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.     

一种实用的等值线型数据网格化方法

数据网格化通常包括三大类:测线型数据网格化、等值线型数据网格化和离散点型数据网格化。文中研究的等值线型数据分块存储、网格点八方位搜索插值的网格化方法较好地解决了平面等值线型数字化数据的网格化计算问题,其计算数据量大,实用性强,精度高,计算速度快。     

逐点激发井深设计网格化方法的选取

随着地震勘探的不断深入,地震资料采集要求主频足够高,频带足够宽,能量足够强,而逐点激发井深设计有助于获得满意的地震信号,要设计合理的井深,网格化方法的选取非常重要。这里首先介绍了Surfer软件中十二种网格化方法原理特征,以及在逐点井深设计上的应用;然后运用Surfer软件中不同的插值方法,对BIN12工区岩性数据进行网格的效果对比与网格的密度探讨,选取了最佳的网格化方法;最后针对几种网格化方法进行井深对比试验,通过资料分析,说明了设计的井深合理,选取网格化方法的效果较好。     

Moving Surface Spline Interpolation Based on Green’s Function

Some commonly used interpolation algorithms are analyzed briefly in this paper. Among all of the methods, biharmonic spline interpolation, which is based on Green’s function and proposed by Sandwell, has become the mainstream method for its high precision, simplicity and flexibility. However, the minimum curvature method has two flaws. First, it suffers from undesirable oscillations between data points, which is solved by interpolation with splines in tension. Second, the computation time is approximately proportional to the cube of the number of data constraints, making the method slow for situations with dense data coverage. Focusing on the second problem, this paper introduces the moving surface spline interpolation method based on Green’s function, and the interpolation error equations are deduced. Because the proposed method only chooses the nearest data points by using the merge sort algorithm for interpolating, the computation time is greatly decreased. The optimal number of the nearest points can be determined by using the interpolation error estimation equation. No matter how many data points there are, this method can be implemented without difficulty. Examples show that the proposed method can obtain high interpolation precision and high computation speed at the same time.     

利用MFC和Surfer实现方位加权多重二次曲面插值

受众多因素的影响,我们采集到的地球物理数据是离散的,这也限制了获得更多的地下信息。为了便于后期的解释工作,数据的补插网格化就至关重要。数据网格化是将空间上不均匀分布的数据,按一定方法归算成规则网格节点的代表值。这里依据方位加权二次曲面插值算法,利用MFC对话框界面与Surfer的混合编程,读取实测数据信息后,在可视化界面实现网格化处理过程。最后调用Surfer生成区域等值线图,与其内部插值方法绘制的等值线图进行比较。一方面验证了方位加权二次曲面插值算法的可适性,另一方面通过人机交互的方式简洁快速实现了庞大数据的批处理,便于用户使用。     

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。     

地球物理不规则分布数据的空间网格化法

不规则分布数据的网格化处理是地球物理数据处理和解释的基础问题,是保证许多地球物理数据处理方法得以成功实施的前提.文中介绍了线性插值法、多元二次函数法、普通克里格法、反插值法等4种空间网格化法的原理,并利用这些网格化方法进行了理论模型和实际航磁数据的试验分析.     

High-resolution regional gravity field model of Pakistan based on best residual terrain model (RTM) of topography and interpolation techniques

This study focuses on the development of absolute gravity model for Pakistan based on best possible residual terrain model of gravity using residual terrain modeling technique. The datasets used for the development of model are observed gravity, global gravity models, and Shuttle Radar Topographic Mission (SRTM30) elevation data. The residual terrain modeling technique has been used in the remove-restore procedure for smoothing the observed gravity field. Different topographic elevation models were tested in the model selection and one best possible model with minimum mean and standard deviation was selected for residual terrain effects. Least square collocation technique has been used for quality control and error estimates. The best possible covariance model was established from residual gravity for onward prediction of gravity anomalies at the earth surface for error and prediction analysis. The residual terrain effect of gravity, value of free air anomaly from EGM96, and observed free air anomaly are added to normal gravity to compute the absolute gravity at earth surface. The prediction of these parameters is made by employing Lagrange interpolation with least square adjustment. The results are compared with ～5% randomly selected data points not utilized for the development of covariance function and/or model development. Spline interpolation technique has also been used for the prediction of gravity field-related parameters. Lagrange interpolation exhibits relatively superior results over spline-based interpolation. This is as per expectation due to the reason that additional gridding for spline interpolation filters the signal part as well. This fact is evident from the results of spline interpolation of Grid-I and Grid-II with relatively better prediction results in Grid-I. This version of the model is capable of prediction having limiting error of 30 mGal. The predicted results show that 96.16% of prediction data falls within above-mentioned limit with Lagrange interpolation technique with least square adjustment for whole Pakistan area. The adverse effect of gridding is absent in case of Grid-I due to relatively flat areas and predicted data matches totally with control values for both spline as well as Lagrange interpolations. However, in case of Grid-II which includes high mountains of Himalaya, gridding effect is present and the accuracy of the predicted results falls to ～92%. The computed results have been compared with absolute values predicted using EGM96 and EGM2008 models as well. The gravity field recovered with PAKGM model is much better, i.e., ～ 96.16%, than both with EGM96 and EGM2008 which is about 85% only.     

交叉验证在离散数据网格化时的应用

根据数据的特点,选择不同的算法和参数对离散数据进行网格化,所得网格化数据对原始数据的反映程度不同。因此,在网格化时,可以利用交叉验证(Cross Validation)对不同的网格化方法进行定量的评估和比较,以选择最能尊重原始数据的网格化算法和参数。     

等值线绘图软件SURFER7.O中九种插值法介绍

SURFER软件是一个功能强大的绘制等值线图及三维立体图软件包，能迅速地将离散的测量数据通过插值转换为连续的数据曲面。SURFER7．0提供的内插方法多达九种，其中每一种内插方法都有其意义及相关的参数设置。作者在本文中主要介绍了该软件中的九种内插方法，并进行了应用示例。     

Nested gridding and streamline-based simulation for fast reservoir performance prediction

Detailed reservoir models routinely contain 10⁶–10⁸ grid blocks. These models often cannot be used directly in a reservoir simulation because of the time and memory required for solving the pressure grid on the fine grid. We propose a nested gridding technique that efficiently obtains an approximate solution for the pressure field. The domain is divided into a series of coarse blocks, each containing several fine cells. Effective mobilities are computed for each coarse grid block and the pressure is then found on the coarse scale. The pressure field within each coarse block is computed using flux boundary conditions obtained from the coarse pressure solution. Streamline-based simulation is used to move saturations forward in time. We test the method for a series of example waterflood problems and demonstrate that the method can give accurate estimates of oil production for large 3D models significantly faster than direct simulation using streamlines on the fine grid, making the method overall approximately up to 1,000 times faster than direct conventional simulation.     

SURFER软件图形数据的进一步处理和利用

ＳＵＲＦＥＲ软件可将原始数据网格化，自动生成等值线图形，使用方便，但它的图形编辑功能较弱，无中文标注功能，多图层操作功能方面也不能令人满意，为了弥补这些不足，需对其数据做进一步处理。本文讨论了ＳＵＲＦＥＲ软件数据的进一步处理和利用中的一些问题，作为例子，着重探讨ＳＵＲＦＥＲ软件与其它系统的接口     

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.     

油藏数值模拟中地质模型的建模流程与方法

围绕油藏数值模拟过程中三维地质模型的建模技术进行了分析与探讨,详细介绍了基于角点网格模型的建模方法,给出了相应的实现步骤。其主要流程是:首先根据断层数据构造断层模型,在断层模型的基础上构建骨架模型;然后在骨架模型约束下采用地层恢复技术实现含断层的地层模型;最后基于结构模型插值物性参数完成属性模型。以塔河油田缝洞型油藏为例,对建模流程和技术的可行性进行验证,结果表明,其建模结果与专业地质建模软件Petrel相符。     

An interpolation method taking into account inequality constraints: I. Methodology

Several kinds of data can provide information about a variable measured on a one- or two-dimensional space; at some points, the value is known to be equal to a certain number. At other points, the only information may be that the variable is greater or smaller than a given value. The theory of splines provides interpolating functions that can take into account both equality and inequality data. These interpolating functions are presented. The parallel between splines and kriging is reviewed, using the formalism of dual kriging. Coefficients of dual kriging can be obtained directly by minimizing a quadratic form. By adding some inequality constraints to this minimization, an interpolating function may be calculated which takes into account inequality data and is more general than a spline. The method is illustrated by some simple one-dimensional examples.Work performed at Sohio Petroleum Company     

Velocity interpolation and streamline tracing on irregular geometries

Streamline tracing on irregular grids requires reliable interpolation of velocity fields. We propose a new method for direct streamline tracing on polygon and polytope cells. While some numerical methods provide a basis function that can be used for interpolation, other methods provide only the fluxes at the faces of the elements. We introduce the concept of full- and raw-field methods. Full-field methods have built-in interpolation but are often not defined on general grids such as polygonal and polyhedral grids which we examine here. Also, reliability issues may arise on non-simplicial meshes in terms of not being able to reproduce constant velocity fields. We propose an interpolation in H(div) and H(curl) valid on general grids that is based on barycentric coordinates and that reproduces uniform flow. The interpolation can be used to compute the streamline directly on the complex cell geometry. The method generalizes to convex polytopes in 3D, with a restriction on the polytope topology near corners that is shown to be satisfied by several popular grid types. Numerical results confirm that the method is applicable to general grids and preserves uniform flow.     

MAPGIS二次开发平台下GRD网格法实现渐变色填充剖平图

首先阐述了GRD网格法进行渐变色填充剖平图的基本原理,然后详细介绍了利用MAPGIS二次开发平台提供SaveDemToSurfGrd（）函数,在VisualBasic下实现将测量数据转换成标准的GRD网格数据,并在MAPGIS影像处理模块中,将GRD文件转换成Msi影像文件,实现渐变色填充剖平图的程序实现过程,最后总结出该方法的优缺点。     

Application of FFT-based Algorithms for Large-Scale Universal Kriging Problems

Looking at kriging problems with huge numbers of estimation points and measurements, computational power and storage capacities often pose heavy limitations to the maximum manageable problem size. In the past, a list of FFT-based algorithms for matrix operations have been developed. They allow extremely fast convolution, superposition and inversion of covariance matrices under certain conditions. If adequately used in kriging problems, these algorithms lead to drastic speedup and reductions in storage requirements without changing the kriging estimator. However, they require second-order stationary covariance functions, estimation on regular grids, and the measurements must also form a regular grid. In this study, we show how to alleviate these rather heavy and many times unrealistic restrictions. Stationarity can be generalized to intrinsicity and beyond, if decomposing kriging problems into the sum of a stationary problem and a formally decoupled regression task. We use universal kriging, because it covers arbitrary forms of unknown drift and all cases of generalized covariance functions. Even more general, we use an extension to uncertain rather than unknown drift coefficients. The sampling locations may now be irregular, but must form a subset of the estimation grid. Finally, we present asymptotically exact but fast approximations to the estimation variance and point out application to conditional simulation, cokriging and sequential kriging. The drastic gain in computational and storage efficiency is demonstrated in test cases. Especially high-resolution and data-rich fields such as rainfall interpolation from radar measurements or seismic or other geophysical inversion can benefit from these improvements.     

Neighborhood discontinuities in bivariate interpolation of scattered observations

Because of the need for computational efficiency, bivariate interpolation methods applied to scattered observations often involve two stages. Initially the variable is estimated at regular grid nodes using a running subset of data (usually of fixed number). This, however, will produce discontinuities in the interpolated surface. Thus a second stage, curvilinear interpolation technique, is applied to estimated values to smooth out the effect of discontinuities. Such problems can be overcome efficiently in processing large data sets by interpolating over natural neighbor subsets. Interpolation procedures that generate discontinuities in the interpolated surface are inappropriate for geological applications, where dislocations due to structural complications may be present.     

The Second-Order Stationary Universal Kriging Model Revisited

Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper.