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Applied Stochastic Hydrogeology$
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Yoram Rubin

Print publication date: 2003

Print ISBN-13: 9780195138047

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195138047.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 20 October 2021

Estimation and Simulation

Estimation and Simulation

Chapter:
3 (p.60) Estimation and Simulation
Source:
Applied Stochastic Hydrogeology
Author(s):

Yoram Rubin

Publisher:
Oxford University Press
DOI:10.1093/oso/9780195138047.003.0008

Two important applications of the SRF concept developed in chapter 2 are point estimation and image simulation. Point estimation considers the SRF Z at an unsampled location, x0, and the goal is to get an estimate for z at x0 which is physically plausible and is optimal in some sense, and to provide a measure of the quality of the estimate. The goal in image simulation is to create an image of Z over the entire domain, one that not only is in agreement with the measurements at their locations, but also captures the correlation pattern of z. We start by considering a family of linear estimators known as kriging. Its appeal is in its simplicity and computational efficiency. We then proceed to discuss Bayesian estimators and will show how to condition estimates on “hard” and “soft” data, and we shall conclude by discussing a couple of simple, easy-to-implement image simulators. One of the simulators presented can be downloaded from the Internet. Linear regression aims at estimating the attribute z at x0: z0 = z(x0), based on a linear combination of n measurements of z: zi = z(xi), i = 1 , . . . ,n. The estimator of z(x0) is z*0, and it is defined by What makes this estimator “linear” is the exclusion of powers and products of measurements. However, nonlinearity may enter the estimation process indirectly, for example, through nonlinear transformation of the attribute. The challenge posed by(3.1) is to determine optimally the n interpolation coefficients λi, and the shift coefficient λ0. The actual estimation error is z*0 - z0; it is unknown, since z0 is unknown, and so no meaningful statement can be made about it. As an alternative, we shall consider the set of all equivalent estimation problems: in this set we maintain the same spatial configuration of measurement locations, but allow for all the possible combinations, or scenarios, of z values at these locations, including x0. We have replaced a single estimation problem with many, but we have improved our situation since now we know the actual z value at x0 and this will allow a systematic approach.

Keywords:   Anisotropy, Co-kriging, Estimation error, Indicators, Lagrange multipliers, Microscale, Point estimation, Screening, Unbiasedness, Variance-covariance matrix

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