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1、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164648.4784648127164658.9784658130164658.9784648.5128164649.4784658.5127164649.4784649127164649.9584659126164649.9584649.5126164650.4584659.5126164650.4584650126164650.9584660125164650.9584650.5125164651.4584660.51
2、25164651.45846511251646528466112416465284651.5124164652.4884661.5124164652.4884652124164652.9784650.5124164652.4884657.5129164653.4784651123164652.9784652.5124164653.9784651.5126164653.4784653125164654.4784652127164653.9784653.5126164654.9584652.5128164654.4784654129164658.9784658130164654.9584654.5
3、128164649.4784658.5127164655.4584655126164656.9884656.5127164655.9584655.5127164657.4884657126164656.4484656129164648.4784657.5127164654.368884653128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 MethodsInverse Distance to a Power（距离倒数加权）Kriging （克立格法）Minimum Curvature （
4、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 Model in Ar
5、cGIS 9反距离加权插值 全局多项式内插确定性内插方法 全局多项式内插 局部多项式内插 内插方法J径向基函数方法f克立格内插方法地统计内插方法廿闩古工故击土千 协同克立格内插2.1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：Z (x, y) = 气Z (气.，y)i=1其中，Z(x )为未知点x处的预测值，Z(x)为已知点x处的值，n为
6、样点的数量， 00ii力为样点的权重值，其计算公式为：人=d/ Edi=1式中di0为未知点与各已知点之间的距离，p是距离的幕。样点在预测过程中受参 数p的影响，幕越高，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。Z 3, y) = f 3, y)
7、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，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
8、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)： B(h) = (h2 xR2)ln(h2 xR2)R h张力样条函数
9、(spline with tension):B(h) = ln( ? ) + K (R - h)2 + C规则样条函数(completely regularized spline):B(h) = (-1)n -r2n = ln兽)2 + E (昱)2 + Cn!n212 en =1高次曲面样条函数(multiquadric spline): B(h) = 土2 xR2反高次曲面样条函数(inverse multiquadric spline)： B(h)=；】-. h2 x R2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，K0为修正贝塞尔函数，与为指数积分
10、函数，Ce为Euler常数，其值 约为 0.577215。Radial Basis Functioninterpolation is a diverse group of data interpolation methods. In terms of the ability to fit your data and to produce a smooth surface, theMultiquadric method is considered by many to be the best. All of the Radial Basis Functionmethods are exact i
11、nterpolators, 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 inKriging. The basis kernel functions define the optimal set of weights to apply
12、to 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 MultiquadricMultilogMultiquadraticNatural Cubic SplineThin Plate Spline where: h is the anisotropically rescaled, relative
13、 distance from the point to the nodeR2 is the smoothing factor specified by the userDefault R2 ValueThe default value for R in the Radial Basis Function gridding algorithm is calculated as follows:(length of diagonal of the data extent2 / (25 * number of data points)Specifying Radial Basis Function
14、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 Functionin the Gridding Method group.4. Click the AdvancedOptionsbutton to display the Radial Basis Advanced Options dialog.5. In the Gener
15、al page, you can specify the function parameters for the gridding operation. The Basis Functionlist 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 variogram in Kr
16、iging Experience indicates that the Multiquadricbasis 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 Parameteris a shaping or smoothing factor. The larger the R2 Parametershaping factor, t
17、he 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 with Linear I
18、nterpolationThe Triangulation with Linear Interpolationethod 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 triangles. The result
19、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 triangle. All grid n
20、odes 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 Interpolatioworks best when your data are evenly distributed over the grid area. Data sets that contain sparse a
21、reas 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 of bending.The Min
22、imum 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 equation must have nine
23、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 exactly.Minimum Curvatur
24、e 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 maximum number of i
25、terations 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. The iterations con
26、tinue 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 grid nodes generate
27、d 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 piece of sheet metal t
28、hrough 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 example, a high tension
29、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 Relaxation Factor,The
30、 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, but the more likely it
31、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 quite popular in some
32、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 s
33、ize. The area associated with the targets Thiessen polygon that was taken from an existing polygon is called the borrowed area.The Natural Neighbor interpolation algorithm uses a weighted average of the neighboring observations, where the weights are proportional to the borrowed area.2.7最近邻法The Near
34、est Neighbor gridding method assigns the value of the nearest point to each grid node. This method is useful when data are already evenly spaced, but need to be converted to a Surfer grid file. Alternatively, in cases where the data are nearly on a grid with only a few missing values, this method is
35、 effective for filling in the holes in the data.2.8多项式回归方法You can select the type of polynomial regression to apply to your data from the Surface Definition group. As you select the different types of polynomials, a generic polynomial form of the equation is presented in the dialog, and the values i
36、n the Parameters group change to reflect the selection. The available choices are:Simple planar surfaceBi-linear saddleQuadratic surfaceCubic surfaceUserdefined polynomialThe Parameters group allows you to specify the maximum powers for the X andY component in the polynomial equation. As you change
37、the Parameters values, the options are changed in the Surface Definition group to reflect the defined parameters.The Max X Order specifies the maximum power for the X component in the polynomial equation.The Max Y Order specifies the maximum power for the Y component in the polynomial equation.The M
38、ax Total Order specifies the maximum sum of the Max X Order and MaxY Order powers. All of the combinations of the X and Y components are included in the polynomial equation as long as the sum of the two powers does not exceed the Max Total Order value.Data MetricsThe collection of data metrics gridd
39、ing methods creates grids of information about the data on a node-by-node basis. The data metrics gridding methods are not, in general, weighted average interpolators of the Z-values. For example, you can obtain information such as:the number of data points used to interpolate each grid node. If the
40、 number of data points used are fairly equal at each grid node, then the quality of the grid at each grid node can be interpreted.the standard deviation, variance, coefficient of variation, and median absolute deviation of the data at each grid node. These are measures of the variability in space of
41、 the grid, and are important information for statistical analysis.the distance to the nearest data point. For example, if the XY values of a data set are sampling locations, use the Distance to Neardta metric to determine locations for new sampling locations. A contour map of the distance to the nea
42、rest data point, quantifies where higher sampling density may be desired.Data metrics use the local data set including breaklines, for a specific grid node for the selected data metric. The local data set is defined by the search parameters. These search parameters are applied to each grid node to d
43、etermine the local data set. In the following descriptions, when computing the value of a grid node (r,c), the local data set S(r,c) consists of data within the specified search parameters centered at the specific grid node only. The set of selected data at the current grid node (r,c), can be repres
44、ented by S(r,c), where S(r,c) = (xi, yi, zi),i = 1,., nwhere n = number of data points in the local data setThe Z(r,c) location refers to a specific node within the grid.There are five groups of data metrics, Z Order Statistics, Z Moment Statistics, Other Z Statistics, Data Location Statistics, and
45、Terrain Statistics.Moving AverageThe Moving Average gridding method assigns values to grid nodes by averaging the data within the grid nodes search ellipse.Search EllipseTo use Moving Average define a search ellipse and specify the minimum number of data to use. For each grid node, the neighboring d
46、ata are identified by centering the search ellipse on the node. The output grid node value is set equal to the arithmetic average of the identified neighboring data. If there are fewer than the specified minimum number of data within the neighborhood, the grid node is blanked.2.9多项式回归方法2.6克立格内插方法2.5
47、.1基本理论克立格(Kriging)内插方法最早始于采矿领域，它将传统矿产储量计算方法和统计学 结合起来，对空间域或时空域上的区域化变量(Regionalized Variable)进行研究，其研究的 主体内容是对数据的结构分析和插值表面的建立及质量评价。2.5.1.1基本假设条件介绍地质统计学中结构分析研究工具一一变异函数之前，先来看看地质统计学中提出 的两个基本假设条件1 (1)二阶平稳假设(Second-Order Stationary)，它必须满足两个条 件，一是研究区域内区域化变量的数学期望存在，且等于常数，二是区域化变量的协方差存 在且相同，即与变量位置无关，仅依赖于变量间的距离，
48、即：(EZ(x) = m EZ(x)-mZ(x + h)-m = C(h)其中，Z(x)为区域化变量，h为变量间的距离，m为常数；(2)内蕴假设(Intrinsic Hypothesis)， 当区域化变量Z(x)的增量Z(x) - Z(x + h)满足下列两个条件:一是在整个研究区域内对任意x和h都有EZ(x) - Z(x + h) = 0，二是增量Z(x) - Z(x + h)的方差函数存在且平稳 (不依赖于x)，即：VarZ(x) - Z(x + h)=EZ(x) - Z(x + h)2 - EZ(x) - Z(x + h)2 = EZ(x) - Z(x + h)2则称Z(x)满足内蕴假设。在地质统计学中，普通克
