Multiple-point Geostatistics E-Book

Gregoire Mariethoz

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 599

Veröffentlichungsjahr: 2014

Beschreibung

This book provides a comprehensive introduction to multiple-point geostatistics, where spatial continuity is described using training images. Multiple-point geostatistics aims at bridging the gap between physical modelling/realism and spatio-temporal stochastic modelling. The book provides an overview of this new field in three parts. Part I presents a conceptual comparison between traditional random function theory and stochastic modelling based on training images, where random function theory is not always used. Part II covers in detail various algorithms and methodologies starting from basic building blocks in statistical science and computer science. Concepts such as non-stationary and multi-variate modeling, consistency between data and model, the construction of training images and inverse modelling are treated. Part III covers three example application areas, namely, reservoir modelling, mineral resources modelling and climate model downscaling. This book will be an invaluable reference for students, researchers and practitioners of all areas of the Earth Sciences where forecasting based on spatio-temporal data is performed.

Leseprobe

Multiple-point geostatistics

Stochastic modeling with training images

Gregoire Mariethoz

Faculty of Geosciences and Environment University of Lausanne, Switzerland

Jef Caers

Energy Resources Engineering Department Stanford University, USA

This edition first published 2015 © 2015 by John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Mariethoz, Gregoire, author. Multiple-point geostatistics : stochastic modeling with training images / Gregoire Mariethoz and Jef Caers. pages cm Includes index.

Summary: “The topic of this book concerns an area of geostatistics that has commonly been known as multiple-point geostatistics because it uses more than two-point statistics (correlation), traditionally represented by the variogram, to model spatial phenomena”–Provided by publisher.

ISBN 978-1-118-66275-5 (hardback) 1. Geology–Statistical methods. 2. Geological modeling. I. Caers, Jef, author. II. Title. QE33.2.S82M37 2015 551.01′5195–dc23

2014035660

A catalogue record for this book is available from the British Library.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Cover image: Courtesy of NASA Earth Observatory.

CONTENTS

Preface

Acknowledgments

Part I: Concepts

Chapter 1: Hiking in the Sierra Nevada

1.1 An imaginary outdoor adventure company: Buena Sierra

1.2 What lies ahead

Chapter 2: Spatial estimation based on random function theory

2.1 Assumptions of stationarity

2.2 Assumption of stationarity in spatial problems

2.3 The kriging solution

2.4 Estimating covariances

2.5 Semivariogram modeling

2.6 Using a limited neighborhood

2.7 Universal kriging

2.8 Semivariogram modeling for universal kriging

2.9 Simple trend example case

2.10 Nonstationary covariances

2.11 Assessment

References

Chapter 3: Universal kriging with training images

3.1 Choosing for random function theory or not?

3.2 Formulation of universal kriging with training images

3.3 Positive definiteness of the sop matrix

3.4 Simple kriging with training images

3.5 Creating a map of estimates

3.6 Effect of the size of the training image

3.7 Effect of the nature of the training image

3.8 Training images for nonstationary modeling

3.9 Spatial estimation with nonstationary training images

3.10 Summary of methodological differences

References

Chapter 4: Stochastic simulations based on random function theory

4.1 The goal of stochastic simulations

4.2 Stochastic simulation: Gaussian theory

4.3 The sequential Gaussian simulation algorithm

4.4 Properties of multi-Gaussian realizations

4.5 Beyond Gaussian or beyond covariance?

References

Chapter 5: Stochastic simulation without random function theory

5.1 Direct sampling

5.2 The extended normal equation

5.3 Simulation by texture synthesis

References

Chapter 6: Returning to the Sierra Nevada

Reference

Part II: Methods

Chapter 7: Introduction

Chapter 8: The algorithmic building blocks

2.1 Grid and pointset representations

2.2 Multivariate grids

2.3 Neighborhoods

2.4 Storage and restitution of data events

2.5 Computing distances

2.6 Sequential simulation

2.7 Multiple grids

2.8 Conditioning

References

Chapter 9: Multiple-point geostatistics algorithms

3.1 Archetypal MPS algorithm

3.2 Pixel-based algorithms

3.3 Patch-based algorithms

3.4 Qualitative comparison of MPS algorithms

3.5 Postprocessing

References

Chapter 10: Markov random fields

4.1 Markov random field model

4.2 Markov mesh models

4.3 Multigrid formulations

References

Chapter 11: Nonstationary modeling with training images

5.1 Modeling nonstationary domains with stationary training images

5.2 Modeling nonstationary domains with nonstationary training images

References

Chapter 12: Multivariate modeling with training images

6.1 Multivariate and multiple-point relationships

6.2 Multivariate conditional simulation: Implementation issues

6.3 An example

6.4 Multivariate simulation as a filtering problem

References

Chapter 13: Training image construction

7.1 Choosing for training images

7.2 Object-based methods

7.3 Process-based models

7.4 Process-mimicking models

7.5 Elementary training images

7.6 From 2D to 3D

7.7 Training data

7.8 Construction of multivariate training images

7.9 Training image databases

References

Chapter 14: Validation and quality control

8.1 Introduction

8.2 Training image – data validation

8.3 Posterior quality control

References

Chapter 15: Inverse modeling with training images

9.1 Introduction

9.2 Inverse modeling: theory and practice

9.3 Sampling-based methods

9.4 Stochastic search

9.5 Parameterization of MPS realizations

References

Chapter 16: Parallelization

10.1 The need for parallel implementations

10.2 The challenges of parallel computing

10.3 Assessing a parallel implementation

10.4 Parallelization strategies

10.5 Graphical Processing Units

References

Part III: Applications

Chapter 17: Reservoir forecasting – the West Coast of Africa (WCA) reservoir

1.1 Introducing the context around WCA

1.2 Application of MPS to the WCA case

1.3 Alternative modeling workflows

References

Chapter 18: Geological resources modeling in mining

2.1 Context: sustaining the mining value chain

2.2 Stochastic updating of a block model

2.3 An alternative workflow: updating geological contacts

Notes

References

Chapter 19: Climate modeling application – the case of the Murray–Darling Basin

3.1 Introduction

3.2 Presentation of the data set

3.3 Climate model downscaling using multivariate MPS

3.4 Results and validation

References

Index

End User License Agreement

List of Tables

Chapter 1

Table I.1.1

Chapter 3

Table I.3.1

Chapter 4

Table I.4.1

Chapter 8

Table II.8.1

Table II.8.2

Table II.8.3

Chapter 9

Table II.9.1

Chapter 18

Table III.2.1

Table III.2.2

Table III.2.3

Table III.2.4

Table III.2.5

Table III.2.6

Table III.2.7

Table III.2.8

List of Illustrations

Chapter 1

Figure I.1.1 (left) Walker Lake exhaustive digital elevation map (size: 260×300 pixels) grid; and (right) 100 extracted sample data. The colorbar represents elevation in units of ft.

Figure I.1.2 Visualization of the 80 paths taken by hikers of two types: (left) minimal effort; and (right) maximal effort. The color indicates how frequently that portion of the path is taken, with redder color denoting higher frequency.

Figure I.1.3 Histograms of the cumulative elevation gain and path length for the minimal- and maximal-effort hiker. Cumulative elevation gain in units of ft, path length in units of grid cells.

Chapter 2

Figure I.2.1 Example loss functions; the most common choice is the parabola (least squares).

Figure I.2.2 (a) A single unique truth; (b) some sample data taken from it; and (c) its histogram. The goal is to estimate the value at the unsampled location marked with

X

.

Figure I.2.3 (a) Rock density in a homogeneous layer of a carbonate reservoir; and (b) rock density in a heterogeneous deltaic reservoir.

Figure I.2.4 Omnidirectional semivariogram of

Z

.

Figure I.2.5 Semivariogram of the exhaustive Walker Lake data set versus the sample variogram.

Figure I.2.6 (a) Global kriging using all 50 sample data at all estimated locations; (b) a local moving neighborhood; (c) using a minimum of 12 samples; and (d) using penalty. Parts (a–c) are calculated using SGEMS (Remy et al., 2008), and (d) is calculated using the R-package RGeostats.

Figure I.2.7 Simple trend case study: (left) the unknown truth (one realization, see Equation (I.2.38)); and (right) the sample data.

Figure I.2.8 (left) OLS estimate of trend; (middle) a universal kriging estimate based on the OLS estimate of trend; and (right) a universal kriging estimate based on the two-step procedure (using R-package RGeostats).

Chapter 3

Figure I.3.1 An analog data set considered relevant for the domain being modeled. The size of this image (250 × 250) need not be the same as the model grid size.

Figure I.3.2 (a) Artifacts induced by using a moving neighborhood; and (b) corrected by using finite domain kriging and by requiring a minimum of 12 neighboring points.

Figure I.3.3 Comparing the estimates obtained by (a) using a 150 × 150 size training image and (b) using a 250 × 250 size training image.

Figure I.3.4 Minimum error maps (kriging variances) compared to the kriging variance obtained using global ordinary kriging.

Figure I.3.5 (a) Reference case and (b) 100-sample data.

Figure I.3.6 (a–b) Two visually dissimilar training images each with their estimated map from 100 sample data (c–d).

Figure I.3.7 Omnidirectional empirical semivariograms for the two training images (a) Figure I.3.6(a) and (b) Figure I.3.6(b) along with fitted semivariograms using package RGeoS. Both have nugget 0, a range of approximately 35, and sill of 0.985.

Figure I.3.8 (a) An exhaustive DEM deemed representative for the Walker Lake area. The size of the training image is 400 × 400. (b) the result of a simple smoother applied to (a).

Figure I.3.9 (a) A training image for the simple trend case; and (b) its auxiliary variable.