Multiple-point Geostatistics

Stochastic Modeling with Training Images

Professor Gregoire Mariethoz, Jef Caers

• Publisher: Wiley-Blackwell
• Year: 2014
• Print ISBN: 9781118662755
• eBook ISBN: 9781118662939
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geology & Earth Sciences

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.

PART I
Concepts

CHAPTER 1
Hiking in the Sierra Nevada

1.1 An imaginary outdoor adventure company: Buena Sierra

As is the case for any applied science, no geostatistical application is without context. This context matters; it determines modeling choices, parameter choices, and the level of detail required in such modeling. In this short first chapter, we introduce an imagined context that has elements common to many applications of geostatistics: sparse local data, indirect (secondary) or trend information, a transfer function or decision variable, as well as a specific study target. The idea of doing so is to remain general by employing a synthetic example whose elements can be linked or translated into one's own area of application.

Consider an imaginary hiking company, Buena Sierra, a start-up company interested in organizing hiking adventures in the Sierra Nevada Mountains in the area shown in Figure I.1.1(left). The company drops customers over a range of locations to hike over a famous but challenging mountain range and meets them at the other end of that range for pickup. Customers require sufficient supplies in what is considered a strenuous trip over rocky terrain, with high elevation changes on possibly hot summer days. Imagine, however, that this area lies in the vicinity of a military base; hence, no detailed topographic or digital elevation model from satellite observation is available at this point. Instead, the company must rely on sparse point information obtained from weather stations in the area, dotted over the landscape; see Figure I.1.1(right). We consider that the exact elevation of these weather stations has been determined. The company now needs to plan for the adventure trip. This would require determining the quantity of supplies needed for each customer, which would require knowing the length of the path and the cumulative elevation gain because both correlate well with effort. The hike will generally move from west to east. The starting location can be any location on the west side from grid cell (100,1) to grid cell (180,1) (see Figure I.1.2).

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.

To make predictions about path length and cumulative elevation gain, a small routing computer program is written; although it simplifies real hiking, the program is considered adequate for this situation. More advanced routing could be applied, but this will not change the intended message of this imaginary example. The program requires as input a digital elevation map (DEM) of the area gridded on a certain grid. The program has as input a certain point on the west side, then walks by scanning for the direction that has the smallest elevation change. The program simulates two types of hikers: the minimal-effort (lazy) hiker and the maximal-effort (achiever) hiker. In both cases, the program assumes the hiker thinks only locally, namely, follows a path that is based on where they are and what lies just ahead. The minimal hiker takes a path of local least resistance (steepest downhill or least uphill). The achiever hiker takes a path of maximal ascent (or minimal descent). Note that the computer program represents a deterministic transfer function: given a single DEM map, a single starting point, and a specific hiker type, it outputs a single deterministic hiking route. If the actual reference, Walker Lake, is used as input, then given starting locations from grid cell (100,1) to (180,1) on the west side, a total of 80 outcomes are generated. These 80 outcomes can be shown as a histogram; see Figure I.1.3. The resulting path statistics for both minimal ...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Acknowledgments
• Part I: Concepts
• Part II: Methods
• Part III: Applications
• Index
• End User License Agreement