Multiquadric方法在中尺度气象资料客观分析中的应用

利用浙江省中尺度自动站的温度资料,引入已广泛应用于测绘、地球物理、地图绘制等领域的Multiquadric插值法,并与目前气象领域内采用较多的Cressman方法进行比较,分析两者在不同站点分布和数量下对实况场的逼近程度,并对Multiquadric中的自由设定参数进行讨论。结果表明,总体而言,Multiquadric方法的均方根误差小于Cressman方法。当站点数为100时,两者差异达0.3℃;当站点数>1000时,Multiquadric方法相对Cressman的改进已经<1%,两者差异非常小。在站点数较少或是站点间隔较大的情况下,Cressman方法较易产生虚假中心,而Multiquadric方法则更加接近实况。Multiquadric平滑参数λ可以被认为是一种低通滤波器,它将高频波滤去,留下较大尺度的系统,同时冷暖空气温度中心的强度随着平滑强度的增加也逐渐减弱。Multiquadric参数c存在一个上限值,当c大于或小于该值时,客观分析结果表现出不同的敏感度。

用Multiquadric方法实现医学图像的弹性配准

目的 改进医学图像弹性配准的精确度和配准方法的稳定性,并简化配准操作。方法 利用Multiquadric插值方法的可平滑性质,将其与一种半自动标记点选择方法结合,建立一种新配准方法。结果 运用此方法进行医学图像的弹性配准,实现了标准图像与变形图像的快速、准确配准。结论 我们建立的配准方法是一种准确、简便和稳定的方法。

一种新的Multiquadric拟插值

Multiquadric(MQ)是由Hardy提出来的一种径向基函数,至今它已在大地测量学、微分方程数值解等许多方面得到了应用。目前已知的multiquadric拟插值有4种,即,Beatson和Powell的LA、LB和LC以及Wu和Schaback的LD,其中,LB是常数再生的,LC和LD是线性再生的。该文首先给出形如LD的拟插值线性再生时其基函数应具有的性质,然后构造了一种具有线性再生性和保单调性的multiquadric拟插值,并对其逼近误差进行了理论分析。最后,通过两个实例进行数值实验,从算例的结果来看,该拟插值具有良好的逼近精度。

空间分数阶扩散方程的Multiquadric拟插值解法

基于拟插值算子对空间分数阶扩散方程构造了一个新的数值格式.首先在散落点上用三次Multiquadric(MQ)函数的平移构造了一个拟插值算子,分析了此拟插值算子的再生性、保形性和对分数阶导数的收敛性,最后利用上述拟插值算子并结合时间差分格式构造了空间分数阶扩散方程的计算格式.收敛性分析显示:当时间方向用Crank-Nicolson格式时,精度为O(Δt2+h4-α),当时间方向用向后Euler格式时,精度为O(Δt+h4-α),其中Δt为时间步长,h为空间步长.数值结果表明MQ拟插值方法是构造数值格式的一个有效工具.

非矩形定义域上的散乱数据曲面重建的Multiquadric基函数拟合法

传统的Multiquadric基函数拟合散乱数据方法,只能用于定义域呈矩形拓扑的散乱数据集上,而逆向工程中通常遇到的是非矩形定义域上的散乱数据集,因此不能用传统方法。文章提出一种新的Multiquadric基函数拟合非矩形定义域上的散乱数据集。该方法首先找到一个与该散乱数据集所在曲面拓扑等价的参数曲面,通过将散乱数据点(xi,yi,zi)一一映射到此参数曲面,反求出其对应参数(ui,vi),由于(ui,vi)∈犤0,1犦×犤0,1犦,从而将非矩形定义域上的Multiquadric基函数拟合方法转化为传统的方法。将Multiquadric曲面与B-样条曲面进行了比较,指出了Multiquadric曲面优于B-样条曲面之处。

A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

Multiquadric Radial Basis Function Approximation Scheme for Solution of Total Variation Based Multiplicative Noise Removal Model

This article introduces a fastmeshless algorithm for the numerical solution nonlinear partial differential equations(PDE)by Radial Basis Functions(RBFs)approximation connected with the Total Variation(TV)-basedminimization functional and to show its application to image denoising containing multiplicative noise.These capabilities used within the proposed algorithm have not only the quality of image denoising,edge preservation but also the property of minimization of staircase effect which results in blocky effects in the images.It is worth mentioning that the recommended method can be easily employed for nonlinear problems due to the lack of dependence on a mesh or integration procedure.The numerical investigations and corresponding examples prove the effectiveness of the recommended algorithm regarding the robustness and visual improvement as well as peak-signal-to-noise ratio(PSNR),signal-to-noise ratio(SNR),and structural similarity index(SSIM)corresponded to the current conventional TV-based schemes.

Improved geologic surface approximation using a multiquadric method with additional constraints

Geologic surface approximation is profoundly affected by the presence, density and location of scattered geologic input data. Many studies have recognized the importance of utilizing varied sources of information when reconstructing a surface. This paper presents an improved geologic surface approximation method using a multiquadric function and borehole data. Additional information, i.e., inequality elevation and dip-strikes data extracted from outcrops or mining faces, is introduced in the form of physical constraints that control local changes in the estimated surface. Commonly accepted hypothesis states that geologic surfaces can be approximated to any desired degree of exactness by the summation of regular, mathematically defined, surfaces: in particular displaced quadric forms. The coefficients of the multiquadric functions are traditionally found by a least squares method. The addition of physical constraints in this work makes such an approach into a non-deterministic polynomial time problem. Hence we propose an objective function that represents the quality of the estimated surface and that includes the additional constraints by incorporation of a penalty function. Maximizing the smoothness of the estimated surface and its fitness to the additional constraints then allows the coefficients of the multiquadric function to be obtained by iterative methods. This method was implemented and demonstrated using data collected from the 81'st coal mining area of the Huaibei Coal Group.

The Application of Multiquadric Function Fitting to Borehole Strain Time Series Data Processing

Based on the existing continuous borehole strain observation,the multiquadric function fitting method was used to deal with time series data. The impact of difference kernel function parameters was discussed to obtain a valuable fitting result,from which the physical connotation of the original data and its possible applications were analyzed.Meanwhile,a brief comparison was made between the results of multiquadric function fitting and polynomial fitting.

Generator,multiquadric generator,quasi-interpolation and multiquadric quasi-interpolation

The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.

Undersea Buried Pipeline Reconstruction Based on the Level Set and Inverse Multiquadric Regularization Method

The electric inversion technique reconstructs the subsurface medium distribution from acquired data.On the basis of electric inversion,objects buried under the earth or seabed,such as pipelines and unexploded ordnance,are detected and located in a contactless manner.However,the process of accurately reconstructing the shape of the target object is challenging because electric inversion is a nonlinear and ill-posed problem.In this work,we present an inverse multiquadric(IMQ)regularization method based on the level set function for reconstructing buried pipelines.In the case of locating underwater objects,the unknown inversion area is split into two parts,the background and the pipeline with known conductivity.The geometry of the pipeline is represented based on the level set function for achieving a noiseless inversion image.To obtain a binary image,the IMQ is used as the regularization term,which‘pushes’the level set function away from 0.We also provide an appropriate method to select the bandwidth and regularization parameters for the IMQ regularization term,resulting in reconstructed images with sharp edges.The simulation results and analysis show that the proposed method performs better than classical inversion methods.

THE SHAPE PRESERVING AND HIGHACCURACYAPPROXIMATIONWITHMULTIQUADRIC

ＴＨＥＳＨＡＰＥＰＲＥＳＥＲＶＩＮＧＡＮＤＨＩＧＨＡＣＣＵＲＡＣＹＡＰＰＲＯＸＩＭＡＴＩＯＮＷＩＴＨＭＵＬＴＩＱＵＡＤＲＩＣ＊ＷＵＺＯＮＧＭＩＮＡＮＤＬＩＵＪＩＡＮＰＩＮＡｂｓｔｒａｃｔ．Ｔｈｉｓｐａｐｅｒｄｉｓｃｕｓｓｅｓｔｈｅｓｕｆｉｃｉｅｎｔｃ...

A Comprehensive Price Prediction System Based on Inverse Multiquadrics Radial Basis Function for Portfolio Selection

Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction. Here, we propose a comprehensive price prediction (CPP) system based on inverse multiquadrics (IMQ) radial basis function. First, the novel radial basis function (RBF) system based on IMQ function rather than traditional Gaussian (GA) function is proposed and centers on multiple price prediction strategies, aiming at improving the efficiency and robustness of price prediction. Under the novel RBF system, we then create a portfolio update strategy based on kernel and trace operator. To assess the system performance, extensive experiments are performed based on 4 data sets from different real-world financial markets. Interestingly, the experimental results reveal that the novel RBF system effectively realizes the integration of different strategies and CPP system outperforms other systems in investing performance and risk control, even considering a certain degree of transaction costs. Besides, CPP can calculate quickly, making it applicable for large-scale and time-limited financial market.

一种新的积分值型MQ拟插值算子

Multiquadric (MQ)函数作为径向基函数的一种,其作为核函数可逼近任何光滑函数,被广泛应用在拟插值的研究中,现有的MQ拟插值大部分都是以离散函数值为已知条件,而在实际应用中,积分值作为已知条件也比较常见,为了让MQ拟插值得到更广泛的应用,本文提出了一种新的基于积分值的MQ拟插值算子。首先利用连续区间上积分值的线性组合来对节点处的导数值进行逼近,然后根据已有的MQ拟插值进一步得到新的积分值型MQ拟插值算子,并给出了误差估计。最后通过数值实验展示了本文构造的积分值型MQ拟插值算子的逼近效果,说明了该方法的可行性和有效性。As a kind of radial basis function, Multiquadric (MQ) function, as a kernel function, can approximate any smooth function, and is widely used in the research of quasi-interpolation. Most of the existing MQ quasi-interpolation is based on the known condition of the discrete function value, and in practical applications, the integral value is also a common-known condition. In order to make MQ quasi-interpolation more widely used, a new MQ quasi-interpolation operator based on integral value is proposed in this paper. First, the derivative value at the node is approximated by the linear combination of integral values on the continuous interval, and then a new integral value MQ quasi-interpolation operator is obtained according to the existing MQ quasi-interpolation, and the error estimate is given. Finally, the approximation effect of the integral-valued MQ quasi-interpolation operator constructed in this paper is demonstrated by numerical experiments, and the feasibility and effectiveness of the proposed method are demonstrated.

Multiquadric拟插值对高阶导数逼近的稳定性分析 谨以此文致《中国科学》创刊六十周年

基于样本数据来数值模拟函数的高阶导数是数值逼近中遇到的一类重要而且基本的问题,差商方法是数值微分的传统方法.但是在实际问题的求解中,它表现出强烈的不稳定性.在实际应用中,由于差商计算的不稳定性,它仅能用来模拟函数的低阶导数.为了更好地模拟函数的高阶导数,本文利用multiquadric拟插值提出了一种新的方法.并将multiquadric拟插值方法模拟函数导数的稳定性与传统差商方法所得结果进行了对比.数值例子很好地验证了本文的理论.从理论论证和数值例子比较来看,multiquadric拟插值方法比差商方法更为稳定.这个性质也表明,基于散乱甚至有干扰的数据,在逼近函数的高阶导数时,multiquadric拟插值方法是一个有效的工具.

Stability of multiquadric quasi-interpolation to approximate high order derivatives

Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical method is the divided difference method.However,it has been shown strongly unstable in practice.Actually,it can only be used to simulate the lower order derivatives in applications.To simulate the high order derivatives,this paper suggests a new method using multiquadric quasi-interpolation.The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method.Moreover,some numerical examples are presented to confirm the theoretical results.Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method.This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.

A MULTIVARIATE MULTIQUADRIC QUASI-INTERPOLATION WITH QUADRIC REPRODUCTION

In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can reproduce linear polynomials to the scheme quadric polynomials. Furthermore, we give the approximation error of the modified scheme. Our multivariate multiquadric quasi-interpolation scheme only requires information of lo- cation points but not that of the derivatives of approximated function. Finally, numerical experiments demonstrate that the approximation rate of our scheme is significantly im- proved which is consistent with the theoretical results.

Applying Multiquadric Quasi-Interpolation to Solve KdV Equation

Quasi-interpolation is very useful in the study of approximation theory and its applications,since it can yield solutions directly without the need to solve any linear system of equations.Based on the good performance,Chen and Wu presented a kind of multiquadric （MQ） quasi-interpolation,which is generalized from the L D operator,and used it to solve hyperbolic conservation laws and Burgers’ equation.In this paper,a numerical scheme is presented based on Chen and Wu’s method for solving the Korteweg-de Vries （KdV） equation.The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative,and the forward divided difference to approximate the temporal derivative,where the spatial derivative is approximated by the derivative of the generalized L D quasi-interpolation operator.The algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.

径向基函数方法在南泥湾油田勘探中的应用

为明确利用奇特开发井网数据拟合地质曲面拟合的合理性和可靠性,选用多二次曲面函数进行插值计算.该方法具有数学算法简单、计算灵活、易于实现及结果比较准确等诸多优点,适合用于散乱数据的插值.通过实例的分析验证,证明了该方法在建立散乱数据点地质曲面时具有良好的逼近效果,可以反映比较真实的地质现象.

基于短时强降雨探测的拉曼激光雷达测温数据WRF同化方法

为挖掘激光雷达高垂直分辨探测优势,将其引入降雨观测与研究中,重点构建一套测温数据模式同化方法,目的在于评估激光雷达数据对降雨模拟的影响。借助激光雷达测温数据综合多级质量控制技术对雨前探测信号进行反演优化,获取更可靠的温度廓线,这种垂直间隔4 m左右的雷达数据对于降雨是一种新型数据。设计三步实验研究新数据WRF模式(weather research and forecasting model)同化方法。第一步开展控制实验,通过模式检验与参数调试获取最佳模拟方案,为同化实验提供依据。第二步开展常规探测数据同化实验,通过WRF模式同化模块将90组地面、高空测站数据融入模式初始场,TS评分提高了0.07,并成功消除了陕西西南部虚假暴雨中心,但存在空报率偏高等问题。第三步开展激光雷达数据同化实验,重点解决制约模式初始场中尺度信息测站稀少的关键难题,引入MQ法将单点数据扩展为49组格点数据并完成三维变分同化模拟,克服了常规数据同化所致空报率偏高的问题,且模拟雨强更接近实况,同时降水TS评分提高了0.12,漏报率降低了0.09。定性与定量分析均表明借助Multiquadric将激光雷达探测数据融入WRF模式可造成模拟效果提升。本文结果表明激光雷达可用于探测短时强降雨,且探测位置宜设在对流云外围。