地质数据处理插值方法.docx

1、地质数据处理插值方法二维数据场的插值方法1二维数据场描述及处理目的数据场数据(xi,yi,zi), i=1,n, 即某特征在二维空间中的n个预测值列表：x坐标y坐标观测数据x坐标y坐标观测数据164648.4784648127164658.9784658130164658.9784648.5128164649.4784658.5127164649.4784649127164649.9584659126164649.9584649.5126164650.4584659.5126164650.4584650126164650.9584660125164650.9584650.5125164651.

2、4584660.5125164651.45846511251646528466112416465284651.5124164652.4884661.5124164652.4884652124164652.9784650.5124164652.4884657.5129164653.4784651123164652.9784652.5124164653.9784651.5126164653.4784653125164654.4784652127164653.9784653.5126164654.9584652.5128164654.4784654129164658.9784658130164654

3、.9584654.5128164649.4784658.5127164655.4584655126164656.9884656.5127164655.9584655.5127164657.4884657126164656.4484656129164648.4784657.5127164654.368884653128处理目的了解该数据场的空间分布情况处理思路网格化绘制等值线图网格化方法：二维数据插值2空间内插方法Surfer8.0中常用的插值方法Gridding Methods Inverse Distance to a Power(距离倒数加权) Kriging（克立格法） Minimum

4、Curvature（最小曲率法） Modified Shepards Method（改进Shepard方法） Natural Neighbor（近邻法） Nearest Neighbor（最近邻法） Polynomial Regression（多项式回归法） Radial Basis Function（径向基函数法） Triangulation with Linear Interpolation（线性插值三角形法） Moving Average（移动平均法） Data Metrics（数据度量方法） Local Polynomial（局部多项式法）Geostatistics Analyst M

5、odel in ArcGIS 92.1反距离加权插值反距离加权插值（Inverse Distance Weighting，简称IDW），反距离加权法是最常用的空间内插方法之一。它的基本原理是：空间上离得越近的物体其性质越相似，反之亦然。这种方法并没有考虑到区域化变量的空间变异性，所以仅仅是一种纯几何加权法。反距离加权插值的一般公式为：其中，为未知点处的预测值，为已知点处的值，n为样点的数量，为样点的权重值，其计算公式为：式中为未知点与各已知点之间的距离，p是距离的幂。样点在预测过程中受参数p的影响，幂越高, 内插的平滑效果越佳。尽管反距离权重插值法很简单，易于实现，但它不能对内插的结果作精度评

6、价，所得结果可能会出现很大的偏差，人为难以控制。2.2全局多项式插值（趋势分析法）根据有限的样本数据拟合一个表面来进行内插，称之为全局多项式内插方法。一般多采用多项式来进行拟合，求各样本点到该多项式的垂直距离的和，通过最小二乘法来获得多项式的系数，这样所得的表面可使各样本点到表面之间距离的平方和最小。如果表面平滑、无弯曲，使用一次多项式拟合；有一处弯曲的表面则用二次多项式进行拟合；若有两处弯曲则需使用三次多项式，依次类推。全局多项式内插一般适用于表面变化平缓的研究区域，或者仅研究区域内全局性趋势的情况3。2.3局部多项式内插局部多项式内插与全局多项式内插相对应，是用多个多项式拟合表面的一种方法

7、，它更多地用来表现研究区域西部的变异情况。其基本原理与全局多项式内插相同。The Local Polynomial gridding method assigns values to grid nodes by using a weighted least squares fit with data within the grid nodes search ellipse.2.4径向基函数方法径向基函数法属于人工神经网络方法，该方法所拟合的表面都必须经过所有样本数据。径向基函数以某个已知点为中心按一定距离变化的函数，因此在每个数据点都会形成径向基函数，即每个基函数的中心落在某一个数据点上。径向

8、基函数适合于非常平滑的表面，要求样本数据量大，如果数据点少，则内插效果不佳3。同时，径向基函数难以对误差进行估计，也是其缺点之一。常用的径向基函数法，它们分别是：薄盘样条函数（thin-plate spline）：张力样条函数（spline with tension）：规则样条函数（completely regularized spline）：高次曲面样条函数（multiquadric spline）：反高次曲面样条函数（inverse multiquadric spline）：各式子中h为表示由点(x，y)到第i个数据点的距离，R参数是用户指定的平滑因子，为修正贝塞尔函数， 为指数积分函数，

9、为 Euler常数，其值约为0.577215。Radial Basis Function interpolation is a diverse group of data interpolation methods. In terms of the ability to fit your data and to produce a smooth surface, the Multiquadric method is considered by many to be the best. All of the Radial Basis Function methods are exact inte

10、rpolators, so they attempt to honor your data. You can introduce a smoothing factor to all the methods in an attempt to produce a smoother surface. Function TypesThe basis kernel functions are analogous to variograms in Kriging. The basis kernel functions define the optimal set of weights to apply t

11、o the data points when interpolating a grid node. The available basis kernel functions are listed in the Type drop-down list in the Radial Basis Function Options dialog.Inverse Multiquadric Multilog Multiquadratic Natural Cubic Spline Thin Plate Spline where:his the anisotropically rescaled, relativ

12、e distance from the point to the node R2is the smoothing factor specified by the userDefault R2 ValueThe default value for R2 in the Radial Basis Function gridding algorithm is calculated as follows:(length of diagonal of the data extent)2 / (25 * number of data points)Specifying Radial Basis Functi

13、on Advanced Options1. Click on Grid | Data. 2. In the Open dialog, select a data file and then click the Open button.3. In the Grid Data dialog, choose Radial Basis Function in the Gridding Method group. 4. Click the Advanced Options button to display the Radial Basis Advanced Options dialog.5. In t

14、he General page, you can specify the function parameters for the gridding operation. The Basis Function list specifies the basis kernel function to use during gridding. This defines the optimal weights applied to the data points during the interpolation. The Basis Function is analogous to the variog

15、ram in Kriging. Experience indicates that the Multiquadric basis function works quite well in most cases. Successful use of the Thin Plate Spline basis function is also reported regularly in the technical literature. The R2 Parameter is a shaping or smoothing factor. The larger the R2 Parameter shap

16、ing factor, the rounder the mountain tops and the smoother the contour lines. There is no universally accepted method for computing an optimal value for this factor. A reasonable trial value for R2 Parameter is between the average sample spacing and one-half the average sample spacing. Triangulation

17、 with Linear InterpolationThe Triangulation with Linear Interpolation method in Surfer uses the optimal Delaunay triangulation. The algorithm creates triangles by drawing lines between data points. The original points are connected in such a way that no triangle edges are intersected by other triang

18、les. The result is a patchwork of triangular faces over the extent of the grid. This method is an exact interpolator.Each triangle defines a plane over the grid nodes lying within the triangle, with the tilt and elevation of the triangle determined by the three original data points defining the tria

19、ngle. All grid nodes within a given triangle are defined by the triangular surface. Because the original data are used to define the triangles, the data are honored very closely.Triangulation with Linear Interpolation works best when your data are evenly distributed over the grid area. Data sets tha

20、t contain sparse areas result in distinct triangular facets on the map.2.5最小曲率法Minimum Curvature is widely used in the earth sciences. The interpolated surface generated by Minimum Curvature is analogous to a thin, linearly elastic plate passing through each of the data values with a minimum amount

21、of bending.The Minimum Curvature gridding algorithm is solves the specified partial differential equation using a successive over-relaxation algorithm. The interior is updated using a chessboard strategy, as discussed in Press, et al. (1988, p. 868). The only difference is that the biharmonic equati

22、on must have nine different colors, rather than just black and white.Minimum Curvature generates the smoothest possible surface while attempting to honor your data as closely as possible. Minimum Curvature is not an exact interpolator, however. This means that your data are not always honored exactl

23、y. Minimum Curvature produces a grid by repeatedly applying an equation over the grid in an attempt to smooth the grid. Each pass over the grid is counted as one iteration. The grid node values are recalculated until successive changes in the values are less than the Maximum Residuals value, or the

24、maximum number of iterations is reached (Maximum Iteration field). The Maximum Residual parameter has the same units as the data, and an appropriate value is approximately 10% of the data precision. If data values are measured to the nearest 1.0 units, the Maximum Residual value should be set at 0.1

25、. The iterations continue until the maximum grid node correction for the entire iteration is less than the Maximum Residual value. The default Maximum Residual value is given by:Default Max Residual = 0.001 (Zmax - Z min)The Maximum Iteration parameter should be set at one to two times the number of

26、 grid nodes generated in the grid file. For example, when generating a 50 by 50 grid using Minimum Curvature, the Maximum Iteration value should be set between 2,500 and 5,000. The Internal Tension and Boundary Tension ,Qualitatively, the Minimum Curvature gridding algorithm is attempting to fit a p

27、iece of sheet metal through all of the observations without putting any creases or kinks in the surface. Between the fixed observation points, the sheet bows a bit. The Internal Tension is used to control the amount of this bowing on the interior: the higher the tension, the less the bowing. For exa

28、mple, a high tension makes areas between observations look like facets of a gemstone. The Boundary Tension controls the amount of bowing on the edges. The range of values for Internal Tension and Boundary Tension are 0 to 1. By default, the Internal Tension and the Boundary Tension are set to 0. the

29、 Relaxation Factor,The Relaxation Factor is as described in Press et al. (1988). In general, the Relaxation Factor should not be altered. The default value (1.0) is a good generic value. Roughly, the higher the Relaxation Factor (closer to two) the faster the Minimum Curvature algorithm converges, b

30、ut the more likely it will not converge at all. The lower the Relaxation Factor (closer to zero) the more likely the Minimum Curvature algorithm will converge, but the algorithm is slower. The optimal Relaxation Factor is derived through trial and error.2.6近邻法The Natural Neighbor gridding method is

31、quite popular in some fields. What is Natural Neighbor interpolation? Consider a set of Thiessen polygons (the dual of a Delaunay triangulation). If a new point (target) were added to the data set, these Thiessen polygons would be modified. In fact, some of the polygons would shrink in size, while none would increase in size. The area associated with the targets Thiessen polygon that was taken from an existing polygon is called the borrowed ar