Interpolation with Splines in Tension: A Green's Function Approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances; however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.     

地球物理勘探中几种二维插值方法的误差分析

为了研究采样和网格化方法对地球物理数据成图精度的影响,为野外数据采集布设提供一定的依据,采用数值模拟确定重力异常场场值,通过不同采样间距和不同插值方法计算重力异常绝对误差均方根值和节点处的绝对误差值,对比不同插值方法的误差,得到了如下认识：1）对于同一插值方法而言,存在小间距绝对误差均方根值小于大间距绝对误差均方根值的关系。2）对不同的插值方法而言:当采样间距小于最小异常地质体尺度时,绝对误差均方根值由小到大的顺序是径向基函数法、改进的谢别德法、克里金插值法、自然邻点法、反距离加权插值法、最近邻点法、最小曲率法,并且线性插值三角网法与自然邻点法具有几乎相同的数值;当采样间距大于最小异常地质体尺度时,绝对误差均方根值由小到大的顺序是径向基函数法、改进的谢别德法、克里金插值法、自然邻点法、最小曲率法、最近邻点法、反距离加权插值法,并且线性插值三角网法和自然邻点法具有几乎相同的数值。3）从绝对误差均方值看,径向基函数方法、改进的谢别德方法和克里金方法数值较小,其中径向基函数值绝对误差均方根值最小。4）从节点处绝对误差值来看,径向基函数方法、克里金方法、改进的谢别德方法相对其他插值方法具有更小的误差,不存在局部误差较小或较大的情况,是相对较好的插值方法,并且径向基函数方法是最好的。     

降雨资料Kriging与IDW插值对比分析—以漓江流域为例

降水空间化信息在很多领域都具有重要意义,而进行空间插值方法形成降水空间化信息是当代较为常用的方式。面对众多的插值方法其插值精度成为我们是否采用的关键,就kirging插值法与IDW(Inverse Distance Weighting)插值进行研究以探讨其插值效果,为此类科学研究提供依据。研究以漓江流域内各气象站点降水统计资料为基础分别采用上述两种方法进行插值处理,通过与预留实测站点比较评判方法的插值效果的优劣。结果表明,多年月平均降水量作为时间步长时,降水丰沛的月份使用kriging插值法较优于IDW插值法,而枯水月份则使用IDW插值法较优于kriging插值法。     

Constrained Smoothing of Noisy Data Using Splines in?Tension

A method of fitting a smooth cubic spline curve through noisy data points is presented. Overshoots of the spline curve between data points were prevented by applying tension to the fit using a quadratic spring approximation, which allowed a linear inverse theory approach to be adopted. Error-bars in the measured data were mapped through the inversion process to give the covariance of the fitted curve. This is an improvement over previous methods, which largely neglect the effect of data errors on the fit. Another improvement is to impose fixed constraints on the fit by simultaneously applying the method of Lagrange multipliers. The effect of these constraints on the covariance of the fitted curve is quantified using results from linear algebra. Example applications to synthetic data and a record of magnetic inclination from Hawaii are given.     

与距离成反比加权法的数据插值方法及程序实现

总结了各种插值方法及其特点，详细阐述了与距离成反比加权法插值方法，在原有方法的基础上提出了新的改进后的与距离成反比加权法插值方法，并由此引入了影响域、影响点集和数据块交集的概念。开发出基于VC 的二维平面和三维空间散乱数据改进的与距离成反比加权法插值算法类，并嵌入电磁资料处理系统软件，为海量电磁资料数据、重力场数据的插值提供了便利。     

F-K反偏移中的插值映射

均匀介质中的F-K反偏移是实现一般介质下F-K反偏移的基础,而均匀介质中的F-K反偏移的实质是一个插值映射过程。根据插值对象为复函数的特点,采用了实部虚部分别插值、去振荡因子法、模和相位分别插值、直接频率域插值4种插值方法对点脉冲模型进行了试算。结果表明:截断的直接频率域插值在适应性和精度综合衡量下为最佳;去振荡因子法精度虽高,但可应用的模型太具理想,不适合一般模型中的应用;实部虚部分别插值与模和相位分别插值两者精度较低。     

Interpolation by regularized spline with tension: I. Theory and implementation

Bivariate and trivariate functions for interpolation from scattered data are derived. They are constructed by explicit minimization of a general smoothness functional, and they include a tension parameter that controls the character of the interpolation function (e.g., for bivariate case the surface can be tuned from a membrane to a thin steel plate), Tension can be applied also in a chosen direction, for modeling of phenomena with a simple type of anisotropy. The functions have regular derivatives of all orders everywhere. This makes them suitable for analysis of surface geometry and for direct application in models where derivatives are necessary. For processing of large datasets (thousands of data points), which are now common in geosciences, a segmentation algorithm with a flexible size of overlapping neighborhood is presented. Simple examples demonstrating flexibility and accuracy of the functions are presented.On leave from the Department of Physical Geography and Cartography, Comenius University, Mlynská dolina, Bratislava, Czechoslovakia.On leave from the Institute of Physics, Dúbravská cesta 9, Bratislava, Czechoslovakia.     

不同雨量次降雨空间插值对比——以四川省南充市降雨为例

为对比分析大雨、中雨、小雨条件下,不同空间插值方法模拟地市州尺度降雨的差异,基于四川省南充市9个县级辖区三次24h降雨量数据,采用反距离加权(Inverse Distance Weighting,IDW)、张力样条函数(Spline with Tension,ST)、局部多项式(Local Polynomial Interpolation,LPI)、ANUDEM四种插值方法,从插值平均误差(ME)、中误差(RMSE)角度进行了对比分析。结果显示,按ME排序,大雨、中雨时LPISTIDWANUDEM,小雨时IDW、ANUDEM、ST基本相似,LPI最大;三种降雨条件下四种插值方法 ME均小于0.5mm。从RMSE看,大雨、中雨、小雨时ANUDEM插值RMSE为1.79mm、3.07mm和0.05mm,显著小于IDW、LPI和ST;三种插值方法之间差异微小,大雨、中雨、小雨时均接近13mm、8mm和0.5mm。在降雨量等级为大雨和中雨时,ANUDEM插值方法优于其他插值方法,而在降雨量等级为小雨时,四种插值方法差异较小。     

A Comparative Study for 3D Surface Modeling of Coal Deposit by Spatial Interpolation Approaches

The planning stages of mining require comprehensive and detailed analyses. The proper determination of the orebody boundary is one of the most important points to provide optimum model structure and projections. The limits can be determined by different methods based on the site geology. Although some three dimensional (3D) models were proposed for providing detailed information concerning a mine deposit, developing a solid model via a 3D approach is novelty. In other words, surface modeling should be performed for creating a solid model and determining limits of the deposit. In this way, sensible generation of the surface model can be achieved. This study investigated the estimation capability of the polynomial approach, which is a novel spatial interpolation method, for modeling a coal deposit surface. The performance of the proposed model was compared with some conventional methods in the literature. The results showed that the polynomial interpolation method is an effective method to employ for surface modeling of a mine deposit.     

Smoothing and interpolation by kriging and with splines

Let scalar measurements at distinct points x₁, , x_n be y₁, , y_n.We may look for a smooth function f(x)that goes through or near the points (x_i, y_i).Kriging assumes f(x)is a random function with known (possibly estimable) covariance function (in the simplest case). Splines assume a definition of the smoothness of a nonrandom function f(x).An elementary explanation is given of the fact that spline approximations are special cases of the solution of a kriging problem.     

基于多元自适应回归样条的青藏高原温泉区域的冻土分布制图

以探地雷达、电磁测深、钻探等技术方法获得野外数据及数字高程（DEM）遥感数据为基础,通过聚类分析和相关性分析对高程、坡度、坡向等因素对多年冻土分布的影响进行了定量化研究.利用非线性的多元自适应回归样条（MARS）方法建立了基于高程、太阳辐射的多年冻土分布模型,通过自身的交叉验证及对比年平均地温模型和逻辑回归模型的总体分...     

区域积温插值的GIS方法

阐述了积温期分解对区域积温插值GIS方法的改进.运用该方法对台站各旬日均温纪录进行分解,得到>0℃期间的积温期长度与对应的气温下降累计值,发现它们之间呈现很好的线性关系,而积温值与积温期之间用指数回归拟合曲线代表更切合.在气温垂直递减率的基础上,用上述直线关系为参数,构建基于GIS(ARC/INFO地理信息系统)GRID模块的插值方法,依据高分辨率的DEM与台站海拔高度TIN之间的高度差进行高度订正以生成多山区域的积温数据库.与改进前的GIS插值相比,该方法可使甘肃省区域内积温插值的相对精度提高近3倍.这种方法可以在甘肃省退耕还林还草等生态建设信息化管理中得到应用.     

多目标区域地球化学编图数据整理方法—以海河流域编图为例

多目标区域地球化学编图是科学表达区域土壤地球化学分布和特征的图示方法,大区域的编图一般可能涉及到对不同的工作区、不同实验室和不同时间的测试数据,以及不同景观条件和网度的表层、深层土壤元素数据或不同分度带的网格数据分别进行拼接合并,因此区域编图对原始数据的整理和网格化处理不可避免.而不同的数据整理方法及其计算参数所形成的结果是非常不同的,数据本身的特点也对编图具有很大的影响.为了真实客观地反映区域内元素的地球化学分布特征,数据整理和处理的方法及技术参数的选择是关键技术问题.作者就化探数据处理中常用的几种插值方法进行了对比评价,并分别从网格间距及搜索半径大小等方面做比较,提出了自己在编图中采用的数据处理方法及选用的技术参数.     

Evaluation of Interpolation Accuracy of Neural Kriging with Application to Temperature-Distribution Analysis

An interpolation method based on a multilayer neural network (MNN), has been examined and tested for the data of irregular sample locations. The main advantage of MNN is in that it can deal with geoscience data with nonlinear behavior and extract characteristics from complex and noisy images. The training of MNN is used to modify connection weights between nodes located in different layers by a simulated annealing algorithm (one of the optimization algorithms of the network). In this process, three types of errors are considered: differences in values, semivariograms, and gradients between sample data and outputs from the trained network. The training is continued until the summation of these errors converges to an acceptably small value. Because the MNN trained by this learning criterion can estimate a value at an arbitrary location, this method is a form of kriging and termed Neural Kriging (NK). In order to evaluate the effectiveness of NK, a problem on restoration ability of a defined reference surface from randomly chosen discrete data was prepared. Two types of surfaces, whose semivariograms are expressed by isotropic spherical and geometric anisotropic gaussian models, were examined in this problem. Though the interpolation accuracy depended on the arrangement pattern of the sample locations for the same number of data, the interpolation errors of NK were shown to be smaller than both those of ordinary MNN and ordinal kriging. NK can also produce a contour map in consideration of gradient constraints. Furthermore, NK was applied to distribution analysis of subsurface temperatures using geothermal investigation loggings of the Hohi area in southwest Japan. In spite of the restricted quantity of sample data, the interpolation results revealed high temperature zones and convection patterns of hydrothermal fluids. NK is regarded as an interpolation method with high accuracy that can be used for regionalized variables with any structure of spatial correlation.     

Shearlet域稀疏约束地震数据重建

在地震数据处理流程中,通常对不规则的、稀疏的或者缺失的地震数据进行插值处理,通过插值方法来避免多次波的预测错误和成像假频等现象,使地震数据处理更加精准。Shearlet变换是一种多尺度变换,具有最佳的稀疏性、方向性以及局部化特性。将Shearlet变换与基于Landweber加速下降迭代方法结合起来对地震数据进行插值,在保证求解精度的同时提高了计算效率。信号和噪声在Shearlet域具有不同的分布特点,通过阈值法压制随机噪声,可提高算法的抗噪性。此外,采用jitter采样的方式,更好地压制了假频信息。理论和实际地震数据验证了该方法的有效性。     

ArcGIS中几种空间内插方法的比较

ArcGIS软件的GA(Geostatistical Analyst)分析扩展模块是一个功能强大、简单易用的数据分析与表面建模工具,应用领域广泛。这里首先简单地介绍了该模块中四类主要的空间内插方法,随后结合该模块的实际界面介绍了该方法的使用,并通过示例比较了几种空间内插方法的优缺点。GA分析扩展模块能用少量的采样数据点的测量值生成连续表面,预测表面的生成、预测表面的准确度和预测表面的误差估计对建模以及分析有重大的意义。     

东北三省月降水量的时空克里金插值研究

为了实现对时空场任意点进行时空插值,以东北三省1999~2008年的月降水数据为研究对象,选用了一类比较实用的积和式时空变异函数模型进行时空克里金插值。每个站点的降水量都是时间序列,插值之前对各站点的降水进行了时序分解和去季节项处理。在分别得到纯空间和纯时间的变异函数基础上建立时空变异函数。将二维的普通克里金插值扩展为三维,并对1999年10月各站点进行降水量估计,同时与单纯空间克里金插值效果进行比较。结果表明时空插值效果理想,因为同时考虑了空间、时间的相关性,插值精度较空间克里金更高。     

基于空间网格插值的重油油藏资源量计算方法——以扎格罗斯白垩系重油层资源评价为例

重油具有黏度大、大规模连续聚集、储层非均质性强、难于开采等特征,利用传统的体积法评价其资源潜力无法揭示重油资源在空间三维尺度上的富集差异性或预测"甜点区"的分布。空间网格插值法基于地理信息系统技术,以成藏组合为基本评价单元,划定各单元分布面积,统计和分析每个成藏组合储层的厚度、含油饱和度、孔隙度、采收率4个关键参数的空间分布规律,并根据实际情况选择反距离加权插值、克里金插值等方法进行插值,实现关键参数网格图的空间叠加计算,最终计算出每个成藏组合的重油资源丰度图,定量表征重油资源的空间分布,厘定重油"甜点区"。将该方法应用于中东扎格罗斯褶皱带重油资源潜力评价,揭示三叠系-侏罗系Butmah成藏组合、侏罗系成藏组合、白垩系成藏组合以及第三系成藏组合总可采资源潜力为140亿t。其中,未来有利勘探目标为白垩系成藏组合,可采资源量为85亿t。初步预测"甜点区"位于盆地西南部。     

A Fractal Interpolatory Approach to Geochemical Exploration Data Processing

Traditional interpolation procedures used for processing geochemical data treat the data as a continuous smooth surface. In this paper, we proposed a fractal spatial interpolatory procedure based on the concept of the fractional delta variance. The method is suitable for processing the geochemical data that are measured at irregularly spaced discrete points, without resort to gridding procedures. The value at each interpolatory point is a function of the fractal disturbance that is related to the fractal dimension determined from the original data set, thus enhancing reconstruction of the natural spatial distribution of element concentration. The proposed procedure has been applied to the copper geochemical data measured from irregularly-spaced floodplain sediment samples in China. We compared geochemical maps created using different interpolatory procedures, including the new fractal method, the Kriging, and the weighted average, against the actual spatial distribution of copper mineral deposits in China, and found that the new fractal method detected more of known Cu-deposits from the test area than the other two methods.     

Dual Kriging with Local Neighborhoods: Application to the Representation of Surfaces

Ordinary kriging, in its common formulation, is a discrete estimator in that it requires the solution of a kriging system for each point in space in which an estimate is sought. The dual formulation of ordinary kriging provides a continuous estimator since, for a given set of data, only a kriging system has to be estimated and the resulting estimate is a function continuously defined in space. The main problem with dual kriging up to now has been that its benefits can only be capitalized if a global neighborhood is used. A formulation is proposed to solve the problem of patching together dual kriging estimates obtained with data from different neighborhoods by means of a blending belt around each neighborhood. This formulation ensures continuity of the variable and, if needed, of its first derivative along neighbor borders. The final result is an analytical formulation of the interpolating surface that can be used to compute gradients, cross-sections, or volumes; or for the quick evaluation of the interpolating surface in numerous locations.