Volume 8, Issue 6, November 2019, Page: 296-305
Kriging and Simulation in Gaussian Random Fields Applied to Soil Property Interpolation
Tofik Mussa Reshid, Department of Statistics, Werabe University, Werabe, Ethiopia
Received: Mar. 28, 2019;       Accepted: Oct. 24, 2019;       Published: Dec. 30, 2019
DOI: 10.11648/j.ajtas.20190806.21      View  132      Downloads  65
Abstract
Spatial modeling is increasingly prominent in many fields of science as statisticians attempt to characterize variability of the processes that are spatially indexed. This paper shows that the Gaussian random field framework is useful for characterizing spatial statistics for soil properties. A sample of soil properties in 94 spatial locations are taken from a field (186.35m×211.44m) wide in northern Ethiopia, Karsa-Malima. We use observations of organic carbon (OC) from the site in our study. Box-Cox transformation is used because of OC follows non-Gaussian distributions. We develop ordinary kriging which is universal kriging with unknown trend models which enables us to predict any point within the field even outside the field up to the “Range” of the model. In this thesis work we predict 100×100 grids (10000 points) using kriging interpolation models. More over in each of these 10000 locations 1000 conditional simulations are made. Interestingly prediction using universal kriging and mean of conditional simulations agree in expectation and kriging variance. For covariance and/or variogram modeling and for parameter estimation we used least square principle and maximum likelihood estimation method. The classical geostatistical approach known as kriging is used as a spatial model for spatial prediction with associated spatial variances. Moreover, conditional simulation is performed. From ordinary kriging model results, predictions are accurate when predictions are close to observation locations. Prediction variance in the observed locations is close to the nugget effect.
Keywords
Gaussian Random Fields, Kriging, Interpolation, Variogram, Range
To cite this article
Tofik Mussa Reshid, Kriging and Simulation in Gaussian Random Fields Applied to Soil Property Interpolation, American Journal of Theoretical and Applied Statistics. Vol. 8, No. 6, 2019, pp. 296-305. doi: 10.11648/j.ajtas.20190806.21
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Omre H.; 1984: Introduction to Geostatistical Theory and Examples of Practical Application; Note, pp. 1-20.
[2]
Wikle C. K, and Royle J. A; 2010: Spatial Statistical Modeling in Biology, Encyclopedia of Life Support System (EOLSS), Eolss Publisher, [http://www.eolss.net] (accessed on 11, 2011), pp. 1-27.
[3]
Santra P, Chopra U. K and Chakraborty D; 2008: Spatial variability of soil properties and its application in predicting surface map of hydraulic parameters in an agricultural farm, Current science, Vol. 95, No. 7, pp. 1-9.
[4]
Kempen M, Heckelei T, Britz W, Leip A, Koeble R, Marchi G; 2005: Computation of a European Spatial Land Use Map the Underlying Statistical Procedures; Agricultural and Resource Economics, Discussion Paper, pp. 1-16.
[5]
Bohling G; 2005: Kriging, C and PE 940, Note, [http://people.ku.edu/∼gbohling/cpe940] (accessed on 3, 2012); pp. 1-20.
[6]
Alan E. G, Peter J. D, Montserrat F. and Petter G.; 2010: Hand Book of Spatial statistics; Modern Statistical Method; Chapman and Hall.
[7]
Chiles; J-P. and Delfiner P; 2012: Geostatistics: Modeling Spatial Uncertainity; 2.
[8]
Matheron G.; 1965: Les Variables Regionalisees of lear Estimation. Une Application de la Theorie des Functiones Alatoires aux science de la Natwe; Masson, Paris.
[9]
Paulo J. Ribeiro and Peter J. Diggle; may 2, 2016: analysis of geostatistical data; version1.7-5.2.
[10]
Kevin P Murphy; 2007: Conjugate Bayesian Analysis of the Gaussian Distribution; Note; pp. 1-29.
[11]
Simpson D, Lindgren F and Rue H; 2011: Markovian Gaussian Model in Spatial Statistics; Statistics, No. 9, pp. 1-17.
[12]
Rue H and Martino S; 2006: Approximate Bayesian Inference for Hierarchical Gaussian Markov Random Field Models, Statistical Planning and Inference, Vol. 137, pp. 1-8.
[13]
Vijay Kumar and Remadevi; 2006: Kriging of Ground Water Levels-A Case Study; Journal of Spatial Hydrology Vol. 6 No. 1.
[14]
Paulo J. Ribeiro and Peter J. Diggle; 2001: geoR: A Package for Geostatistical analysis, Vol.1/2.
[15]
Firas Ajil Jassim, Fawzi Hasan Altaany; 2013: Image Interpolation Using Kriging Technique for Spatial Data; Canadian Journal on Image Processing and Computer Vision Vol. 4 No. 2.
[16]
Andreas Lichtenstern; 2013: Kriging Method in Spatial Statistics; Bachelor’s Thesis.
[17]
Rue H and Follestad T; 2003: Gaussian Markov Random Field Models with Applications; Spatial Statistics; pp. 1-20.
[18]
Gao Gu M and Zhu H; 2000: Maximum Likelihood Estimation for Spatial Models by Markov Chain Monte Carlo Approximation; J. R statist.soc.; Vol. 63, part 2, pp. 339-355.
[19]
Ethan B and Michael L; 2008: Estimating Deformations of Isotropic Gaussian Random Fields on the Plane; The Anals of Statistics; Vol. 36, No. 2,; pp. 1-24.
[20]
Jo Eidsvik; 2011: Spatial statistics, Parameter Estimation and Kriging for Gaussian Random Fields; Note, pp 1-12.
[21]
Michael Sherman.; 2011: Spatial Statistics and Spatio-Temporal Data, Covariance Function and Directional Properties; Jhon Wiley and sons Ltd publication
[22]
Derya Ozturk and Fatmagul Kilic; 2016: Geostatistical Approach for Spatial Interpolation of Meteorological data; Anal of the Brazilian Academy of science.
[23]
Jack P. C. Kleijnen; 2017: Kriging: Method and application.
[24]
Tomislav Malvic et al.; 2019: Kriging with a Small Number of Data Points Supported by Jack-Knifing, a case Study in the Sava Depression (Northern Croatia), Geosciences article.
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