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Geostatistics

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).

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Geostatistics

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  1. Geostatistics GLY 560: GIS for Earth Scientists

  2. 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 GLY560: GIS

  3. Estimator of Error • We need to develop a good estimate of an unknown. Say we have three estimates of an unknown: UB Geology GLY560: GIS

  4. 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 GLY560: GIS

  5. Estimator of Error • Note that the variance can be written more generally as: • Such an estimator is called “linear” UB Geology GLY560: GIS

  6. 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 GLY560: GIS

  7. 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 GLY560: GIS

  8. Realizations UB Geology GLY560: GIS

  9. 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 GLY560: GIS

  10. Experimental Variogram After Kitanidis (Intro. To Geostatistics) UB Geology GLY560: GIS

  11. Experimental Variogram After Kitanidis (Intro. To Geostatistics) UB Geology GLY560: GIS

  12. Small-Scale Variation: Discontinuous Case Correlation smaller than sampling scale: Z2 = cos (2 p x / 0.001) After Kitanidis (Intro. To Geostatistics) UB Geology GLY560: GIS

  13. Small-Scale Variation:Parabolic Case Correlation larger than sampling scale: Z2 = cos (2 p x / 2) After Kitanidis (Intro. To Geostatistics) UB Geology GLY560: GIS

  14. 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 GLY560: GIS

  15. 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 Uncorrelated g(h) Correlated at a max distance h UB Geology GLY560: GIS

  16. Range Semi-Variogram function Sill Nugget Separation Distance Variogram for Stationary Dataset • Range: maximum distance at which data are correlated • Nugget: distance over which data are absolutely correlated or unsampled • Sill: maximum variance (g(h)) of data pairs UB Geology GLY560: GIS

  17. Variogram Models UB Geology GLY560: GIS

  18. 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 GLY560: GIS

  19. Exponential Universal Circular Examples of Kriging UB Geology GLY560: GIS

  20. 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 GLY560: GIS

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