[ArcGIS小白记]Kriging插值

ArcToolBox>>Spatial Analyst Tools>>Interpolation>>Kriging

引用来源:http://bbs.esrichina-bj.cn/ESRI/viewthread.php?tid=55815

地统计与确定性插值的最大区别在于,地统计插值引入了概率模型,即地统计插值认为从一个统计模型不可能完全精确地得出预测值,所以在进行预测时,应该给出预测值的误差,即预测值在一定概率内合理。

通常所说的地统计插值是指克里格插值法(Kriging)
Z
(s)=μ(s)+ε(s)
S表示不同的位置点,可以是用经纬度表示的空间坐标。
Z(s)是该位置点的属性值。μ(s)为确定趋势值,ε(s)
为自相关随机误差。

当要考虑多个协同变量的情况下,可采用协克里格插值法(Co-Kriging).则其计算公式将变为:
Zj(s)=μj(s)+εj(s)
表示的是第j个变量的情况。在协克里格中,只对主变量进行预估,但将在插值预估时引入不同变量间的随机误差项εj(s)的交叉相关性值,从而构建协同克里格模型。

克里格插值是一个最优的无偏估计法。

获得预测图并不要求数据呈正态分布。但当数据呈正态分布时,克里格插值法将是无偏估计法中效果最好的一种方法。

因此,在进行克里格插值前,可先对非正态分布的数据进行转换,包括Log对数转换,Box-Cox转换,使之呈正态分布,然后再进行插值。
根据样点数据统计特征的不同可将克里格分成多种不同的插值法:

当样点数据是二进制值时,用指示克里格插值法进行概率预测;

对样点数据进行了未知函数变换后,可用该变换函数进行析取克里格插值;

当样点数据的趋势值μ(s)是一个未知常量时,用普通克里格;

当样点数据的趋势可用一个多项式进行拟合,但回归系数未知时,用泛克里格插值法;

当样点数据的趋势已知时,用简单克里格插值法;

其中最常用的是普通克里格与泛克里格插值法;当加入了协变量进行插值时,则叫作协同普通克里格插值法和协同泛克里格插值法。

同反距离权重插值法IDW一样,克里格插值法同样可以表示为:
Z(x0)=
∑λi Z(xi)
Z(x0)
为待估点的值, Z(xi) 为待估点周围的已知样点值,λi为第i个已知点的权重

所不同之处在于,IDW的权重为待估点与已知样点间距离的u次幂的倒数,而克里格的权重值不仅考虑待估点与已知样点、已知样点之间的距离,还考虑了其空间分布的方位。通过半变异函数来赋权重值。

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