Systematic examination of ‘precursor waves’ of strong earthquakes by using the wavelet method

Based on the borehole strainmeter recordings immediately before the strong earthquakes （Ms〉7.5） during the period from 2001 to 2005 observed at the Taian observation station, Shandong Province, the authors make a systematic and objective examination of the precursor waves of the quakes. The effects of Earth tides with periods larger than 128 min are eliminated through high-pass filtering; and atmospheric-pressure inferences are removed by linear regression. The 2-128 min signals are then separated into six frequency bands by employing the wavelet method. Results indicate that the wavelet method is capable of picking out information of weak variations in the signals. According to the characteristics of the ＇precursor waves＇ obtained from wavelet transformation, the method of overrun ratio analysis is put forward for examination. All the detailed components of the wavelets have been analyzed. For the time series of the overrun rate in all these components, statisitical calculation has been made for the slopes of fit curves, and mean values and standard deviations were obtained and positive-negative slope ratios were analyzed. The three statistical data show that ＇precursor waves＇ are not widely recorded by borehole strainmeter within 15 days before remote strong earthquakes.

A novel wavelet method for electric signals analysis in underwater arc welding

Electric signals are acquired and analyzed in order to monitor the underwater arc welding process. Voltage break point and magnitude are extracted by detecting arc voltage singularity through the modulus maximum wavelet （MMW） method. A novel threshold algorithm, which compromises the hard-threshold wavelet （HTW） and soft-threshold wavelet （STW） methods, is investigated to eliminate welding current noise. Finally, advantages over traditional wavelet methods are verified by both simulation and experimental results.

Theory and Application of Natural-Based Wavelet Method

Wavelet method is often used in analyzing trend and period of time sequence. When using wavelet method one serious problem is different chosen wavelet basis and scale would lead to different results. Sometimes, the results vary greatly. To overcome this problem and to improve the accuracy and efficiency, a new method denoted by Natural-based Wavelet Method is introduced and extended. It can be proved that the proposed method in fact is a special class of discrete wavelet. At first, two numerical examples are designed to show the capacity of the novel method. Second, this method is applied to a precipitation series. According to wavelet analysis and short-range precipitation prediction, this precipitation exists a faintly increasing trend. Through the analysis, the studied precipitation has two major periods: 11 and 41 years. The results validate that the Natural-based Wavelet Method used in application of multi-complicated observed data is suitable. It is easy to write the source code of the proposed method. When new information appears, new information can be easily added into the original sequence, this is another advantage of the proposed method.

A high-order accurate wavelet method for solving Schrdinger equations with general nonlinearity

A sampling approximation for a function defined on a bounded interval is proposed by combining the Coiflet-type wavelet expansion and the boundary extension technique. Based on such a wavelet approximation scheme, a Galerkin procedure is developed for the spatial discretization of the generalized nonlinear Schr6dinger （NLS） equa- tions, and a system of ordinary differential equations for the time dependent unknowns is obtained. Then, the classical fourth-order explicit Runge-Kutta method is used to solve this semi-discretization system. To justify the present method, several widely considered problems are solved as the test examples, and the results demonstrate that the proposed wavelet algorithm has much better accuracy and a faster convergence rate in space than many existing numerical methods.

A WAVELET METHOD FOR BENDING OF CIRCULAR PLATE WITH LARGE DEFLECTION

A wavelet method for solving strongly nonlinear boundary value problems is described, which has been demonstrated early to have a convergence rate of order 4, almost independent of the nonlinear intensity of the equations. By using such a method, we study the bending problem of a circular plate with arbitrary large deflection. As the deflection increases, the bending behavior usually exhibits a so-called plate-to-membrane transition. Capturing such a transition has ever frustrated researchers for decades. However, without introducing any addi- tional treatment, we show in this study that the proposed wavelet solutions can naturally cover the plate-membrane transition region as the plate deflection increases. In addition, the high accuracy and efficiency of the wavelet method in solving strongly nonlinear problems is numerically confirmed, and applicable scopes for the linear, the membrane and the yon Karman plate theories are identified with respect to the large deformation bending of circular plates.

Ship hull optimization based on wave resistance using wavelet method

The ship hull surface optimization based on the wave resistance is an important issue in the ship engineering industry. The wavelet method may provide a convenient tool for the surface hull optimization. As a preliminary study, we use the wavelet method to optimize the hull surface based on the Michel wave resistance for a Wigley model in this paper. Firstly, we express the model＇s surface by the wavelet decomposition expressions and obtain a reconstructed surface and then validate its accuracy. Secondly, we rewrite the Michel wave resistance formula in the wavelet bases, resulting in a simple formula containing only the ship hull surface＇s wavelet coefficients. Thirdly, we take these wavelet coefficients as optimization variables, and analyze the main wave resistance distribution in terms of scales and locations, to reduce the number of optimization variables. Finally, we obtain the optimal hull surface of the Wigley model through genetic algorithms, reducing the wave resistance almost by a half. It is shown that the wavelet method may provide a new approach for the hull optimization.

Nonlinear Wavelet Methods for High-dimensional Backward Heat Equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Reginska＇s work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.

Application of wavelet to strength log from scratch test

This paper proposes a methodology to construct logs of rock strength from the cutting force signal recorded in scratch tests conducted in the ductile regime.The approach,which is based on the application of discrete wavelet transforms,recognizes the existence of two length scales[c and[r.The strength length scale[c represents the length over which it is meaningful to measure strength,while the repeatability length scale[r is related to the resolution at which the force signal must be observed to become insensitive to the stochastic micro-failure events triggered by the motion of the cutter.It is postulated that the original cutting force signal,assumed to be sampled at a high enough frequency,can be decomposed into a deterministic signal intrinsic to the rock and a stochastic one resulting from discrete rock failure events.The technique of multiresolution analysis based on the maximal overlap discrete wavelet transform is applied as a low-pass filter to the original cutting force signals so as to wipe out the high-frequency components associated with the stochastic rock failure events.A criterion to determine the optimum cutoff frequency of the low-pass filter and the corresponding repeatability length scale is discussed in terms of the correlation coefficients between different filtered signals.It is shown that the low-pass filtered signals obtained at the optimum cutoff frequency have two salient features:(i)repeatability over different tests conducted at the same depth of cut on the same sample,and(ii)variability along the cutting distance.The excellent repeatability reveals that the deterministic background trend of the original force signals is relevant to the rock strength property,and the variability of the background trend captures the spatial variation of the rock strength.

B-Spline Wavelet on Interval Finite Element Method for Static and Vibration Analysis of Stiffened Flexible Thin Plate

A new wavelet finite element method(WFEM)is constructed in this paper and two elements for bending and free vibration problems of a stiffened plate are analyzed.By means of generalized potential energy function and virtual work principle,the formulations of the bending and free vibration problems of the stiffened plate are derived separately.Then,the scaling functions of the B-spline wavelet on the interval(BSWI)are introduced to discrete the solving field variables instead of conventional polynomial interpolation.Finally,the corresponding two problems can be resolved following the traditional finite element frame.There are some advantages of the constructed elements in structural analysis.Due to the excellent features of the wavelet,such as multi-scale and localization characteristics,and the excellent numerical approximation property of the BSWI,the precise and efficient analysis can be achieved.Besides,transformation matrix is used to translate the meaningless wavelet coefficients into physical space,thus the resolving process is simplified.In order to verify the superiority of the constructed method in stiffened plate analysis,several numerical examples are given in the end.

The periodicity of 3C 273's radio light curve at 15 GHz found by the wavelet method

We analyze the radio light curve of 3C 273 at 15 GHz from 1963 to 2006 taken from the database of the literature,and find evidence of quasi-periodic activity.Using the wavelet analysis method to analyze these data,our results indicate that:(1) There is one main outburst period of P1=8.1±0.1 year in 3C 273.This period is in a good agreement with Ozernoi's analysis in optical bands.(2) Based on the possible periods,we expect the next burst in 2014 October.

A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media

In this paper, we consider numerical simulation of wave propagation in fluidsaturated porous media. A wavelet finite-difference method is proposed to solve the 2-D elastic wave equation. The algorithm combines flexibility and computational efficiency of wavelet multi-resolution method with easy implementation of the finite-difference method. The orthogonal wavelet basis provides a natural framework, which adapt spatial grids to local wavefield properties. Numerical results show usefulness of the approach as an accurate and stable tool for simulation of wave propagation in fluid-saturated porous media.

The Adaptive Wavelet Collocation Method and Its Application in Front Simulation

The adaptive wavelet collocation method （AWCM） is a variable grid technology for solving partial differential equations （PDEs） with high singularities. Based on interpolating wavelets, the AWCM adapts the grid so that a higher resolution is automatically attributed to domain regions with high singularities. Accuracy problems with the AWCM have been reported in the literature, and in this paper problems of efficiency with the AWCM are discussed in detail through a simple one-dimensional （1D） nonlinear advection equation whose analytic solution is easily obtained. A simple and efficient implementation of the AWCM is investigated. Through studying the maximum errors at the moment of frontogenesis of the 1D nonlinear advection equation with different initial values and a comparison with the finite difference method （FDM） on a uniform grid, the AWCM shows good potential for modeling the front efficiently. The AWCM is also applied to a two-dimensional （2D） unbalanced frontogenesis model in its first attempt at numerical simulation of a meteorological front. Some important characteristics about the model are revealed by the new scheme.

Application of Wavelet Finite Element Method to Simulation of the Temperature Field of Copier Paper

Simulation of the temperature field of copier paper in copier fusing is very important for improving the fusing property of reprography. The temperature field of copier paper varies with a high gradient when the copier paper is moving through the fusing rollers. By means of conventional shaft elements, the high gradient temperature variety causes the oscillation of the numerical solution. Based on the Daubechies scaling functions, a kind of wavelet based element is constructed for the above problem. The temperature field of the copier paper moving through the fusing rollers is simulated using the two methods. Comparison of the results shows the advantages of the wavelet finite element method, which provides a new method for improving the copier properties.

A wavelet multiscale method for inversion of Maxwell equations

This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to the size of scale from the shortest to the longest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision.

Multi-symplectic wavelet splitting method for the strongly coupled Schrodinger system

We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.

Discontinuous Legendre Wavelet Galerkin Method for One-Dimensional Advection-Diffusion Equation

This paper presents discontinuous Legendre wavelet Galerkin (DLWG) approach for solving one-dimensional advection-diffusion equation (ADE). Variational formulation of this type equation and corresponding numerical fluxes are devised by utilizing the advantages of both the Legendre wavelet bases and discontinuous Galerkin (DG) method. The distinctive features of the proposed method are its simple applicability for a variety of boundary conditions and able to effectively approximate the solution of PDEs with less storage space and execution. The results of a numerical experiment are provided to verify the efficiency of the designed new technique.

ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations ( ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.

APPLICATION OF WAVELET THEORY IN RESEARCHON WEIGHT FUNCTION OF MESHLESS METHOD

Multiresolution analysis of wavelet theory can give an effective way to describe the information at various levels of approximations or different resolutions, based on spline wavelet analysis,so weight function is orthonormally projected onto a sequence of closed spline subspaces, and is viewed at various levels of approximations or different resolutions. Now, the useful new way to research weight function is found, and the numerical result is given.

Wavelet Collocation Method for Solving Elliptic Singularly Perturbed Problem

Wavelet collocation method is used to solve an elliptic singularly perturbed problem with two parameters. The B-spline function is used as a single mother wavelet, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problem and the result shows the reliability and efficiency of the method.