Geostatistics GLY 560: GIS for Earth Scientists

2/22/2016UB Geology GLY560: GIS 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)

2/22/2016UB Geology GLY560: GIS Estimator of Error We need to develop a good estimate of an unknown. Say we have three estimates of an unknown:

2/22/2016UB Geology GLY560: 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”.

2/22/2016UB Geology GLY560: GIS Estimator of Error Note that the variance can be written more generally as: Such an estimator is called “linear”

2/22/2016UB Geology GLY560: 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)

2/22/2016UB Geology GLY560: 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

2/22/2016UB Geology GLY560: GIS Realizations

2/22/2016UB Geology GLY560: 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

2/22/2016UB Geology GLY560: GIS Experimental Variogram After Kitanidis (Intro. To Geostatistics)

2/22/2016UB Geology GLY560: GIS Experimental Variogram After Kitanidis (Intro. To Geostatistics)

2/22/2016UB Geology GLY560: GIS Small-Scale Variation: Discontinuous Case Correlation smaller than sampling scale: Z 2 = cos (2  x / 0.001) After Kitanidis (Intro. To Geostatistics)

2/22/2016UB Geology GLY560: GIS Correlation larger than sampling scale: Z 2 = cos (2  x / 2) Small-Scale Variation: Parabolic Case After Kitanidis (Intro. To Geostatistics)

2/22/2016UB Geology GLY560: 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)

2/22/2016UB Geology GLY560: GIS Stationarity and the Variogram Under the condition of stationarity, the variogram will tell us over what scale the data are correlated.  (h) h Correlated at any distance Correlated at a max distance Uncorrelated

2/22/2016UB Geology GLY560: GIS Variogram for Stationary Dataset Nugget Range Sill Separation Distance Semi-Variogram function Range: maximum distance at which data are correlated Nugget: distance over which data are absolutely correlated or unsampled Sill: maximum variance (  (h)) of data pairs

2/22/2016UB Geology GLY560: GIS Variogram Models

2/22/2016UB Geology GLY560: 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.

2/22/2016UB Geology GLY560: GIS Examples of Kriging Universal Exponential Circular

2/22/2016UB Geology GLY560: GIS 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