady non-uniform flow scenarios. And also this proposed method is compared with finite difference method and Nelder Mead Simplex. The results show that it can give better results by the new method. It has good noise resistance and provides a new way to identify water quality model parameters.

A Nonstandard Finite Difference Scheme for SIS Epidemic Model with Delay: Stability and Bifurcation Analysis

A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD) method. The dynamics of the obtained discrete system is investigated. First we show that the discrete system has equilibria which are exactly the same as those of continuous model. By studying the distribution of the roots of the characteristics equations related to the linearized system, we can provide the stable regions in the appropriate parameter plane. It is shown that the conditions for those equilibria to be asymptotically stable are consistent with the continuous model for any size of numerical time-step. Furthermore, we also establish the existence of Neimark-Sacker bifurcation (also called Hopf bifurcation for map) which is controlled by the time delay. The analytical results are confirmed by some numerical simulations.