DOCSLIB.ORG

Conditional Simulation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conditional Simulation Basics in Geostatistics 3 Geostatistical Monte-Carlo methods: Conditional simulation Hans Wackernagel MINES ParisTech NERSC • April 2013 http://hans.wackernagel.free.fr Basic concepts Geostatistics Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 2 / 34 Concepts Geostatistical model The experimental variogram serves to analyze the spatial structure of a regionalized variable z(x). It is fitted with a variogram model which is the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x). Kriging: Best Linear Unbiased Estimation of point values (or spatial averages) at any location of a region. Conditional simulation: generate an ensemble of realizations of the random function, conditional upon data. Statistics not linearly related to data can be computed from this ensemble. Concepts Geostatistical model The experimental variogram serves to analyze the spatial structure of a regionalized variable z(x). It is fitted with a variogram model which is the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x). Kriging: Best Linear Unbiased Estimation of point values (or spatial averages) at any location of a region. Conditional simulation: generate an ensemble of realizations of the random function, conditional upon data. Statistics not linearly related to data can be computed from this ensemble. Concepts Geostatistical model The experimental variogram serves to analyze the spatial structure of a regionalized variable z(x). It is fitted with a variogram model which is the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x). Kriging: Best Linear Unbiased Estimation of point values (or spatial averages) at any location of a region. Conditional simulation: generate an ensemble of realizations of the random function, conditional upon data. Statistics not linearly related to data can be computed from this ensemble. Limitations of linear geostatistics Adequate for Gaussian random functions: in practice the distribution function is often skew. Probing with two-point statistics (covariance function, variogram): other tools are also available. Need for non-linear estimates, e.g. for estimating probability of exceeding: environmental threshold, cut-off grade in mining. Conditional simulation techniques address all these aspects. Gaussian conditional simulation generates an ensemble of realizations on which non-linear statistics can be readily computed. Random functions: spatial correlation structure Stationary random function: mean, variance and spatial distribution function exist, spatial correlation is described by the covariance function: C(h) = E[(Z(x+h) − m) · (Z(x) − m)] the variogram of a stationary random function is given by the formula: γ(h) = C(0) − C(h) Gaussian conditional simulation Classical approach 1 Simulate realizations of a stationary Gaussian random function with known covariance function C(h). 2 Condition the realizations using simple kriging. Gaussian simulation 1) Unconditional simulation of a Gaussian RF Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 7 / 34 Simulation of a Gaussian random function Turning bands method (TBM) The simulation of realizations of a GRF can be done simply: determine the 1D covariance function of a corresponding 2D or 3D covariance model, generate directions θ1; : : : ; θK simulate realizations of 1D processes Y1;:::; YK along lines in those directions, project a given point on the lines and combine the corresponding simulated values to obtain the simulated value of the 3D process at that point: K 1 X Y(x) = p Yk(< x; θk >) for x 2 D: K k=1 1D covariance function corresponding to 2D or 3D isotropic covariance The following formulas rely on Bochner’s theorem, in 3D: Z 1 d C3D(h) = C1D(t h) dt and C1D(h) = (h C3D(h)) 0 dh in 2D: Z π Z π=2 1 dC2D C2D(h) = C1D(h sin θ) dθ and C1D(h) = 1 + h (h sin θ) dθ π 0 0 dh Example: exponential covariance function The 1D model associated to a 3D exponential covariance is: h h C (h) = 1 − exp − with h; a ≥ 0 1D a a Migration method: compute Poisson points, split intervals into halves set to ±1 (mean interval length is 2a): TBM: exponential covariance Gaussian simulation 2) Conditional simulation of a Gaussian RF Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 12 / 34 Best linear unbiased estimation (BLUE): kriging ? Estimation of a value Z at a location x0 in geographical space is performed using a linear combination of weights wα with data at neighboring locations xα, α = 1;:::; n. Kriging with known mean m (simple kriging): n ? X Z (x0) = m + wα (Z(xα) − m) α=1 Conditional simulation of Gaussian RF ? ? ZCS(x) = Z (x) + (ZS(x) − ZS(x)) | {z } | {z } kriged from data simulated kriging error ( − ) + = Conditional simulation and kriging Comparison with kriging Simulation (left) Samples (right) Simple kriging (left) Conditional simulation (right) Conditional simulation and kriging Conditionally on the data, the mean of conditional simulations is equal to the kriging: ? E ZCS(x) j Z(xα); α= 1;:::; n = Z (x); the variance of the conditional simulations is the kriging variance: ? 2 var(ZCS(x) j Z(xα); α= 1;:::; n) = var(ZS(x) − ZS(x)) = σK(x): Gaussian simulation with non-Gaussian data 1 Fit of a Gaussian anamorphosis function Z(x) = '(Y(x)): 2 Transform the data to −1 Gaussian values: Y(xα) = ' (Z(xα)). 3 Fit the variogram of the Gaussian random function Y(x): 4 Simulate realizations YS(x): 5 Condition YS(x) with Y(xα), thus obtaining YCS(x). 6 Transform the result to the initial scale: ZCS(x) = '(YCS(x)): 7 Compute various statistics on the ensemble of realizations. Case study Simulating Yeu island The island is located off the south-west coast of Bretagne Measurements of elevation in the sea (depths). Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 18 / 34 Kriging the elevation data Negative kriging estimates are set to zero (below sea level). Conditional simulation of elevation 9 realizations of Yeu island Simulation profiles along island Probability that elevation is above sea level Estimation of surface (km²) of Yeu island 2 Real From kriging Sim. min Sim. mean Sim. max Surface 23.3 22.9 15.4 23.2 31.9 From conditional simulation the volume is estimated to be: 0.188 km3 (as compared to the value deduced from kriging results: 0.169 km3) Conclusion Summary Gaussian random function simulations Adequate for simulating Gaussian random functions Anamorphosis to apply them to non-Gaussian data Satisfy the need for non-linear estimates, e.g. for estimating probability of exceeding: environmental threshold cut-off grade in mining Generate an ensemble of realisations on which non-linear statistics are readily computed. However, in a number of applications there is a need for stochastic models beyond the random functions framework... Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 24 / 34 Conclusion Summary Gaussian random function simulations Adequate for simulating Gaussian random functions Anamorphosis to apply them to non-Gaussian data Satisfy the need for non-linear estimates, e.g. for estimating probability of exceeding: environmental threshold cut-off grade in mining Generate an ensemble of realisations on which non-linear statistics are readily computed. However, in a number of applications there is a need for stochastic models beyond the random functions framework... Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 24 / 34 References M Armstrong, A G Galli, H Beucher, G Le Loc’h, D Renard, B Doligez, R Eschard, and F Geffroy. Plurigaussian Simulations in Geosciences. Springer-Verlag, Berlin, 2nd edition, 2011. JP Chilès and P Delfiner. Geostatistics: Modeling Spatial Uncertainty. Wiley, New York, 2nd edition, 2012. C Lantuéjoul. Geostatistical Simulation: Models and Algorithms. Springer-Verlag, Berlin, 2002. G. Matheron. Random Sets and Integral Geometry. John Wiley & Sons, New York, 1975. RGeoS case-study Conditional simulation example with RGeoS 9.1.2 Code from the document Doc2D.pdf on the site: http://rgeos.free.fr Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 26 / 34 Loading the soil pollution data set library(RGeoS) data(Exdemo_2D_pollution.table) DAT=Exdemo_2D_pollution.table data.db = db.create(DAT, flag.grid=FALSE,ndim=2,autoname=F) data.db = db.locate(data.db,"Zn","z",1) plot(data.db,pch=21,bg.in="black",title="Zn Sample locations") # suppress two outliers hist(DAT$Zn,n=20) data.db = db.sel(data.db,Zn<20) hist(DAT$Zn[DAT$Zn<20]) Zn Sample locations Histogram of DAT$Zn ● ● ● ● 50 ● ● ● ● 510 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 40 ● ● ● ● ● 505 ● ● ● ● ● ● ● ● ● 30 ● 500 ● ● ● ● ● ● ● ● ● ● ● ● Frequency ● ● 495 ● 20 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 490 ● ● ● 10 ● ● ● ● ● ● ● ● 485 ● 0 110 115 120 125 130 135 140 5 10 15 20 25 30 DAT$Zn Structural analysis data.vario = vario.calc(data.db,lag=1,nlag=10) plot(data.vario,npairdw=TRUE,npairpt=TRUE) data.4dir.vario = vario.calc(data.db,lag=1,nlag=10,dir=c(0,45,90,135)) plot(data.4dir.vario,title="Directional variograms") data.model = model.auto(data.vario, struct=c("Spherical","Exponential"), title="Modelling omni-directional variogram") Modelling omni−directional variogram 184 3.0 204 ● 3.0 187 ● 205 229 ● ● ● 198 2.5
Load more

Recommended publications
  • Getting Started with Geostatistics
    FedGIS Conference February 24–25, 2016 | Washington, DC Getting Started with Geostatistics Brett Rose Objectives • Using geography as an integrating factor. • Introduce a variety of core tools available in GIS. • Demonstrate the utility of spatial & Geo statistics for a range of applications. • Modeling relationships and driving decisions Outline • Spatial Statistics vs Geostatistics • Geostatistical Workflow • Exploratory Spatial Data Analysis • Interpolation methods • Deterministic vs Geostatistical •Geostatistical Analyst in ArcGIS • Demos • Class Exercise In ArcGIS Geostatistical Analyst • interactive ESDA • interactive modeling including variography • many Kriging models (6 + cokriging) • pre-processing of data - Decluster, detrend, transformation • model diagnostics and comparison Spatial Analyst • rich set of tools to perform cell-based (raster) analysis • kriging (2 models), IDW, nearest neighbor … Spatial Statistics • ESDA GP tools • analyzing the distribution of geographic features • identifying spatial patterns What are spatial statistics in a GIS environment? • Software-based tools, methods, and techniques developed specifically for use with geographic data. • Spatial statistics: – Describe and spatial distributions, spatial patterns, spatial processes model , and spatial relationships. – Incorporate space (area, length, proximity, orientation, and/or spatial relationships) directly into their mathematics. In many ways spatial statistics extenD what the eyes anD minD Do intuitively to assess spatial patterns, trenDs anD relationships.
    [Show full text]
  • An Introduction to Spatial Autocorrelation and Kriging
    An Introduction to Spatial Autocorrelation and Kriging Matt Robinson and Sebastian Dietrich RenR 690 – Spring 2016 Tobler and Spatial Relationships • Tobler’s 1st Law of Geography: “Everything is related to everything else, but near things are more related than distant things.”1 • Simple but powerful concept • Patterns exist across space • Forms basic foundation for concepts related to spatial dependency Waldo R. Tobler (1) Tobler W., (1970) "A computer movie simulating urban growth in the Detroit region". Economic Geography, 46(2): 234-240. Spatial Autocorrelation (SAC) – What is it? • Autocorrelation: A variable is correlated with itself (literally!) • Spatial Autocorrelation: Values of a random variable, at paired points, are more or less similar as a function of the distance between them 2 Closer Points more similar = Positive Autocorrelation Closer Points less similar = Negative Autocorrelation (2) Legendre P. Spatial Autocorrelation: Trouble or New Paradigm? Ecology. 1993 Sep;74(6):1659–1673. What causes Spatial Autocorrelation? (1) Artifact of Experimental Design (sample sites not random) Parent Plant (2) Interaction of variables across space (see below) Univariate case – response variable is correlated with itself Eg. Plant abundance higher (clustered) close to other plants (seeds fall and germinate close to parent). Multivariate case – interactions of response and predictor variables due to inherent properties of the variables Eg. Location of seed germination function of wind and preferred soil conditions Mechanisms underlying patterns will depend on study system!! Why is it important? Presence of SAC can be good or bad (depends on your objectives) Good: If SAC exists, it may allow reliable estimation at nearby, non-sampled sites (interpolation).
    [Show full text]
  • Master of Science in Geospatial Technologies Geostatistics
    Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa MasterMaster ofof ScienceScience inin GeospatialGeospatial TechnologiesTechnologies GeostatisticsGeostatistics PredictionsPredictions withwith GeostatisticsGeostatistics CarlosCarlos AlbertoAlberto FelgueirasFelgueiras [email protected]@isegi.unl.pt 1 Master of Science in Geoespatial Technologies PredictionsPredictions withwith GeostatisticsGeostatistics Contents Stochastic Predictions Introduction Deterministic x Stochastic Methods Regionalized Variables Stationary Hypothesis Kriging Prediction – Simple, Ordinary and with Trend Kriging Prediction - Example Cokriging CrossValidation and Validation Problems with Geostatistic Estimators Advantages on using geostatistics 2 Master of Science in Geoespatial Technologies PredictionsPredictions withwith GeostatisticsGeostatistics o sample locations Deterministic x Stochastic Methods z*(u) = K + estimation locations • Deterministic •The z *(u) value is estimated as a Deterministic Variable. An unique value is associated to its spatial location. • No uncertainties are associated to the z*(u)? estimations + • Stochastic •The z *(u) value is considered a Random Variable that has a probability distribution function associated to its possible values • Uncertainties can be associated to the estimations • Important: Samples are realizations of cdf Random Variables 3 Master of Science in Geoespatial Technologies PredictionsPredictions withwith GeostatisticsGeostatistics • Random Variables and Random
    [Show full text]
  • An Introduction to Model-Based Geostatistics
    An Introduction to Model-based Geostatistics Peter J Diggle School of Health and Medicine, Lancaster University and Department of Biostatistics, Johns Hopkins University September 2009 Outline • What is geostatistics? • What is model-based geostatistics? • Two examples – constructing an elevation surface from sparse data – tropical disease prevalence mapping Example: data surface elevation Z 6 65 70 75 80 85 90 95100 5 4 Y 3 2 1 0 1 2 3 X 4 5 6 Geostatistics • traditionally, a self-contained methodology for spatial prediction, developed at Ecole´ des Mines, Fontainebleau, France • nowadays, that part of spatial statistics which is concerned with data obtained by spatially discrete sampling of a spatially continuous process Kriging: find the linear combination of the data that best predicts the value of the surface at an arbitrary location x Model-based Geostatistics • the application of general principles of statistical modelling and inference to geostatistical problems – formulate a statistical model for the data – fit the model using likelihood-based methods – use the fitted model to make predictions Kriging: minimum mean square error prediction under Gaussian modelling assumptions Gaussian geostatistics (simplest case) Model 2 • Stationary Gaussian process S(x): x ∈ IR E[S(x)] = µ Cov{S(x), S(x′)} = σ2ρ(kx − x′k) 2 • Mutually independent Yi|S(·) ∼ N(S(x),τ ) Point predictor: Sˆ(x) = E[S(x)|Y ] • linear in Y = (Y1, ..., Yn); • interpolates Y if τ 2 = 0 • called simple kriging in classical geostatistics Predictive distribution • choose the
    [Show full text]
  • Efficient Geostatistical Simulation for Spatial Uncertainty Propagation
    Efficient geostatistical simulation for spatial uncertainty propagation Stelios Liodakis Phaedon Kyriakidis Petros Gaganis University of the Aegean Cyprus University of Technology University of the Aegean University Hill 2-8 Saripolou Str., 3036 University Hill, 81100 Mytilene, Greece Lemesos, Cyprus Mytilene, Greece [email protected] [email protected] [email protected] Abstract Spatial uncertainty and error propagation have received significant attention in GIScience over the last two decades. Uncertainty propagation via Monte Carlo simulation using simple random sampling constitutes the method of choice in GIScience and environmental modeling in general. In this spatial context, geostatistical simulation is often employed for generating simulated realizations of attribute values in space, which are then used as model inputs for uncertainty propagation purposes. In the case of complex models with spatially distributed parameters, however, Monte Carlo simulation could become computationally expensive due to the need for repeated model evaluations; e.g., models involving detailed analytical solutions over large (often three dimensional) discretization grids. A novel simulation method is proposed for generating representative spatial attribute realizations from a geostatistical model, in that these realizations span in a better way (being more dissimilar than what is dictated by change) the range of possible values corresponding to that geostatistical model. It is demonstrated via a synthetic case study, that the simulations produced by the proposed method exhibit much smaller sampling variability, hence better reproduce the statistics of the geostatistical model. In terms of uncertainty propagation, such a closer reproduction of target statistics is shown to yield a better reproduction of model output statistics with respect to simple random sampling using the same number of realizations.
    [Show full text]
  • ON the USE of NON-EUCLIDEAN ISOTROPY in GEOSTATISTICS Frank C
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Collection Of Biostatistics Research Archive Johns Hopkins University, Dept. of Biostatistics Working Papers 12-5-2005 ON THE USE OF NON-EUCLIDEAN ISOTROPY IN GEOSTATISTICS Frank C. Curriero Department of Environmental Health Sciences and Biostatistics, Johns Hopkins Bloomberg School of Public Health, [email protected] Suggested Citation Curriero, Frank C., "ON THE USE OF NON-EUCLIDEAN ISOTROPY IN GEOSTATISTICS" (December 2005). Johns Hopkins University, Dept. of Biostatistics Working Papers. Working Paper 94. http://biostats.bepress.com/jhubiostat/paper94 This working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commercially reproduced without the permission of the copyright holder. Copyright © 2011 by the authors On the Use of Non-Euclidean Isotropy in Geostatistics Frank C. Curriero Department of Environmental Health Sciences and Department of Biostatistics The Johns Hopkins University Bloomberg School of Public Health 615 N. Wolfe Street Baltimore, MD 21205 [email protected] Summary. This paper investigates the use of non-Euclidean distances to characterize isotropic spatial dependence for geostatistical related applications. A simple example is provided to demonstrate there are no guarantees that existing covariogram and variogram functions remain valid (i.e. positive de¯nite or conditionally negative de¯nite) when used with a non-Euclidean distance measure. Furthermore, satisfying the conditions of a metric is not su±cient to ensure the distance measure can be used with existing functions. Current literature is not clear on these topics. There are certain distance measures that when used with existing covariogram and variogram functions remain valid, an issue that is explored.
    [Show full text]
  • Geostatistics: Basic Aspects for Its Use to Understand Soil Spatial Variability
    1867-59 College of Soil Physics 22 October - 9 November, 2007 GEOSTATISTICS: BASIC ASPECTS FOR ITS USE TO UNDERSTAND SOIL SPATIAL VARIABILITY Luis Carlos Timm ICTP Regular Associate, University of Pelotas Brazil GEOSTATISTICS: BASIC ASPECTS FOR ITS USE TO UNDERSTAND SOIL SPATIAL VARIABILITY TIMM, L.C.1*; RECKZIEGEL, N.L. 2; AQUINO, L.S. 2; REICHARDT, K.3; TAVARES, V.E.Q. 2 1Regular Associate of ICTP, Rural Engineering Department, Faculty of Agronomy, Federal University of Pelotas, CP 354, 96001-970, Capão do Leão-RS, Brazil.; *[email protected]; [email protected] 2 Rural Engineering Department, Faculty of Agronomy, Federal University of Pelotas, CP 354, 96001-970, Capão do Leão-RS, Brazil.; 3 Soil Physics Laboratory, Center for Nuclear Energy in Agriculture, University of São Paulo, CP 96, 13400-970, Piracicaba-SP, Brazil. 1 Introduction Frequently agricultural fields and their soils are considered homogeneous areas without well defined criteria for homogeneity, in which plots are randomly distributed in order to avoid the eventual irregularity effects. Randomic experiments are planned and when in the data analyses the variance of the parameters of the area shows a relatively small residual component, conclusions are withdrawn about differences among treatments, interactions, etc. If the residual component of variance is relatively large, which is usually indicated by a high coefficient of variation (CV), the results from the experiment are considered inadequate. The high CV could be a cause of soil variability, which was assumed homogeneous before starting the experiment. In Fisher´s classical statistics, which is mainly based on data that follows the Normal Distribution, each sampling point measurement is considered as a random variable Zi, independently from the other Zj.
    [Show full text]
  • A Practical Guide to Geostatistical Mapping of Environmental Variables
    A Practical Guide to Geostatistical Mapping of Environmental Variables Tomislav Hengl EUR 22904 EN - 2007 The mission of the Institute for Environment and Sustainability is to provide scientific-technical support to the European Union’s Policies for the protection and sustainable development of the European and global environment. European Commission Joint Research Centre Institute for Environment and Sustainability Contact information: Address: JRC Ispra, Via E. Fermi 1, I-21020 Ispra (VA), Italy Tel.: +39- 0332-785349 Fax: +39- 0332-786394 http://ies.jrc.ec.europa.eu http://www.jrc.ec.europa.eu Legal Notice Neither the European Commission nor any person acting on behalf of the Commission is respon- sible for the use which might be made of this publication. A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server http://europa.eu JRC 38153 EUR 22904 EN ISBN 978-92-79-06904-8 ISSN 1018-5593 Luxembourg: Office for Official Publications of the European Communities © European Communities, 2007 Non-commercial reproduction and dissemination of the work as a whole freely permitted if this original copyright notice is included. To adapt or translate please contact the author. Printed in Italy European Commission EUR 22904 EN — Joint Research Centre — Institute for the Environment and Sustainability Title: A Practical Guide to Geostatistical Mapping of Environmental Variables Author(s): Tomislav Hengl Luxembourg: Office for Official Publications of the European Communities 2007 — 143 pp. — 17.6 Ö 25.0 cm EUR — Scientific and Technical Research series — ISSN 1018-5593 ISBN: 978-92-79-06904-8 Abstract Geostatistical mapping can be defined as analytical production of maps by using field observations, auxiliary information and a computer program that calculates values at locations of interest.
    [Show full text]
  • Projection Pursuit Multivariate Transform
    Paper 103, CCG Annual Report 14, 2012 (© 2012) Projection Pursuit Multivariate Transform Ryan M. Barnett, John G. Manchuk, and Clayton V. Deutsch Transforming complex multivariate geological data to be multivariate Gaussian is an important and challenging problem in geostatistics. A variety of transforms are available to accomplish this goal, but may struggle with data sets of high dimensional and sample sizes. Projection Pursuit Density Estimation (PPDE) is a well-established non-parametric method for estimating the joint PDF of multivariate data. A central component of the PPDE algorithm involves the transformation of the original data towards a multivariate Gaussian distribution. Rather than use the PPDE for its original intended purpose of density estimation, this convenient data transformation will be utilized to map complex data to a multivariate Gaussian distribution within a geostatistical modeling context. This approach is proposed as the Projection Pursuit Multivariate Transform (PPMT), which shows the potential to be very effective on data sets with a large number of dimensions. The PPMT algorithm is presented along with considerations and case studies. Introduction The multivariate Gaussian distribution is very commonly adopted within geostatistics for describing geologic variables. This is due to the mathematical tractability of the multiGaussian distribution [10], which may be either convenient or necessary for geostatistical modeling. As geologic variables are often non-Gaussian in nature, a wide variety of techniques [1] are available to transform them to a multiGaussian form, with their associated back-transforms to reintroduce the original distributions. The widely applied normal score transformation [5,10] will guarantee that variables are made univariate Gaussian; however, multivariate complexities such as heteroscedastic, non-linear, and constraint features may still exist.
    [Show full text]
  • A COMPARISON of CLASSICAL STATISTICS and GEOSTATISTICS for ESTIMATING SOIL SURFACE TEMPERATURE by PARICHEHR HEMYARI Bachelor Of
    A COMPARISON OF CLASSICAL STATISTICS AND GEOSTATISTICS FOR ESTIMATING SOIL SURFACE TEMPERATURE By PARICHEHR HEMYARI 1, Bachelor of Science Pahlavi University Shiraz, Iran 1977 Master of Science Oklahoma State University Stillwater, Oklahoma 1980 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 1984 A COMPARISON OF CLASSICAL STATISTICS AND GEOSTATISTICS FOR ESTIMATING SOIL SURFACE TEMPERATURE Thesis Approved: ii 1218519 J ACKNOWLEDGMENTS Much appreciation and gratitude is extended to all of those who assisted in this research: To my thesis adviser, Dr. D. L. Nofziger, to whom I am deeply indebted for his guidance, assistance, and friendship. To the other members of my committee: Dr. J. F. Stone, Dr. L. M. Verhalen, and Dr. M. S. Keener for their suggestions and constructive criticisms. To Mr. H. R. Gray, Mr. J. R. Williams, Mr. B. A. Bittle, Mr. K. W. Grace, and Mr. D. A. Stone for their technical assistance in accom­ plishing research goals. To Dr. P. W. Santelmann and the Oklahoma State University Agron­ omy Department for my research assistantship and the necessary research funds. Finally, my deepest and sincerest appreciation to my family in Iran, and also to my family away from home, Mr. and Mrs. H. R. Gray. This acknowledgment would not be complete without a very special thanks to God our Lord for keeping me in faith and giving me strength throughout my life. iii TABLE OF CONTENT Chapter Page I. INTRODUCTION 1 II. LITERATURE REVIEW 3 A. Introduction to Geostatistics 3 B.
    [Show full text]
  • A Geostatistical Simulation of a Mineral Deposit Using Uncertain Experimental Data
    minerals Article A Geostatistical Simulation of a Mineral Deposit using Uncertain Experimental Data João Narciso 1,* , Cristina Paixão Araújo 2, Leonardo Azevedo 1 , Ruben Nunes 1 , João Filipe Costa 2 and Amílcar Soares 1 1 CERENA/DECivil, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1049-001 Lisbon, Portugal; [email protected] (L.A.); [email protected] (R.N.); [email protected] (A.S.) 2 Departamento de Engenharia de Minas, Universidade Federal do Rio Grande do Sul, 91509-900 Porto Alegre, Brazil; [email protected] (C.P.A.); [email protected] (J.F.C.) * Correspondence: [email protected]; Tel.: +351-218-417-425 Received: 19 March 2019; Accepted: 19 April 2019; Published: 23 April 2019 Abstract: In the geostatistical modeling and characterization of natural resources, the traditional approach for determining the spatial distribution of a given deposit using stochastic sequential simulation is to use the existing experimental data (i.e., direct measurements) of the property of interest as if there is no uncertainty involved in the data. However, any measurement is prone to error from different sources, for example from the equipment, the sampling method, or the human factor. It is also common to have distinct measurements for the same property with different levels of resolution and uncertainty. There is a need to assess the uncertainty associated with the experimental data and integrate it during the modeling procedure. This process is not straightforward and is often overlooked. For the reliable modeling and characterization of a given ore deposit, measurement uncertainties should be included as an intrinsic part of the geo-modeling procedure.
    [Show full text]
  • Model-Based Geostatistics
    This is page i Printer: Opaque this Model-based Geostatistics Peter J. Diggle and Paulo J. Ribeiro Jr. May 22, 2006 2006 SPRINGER SCIENCE+BUSINESS MEDIA, LLC. All rights reserved. No part of this work may be reproduced in any form without the written permission of SPRINGER SCIENCE+BUSINESS MEDIA, LLC. ii Peter J. Diggle Department of Mathematics and Statistics Lancaster University, Lancaster, UK LA1 4YF [email protected] Paulo J. Ribeiro Jr Departamento de Estat´ıstica Universidade Federal do Paran´a C.P. 19.081 Curitiba, Paran´a, Brasil 81.531-990 [email protected] This is page iii Printer: Opaque this Preface Geostatistics refers to the sub-branch of spatial statistics in which the data consist of a finite sample of measured values relating to an underlying spa- tially continuous phenomenon. Examples include: heights above sea-level in a topographical survey; pollution measurements from a finite network of monitor- ing stations; determinations of soil properties from core samples; insect counts from traps at selected locations. The subject has an interesting history. Orig- inally, the term geostatistics was coined by Georges Matheron and colleagues at Fontainebleau, France, to describe their work addressing problems of spatial prediction arising in the mining industry. See, for example, Matheron (1963, 1971). The ideas of the Fontainebleau school were developed largely indepen- dently of the mainstream of spatial statistics, with a distinctive terminology and style which tended to conceal the strong connections with parallel developments in spatial statistics. These parallel developments included work by Kolmogorov (1941), Mat´ern (1960, reprinted as Mat´ern, 1986), Whittle (1954, 1962, 1963), Bartlett (1964, 1967) and others.
    [Show full text]