地理统计

地理统计（英语：geostatistics，或译作地统计学、地学统计、地质统计学等）是统计学中关注空间或时空数据集的一个分支，最初是从采矿作业中预测矿石品位的概率分布而发展出来的^[1]，目前已应用于石油地质学、水文地质学、水文学、气象学、海洋学、地球化学、地质冶金学（英语：Geometallurgy）、地理学、林业、环境控制、景观生态学、土壤学，以及农业（尤其是精准农业）等多个学科。地理统计应用于地理学的各个分支，特别是涉及疾病传播（流行病学）、商业和军事规划（物流）的实践，还应用于建设高效的空间网络（英语：Spatial network）。地理统计相关算法已融入地理信息系统（GIS）等许多应用场景。

背景

地理统计与插值方法密切相关，但远不止简单的插值问题。地理统计技术依赖基于随机函数（或随机变量）理论的统计模型来模拟与空间估计和模拟相关的不确定性。

许多更简单的插值方法/算法，例如反距离加权、双线性插值和最近邻插值，在地统计学问世前就已经普及。^[2]但地统计学超越了插值问题，将位于未知位置的要研究的现象视作一组相关的随机变量。

令 $Z (x)$ 为特定位置 $x$ 处的感兴趣变量的值。这个值是未知的（例如温度、降雨量、测压水位、地质相等）。尽管可以前往位置 $x$ 测量该数值，但地统计学认为该值在尚未测量时是随机的。然而， $Z (x)$ 又不完全随机，可以用累积分布函数（CDF）定义，而该函数依赖于关于 $Z (x)$ 值的某些已知信息（information）：

F({\mathit {z}},\mathbf {x} )=\operatorname {Prob} \lbrace Z(\mathbf {x} )\leqslant {\mathit {z}}\mid {\text{information}}\rbrace .

通常，如果靠近 $x$ 的某些位置（或位于 $x$ 的邻域中）的 $Z$ 的值已知，则可以通过该邻域来约束 $Z (x)$ 的累积分布函数：如果假设空间是高度连续的（空间自相关），则 $Z (x)$ 必与附近的值相似。相反，若空间连续性很弱，则 $Z (x)$ 可以取任何值。随机变量的空间连续性可以用空间连续性模型来描述；它可以是基于变差函数的地统计学中的参数形式的模型，也可以是非参数形式的，如多点模拟^[3]或伪遗传方法。

研究者可将单个空间模型应用在整个定义域上，借此假设 $Z$ 是一个平稳过程。它表示相同的统计属性适用于整个定义域。许多种地理统计方法提供了将这些平稳性假设的条件放宽的方法。

该框架中，可以区分两个建模目标：

估计 $Z (x)$ 的值，通常使用累积分布函数 $f (z, x)$ 的期望、中位数或众数。其通常表现为估计问题。
考虑每个位置上的每种可能结果，从整个概率密度函数 $f (z, x)$ 中采样。其方法通常是建立几个替代性的 $Z$ ，称为实现（realization）。考虑在 $N$ 维网格节点（或像素）中离散化的域。每个实现都是完整 $N$ 维联合分布函数的样本

F(\mathbf {z} ,\mathbf {x} )=\operatorname {Prob} \lbrace Z(\mathbf {x} _{1})\leqslant z_{1},Z(\mathbf {x} _{2})\leqslant z_{2},...,Z(\mathbf {x} _{N})\leqslant z_{N}\rbrace .

该方法承认插值问题存在多种解法。每个实现都被视作真实变量可能取值的情形。然后，所有与之相关的工作流都在考虑实现的集成，从而考虑允许概率预测的预测集成。因此，地统计学常用于在求解逆问题时生成或更新空间模型。^[4]^[5]

地理统计估计和多重实现方法都存在许多方法。一些参考书提供了该学科的全面概述。^[6]^[2]^[7]^[8]^[9]^[10]^[11]^[12]^[13]^[14]^[15]

方法

估计

克里金法

克里金法（Kriging）是一类地统计技术，用于在缺少观测值的位置，根据在附近位置的观察值插入随机场的值（例如高程z）。

贝叶斯估计

贝叶斯推断是一种统计推断方法，它使用贝叶斯定理在获得更多证据或信息时更新概率模型。贝叶斯推断在地统计学中日益重要。^[16]贝叶斯估计通过空间过程实现克里金法，最常见的是高斯过程，并使用贝叶斯定理更新该过程以计算其后验概率。另有高维贝叶斯地统计学。^[17]

有限差分法

考虑到概率守恒原理，循环差分方程（有限差分方程）可与格网相结合，计算概率，对地质构造的不确定性进行量化。此过程是马尔可夫链和贝叶斯模型的数值替代方法。^[18]

模拟

聚合
分解
Turning bands
科列斯基分解
截断高斯
Plurigaussian
Annealing
光谱模拟
序列指标
序列高斯
Dead Leave
转移概率
马尔可夫链地理统计
马尔可夫网格模型
支持向量机
布尔模拟
遗传模型
伪遗传模型
元胞自动机
多点地统计学

定义和工具

区域化变量理论（英语：Regionalized variable theory）
协方差函数
半方差
变差函数（英语：Variogram）
克里金法（英语：Kriging）
基台值
变程
金块效应（英语：Nugget effect）

参见

参考文献

^ Krige, Danie G. (1951). "A statistical approach to some basic mine valuation problems on the Witwatersrand". J. of the Chem., Metal. and Mining Soc. of South Africa 52 (6): 119–139
^ ^2.0 ^2.1 Isaaks, E. H. and Srivastava, R. M. (1989), An Introduction to Applied Geostatistics, Oxford University Press, New York, USA.
^ Mariethoz, Gregoire, Caers, Jef (2014). Multiple-point geostatistics: modeling with training images. Wiley-Blackwell, Chichester, UK, 364 p.
^ Hansen, T.M., Journel, A.G., Tarantola, A. and Mosegaard, K. (2006). "Linear inverse Gaussian theory and geostatistics", Geophysics 71
^ Kitanidis, P.K. and Vomvoris, E.G. (1983). "A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one-dimensional simulations", Water Resources Research 19(3):677-690
^ Remy, N., et al. (2009), Applied Geostatistics with SGeMS: A User's Guide, 284 pp., Cambridge University Press, Cambridge.
^ Deutsch, C.V., Journel, A.G, (1997). GSLIB: Geostatistical Software Library and User's Guide (Applied Geostatistics Series), Second Edition, Oxford University Press, 369 pp., http://www.gslib.com/ （页面存档备份，存于互联网档案馆）
^ Chilès, J.-P., and P. Delfiner (1999), Geostatistics - Modeling Spatial Uncertainty, John Wiley & Sons, Inc., New York, USA.
^ Lantuéjoul, C. (2002), Geostatistical simulation: Models and algorithms, 232 pp., Springer, Berlin.
^ Journel, A. G. and Huijbregts, C.J. (1978) Mining Geostatistics, Academic Press. ISBN 0-12-391050-1
^ Kitanidis, P.K. (1997) Introduction to Geostatistics: Applications in Hydrogeology, Cambridge University Press.
^ Wackernagel, H. (2003). Multivariate geostatistics, Third edition, Springer-Verlag, Berlin, 387 pp.
^ Pyrcz, M. J. and Deutsch, C.V., (2014). Geostatistical Reservoir Modeling, 2nd Edition, Oxford University Press, 448 pp.
^ Tahmasebi, P., Hezarkhani, A., Sahimi, M., 2012, Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 16(3):779-79742,
^ Schnetzler, Manu. Statios - WinGslib. [2023-05-14]. （原始内容存档于2015-05-11）.
^ Banerjee S., Carlin B.P., and Gelfand A.E. (2014). Hierarchical Modeling and Analysis for Spatial Data, Second Edition. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. ISBN 9781439819173
^ Banerjee, Sudipto. High-Dimensional Bayesian Geostatistics. Bayesian Anal. 12 (2017), no. 2, 583--614. doi:10.1214/17-BA1056R. https://projecteuclid.org/euclid.ba/1494921642 （页面存档备份，存于互联网档案馆）
^ Cardenas, IC. A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields. Engineering Geology. 2023. doi:10.1016/j.enggeo.2023.107001  .

Armstrong, M and Champigny, N, 1988, A Study on Kriging Small Blocks, CIM Bulletin, Vol 82, No 923
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ISO/DIS 11648-1 Statistical aspects of sampling from bulk materials-Part1: General principles
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McGrew, J. Chapman, & Monroe, Charles B., 2000. An introduction to statistical problem solving in geography, second edition, McGraw-Hill, New York.
Merks, J W, 1992, Geostatistics or voodoo science, The Northern Miner, May 18
Merks, J W, Abuse of statistics, CIM Bulletin, January 1993, Vol 86, No 966
Myers, Donald E.; "What Is Geostatistics? （页面存档备份，存于互联网档案馆）
Philip, G M and Watson, D F, 1986, Matheronian Geostatistics; Quo Vadis?, Mathematical Geology, Vol 18, No 1
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外部链接

GeoENVia （页面存档备份，存于互联网档案馆） promotes the use of geostatistical methods in environmental applications, and organizes bi-annual conferences.
[1] （页面存档备份，存于互联网档案馆）, a resource on the internet about geostatistics and spatial statistics
On-Line Library that chronicles Matheron's journey from classical statistics to the new science of geostatistics （页面存档备份，存于互联网档案馆）
[2] （页面存档备份，存于互联网档案馆）
https://web.archive.org/web/20040326205028/http://geostatscam.com/ Is the site of Jan W. Merks, who claims that geostatistics is "voodoo science" and a "scientific fraud"
[3] （页面存档备份，存于互联网档案馆） It is a group for exchanging of ideas and discussion on multiple point geostatistics (MPS).

[1] Krige, Danie G. (1951). "A statistical approach to some basic mine valuation problems on the Witwatersrand". J. of the Chem., Metal. and Mining Soc. of South Africa 52 (6): 119–139

[IandS1989-2] 2.0 ^2.1 Isaaks, E. H. and Srivastava, R. M. (1989), An Introduction to Applied Geostatistics, Oxford University Press, New York, USA.

[3] Mariethoz, Gregoire, Caers, Jef (2014). Multiple-point geostatistics: modeling with training images. Wiley-Blackwell, Chichester, UK, 364 p.

[4] Hansen, T.M., Journel, A.G., Tarantola, A. and Mosegaard, K. (2006). "Linear inverse Gaussian theory and geostatistics", Geophysics 71

[5] Kitanidis, P.K. and Vomvoris, E.G. (1983). "A geostatistical approach to the inverse problem in groundwater modeling (steady state) and one-dimensional simulations", Water Resources Research 19(3):677-690

[6] Remy, N., et al. (2009), Applied Geostatistics with SGeMS: A User's Guide, 284 pp., Cambridge University Press, Cambridge.

[7] Deutsch, C.V., Journel, A.G, (1997). GSLIB: Geostatistical Software Library and User's Guide (Applied Geostatistics Series), Second Edition, Oxford University Press, 369 pp., http://www.gslib.com/ （页面存档备份，存于互联网档案馆）

[8] Chilès, J.-P., and P. Delfiner (1999), Geostatistics - Modeling Spatial Uncertainty, John Wiley & Sons, Inc., New York, USA.

[9] Lantuéjoul, C. (2002), Geostatistical simulation: Models and algorithms, 232 pp., Springer, Berlin.

[10] Journel, A. G. and Huijbregts, C.J. (1978) Mining Geostatistics, Academic Press. ISBN 0-12-391050-1

[11] Kitanidis, P.K. (1997) Introduction to Geostatistics: Applications in Hydrogeology, Cambridge University Press.

[12] Wackernagel, H. (2003). Multivariate geostatistics, Third edition, Springer-Verlag, Berlin, 387 pp.

[13] Pyrcz, M. J. and Deutsch, C.V., (2014). Geostatistical Reservoir Modeling, 2nd Edition, Oxford University Press, 448 pp.

[14] Tahmasebi, P., Hezarkhani, A., Sahimi, M., 2012, Multiple-point geostatistical modeling based on the cross-correlation functions, Computational Geosciences, 16(3):779-79742,

[15] Schnetzler, Manu. Statios - WinGslib. [2023-05-14]. （原始内容存档于2015-05-11）.

[16] Banerjee S., Carlin B.P., and Gelfand A.E. (2014). Hierarchical Modeling and Analysis for Spatial Data, Second Edition. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. ISBN 9781439819173

[17] Banerjee, Sudipto. High-Dimensional Bayesian Geostatistics. Bayesian Anal. 12 (2017), no. 2, 583--614. doi:10.1214/17-BA1056R. https://projecteuclid.org/euclid.ba/1494921642 （页面存档备份，存于互联网档案馆）

[18] Cardenas, IC. A two-dimensional approach to quantify stratigraphic uncertainty from borehole data using non-homogeneous random fields. Engineering Geology. 2023. doi:10.1016/j.enggeo.2023.107001  .

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