Geostatistics GLY 560 GIS for Earth Scientists Introduction

Geostatistics GLY 560: GIS for Earth Scientists

Introduction Premise: One cannot obtain error-free estimates of unknowns (or find a deterministic model) Approach: Use statistical methods to reduce and estimate the error of estimating unknowns (must use a probabilistic model) UB Geology GLY 560: GIS

Estimator of Error • We need to develop a good estimate of an unknown. Say we have three estimates of an unknown: UB Geology GLY 560: GIS

Estimator of Error • An estimator that minimizes the mean square error (variance) is called a “best” estimator • When the expected error is zero, then the estimator is called “unbiased”. UB Geology GLY 560: GIS

Estimator of Error • Note that the variance can be written more generally as: • Such an estimator is called “linear” UB Geology GLY 560: GIS

BLUE An estimator that is • Best: minimizes variance • Linear: can be expressed as the sum of factors • Unbiased: expects a zero error …is called a BLUE (Best Linear Unbiased Estimator) UB Geology GLY 560: GIS

BLUE • We assume that the sample dataset is a sample from a random (but constrained) distribution • The error is also a random variable • Measurements, estimates, and error can all be described by probability distributions UB Geology GLY 560: GIS

Realizations UB Geology GLY 560: GIS

Experimental Variogram • Measures the variability of data with respect to spatial distribution • Specifically, looks at variance between pairs of data points over a range of separation scales UB Geology GLY 560: GIS

Experimental Variogram After Kitanidis (Intro. To Geostatistics) UB Geology GLY 560: GIS

Experimental Variogram After Kitanidis (Intro. To Geostatistics) UB Geology GLY 560: GIS

Small-Scale Variation: Discontinuous Case Correlation smaller than sampling scale: Z 2 = cos (2 p x / 0. 001) After Kitanidis (Intro. To Geostatistics) UB Geology GLY 560: GIS

Small-Scale Variation: Parabolic Case Correlation larger than sampling scale: Z 2 = cos (2 p x / 2) After Kitanidis (Intro. To Geostatistics) UB Geology GLY 560: GIS

Stationarity • Stationarity implies that an entire dataset is described by the same probabilistic process… that is we can analyze the dataset with one statistical model (Note: this definition differs from that given by Kitanidis) UB Geology GLY 560: GIS

Stationarity and the Variogram • Under the condition of stationarity, the variogram will tell us over what scale the data are correlated. Correlated at any distance g(h) Uncorrelated Correlated at a max distance h UB Geology GLY 560: GIS

Variogram for Stationary Dataset Semi-Variogram function • Range: maximum distance at which data are correlated • Nugget: distance over which data are absolutely correlated or unsampled Range Sill Nugget Separation Distance UB Geology GLY 560: GIS • Sill: maximum variance (g(h)) of data pairs

Variogram Models UB Geology GLY 560: GIS

Kriging • Kriging is essentially the process of using the variogram as a Best Linear Unbiased Estimator (BLUE) • Conceptually, one is fitting a variogram model to the experimental variogram. • Kriging equations may be used as interpolation functions. UB Geology GLY 560: GIS

Examples of Kriging Universal Exponential UB Geology GLY 560: GIS Circular

Final Thoughts • Kriging produces nice (can be exact) interpolation • Intelligent Kriging requires understanding of the spatial statistics of the dataset • Should display experimental variogram with Kriging or similar methods UB Geology GLY 560: GIS