Basic geostatistics

1. Basic geostatistics Austin Troy

2. Interpolation

3. Three methods in Arc GIS • IDW • SPLINE • Kriging

4. Inverse Distance Weighting • Weighted moving average • Weights by distance • Assumes unknown value influenced more by nearby than far away points, but we can control how rate of decay • Validity testing requires taking additional observations. • Sensitive to sampling, with circular patterns

5. Inverse Distance Weighting • Where λi are given by some weighting fn and • Common form of weighting function is d-p yielding:

6. IDW-How it works • Z value field: numeric attribute to be interpolated • Power: determines relationship of weighting and distance; where p= 0, no decrease in influence with distance; as p increases distant points becoming less influential in interpolating Z value at a given pixel • Neighborhood type: standard or smooth

7. Neighborhood types: Standard • Specifications • # neighbors to include • Ellipse height/width • Sector type/ angles: to ensure inclusion of obs. from all directions

8. Neighborhood types: Smooth • 3 Ellipses • Smoothing factor: size of gaps between ellipses • Inner : same weighting as “standard” • In between: weights multiplied by sigmoidal value from 1 (inner edge) to 0 (outer edge).

9. IDW Weights

10. IDW- P parameter • What is the best P to use? • Minimize Root Mean Squared Prediction Error (RMSPE) • Use cross validation to check. • You can look for an optimal P by testing your sample point data against a validation data set

11. Plot of model fits The blue line indicates degree of spatial autocorrelation (required for interpolation). The closer to the dashed (1:1) line, the more perfectly autocorrelated. Where horizontal, indicates data independence Mean pred. Error near zero means unbiased

12. Plot of model errors

13. Spline Method • Another option for interpolation method • This fits a curve through the sample data assign values to other locations based on their location on the curve • Thin plate splines create a surface that passes through sample points with the least possible change in slope at all points, that is with a minimum curvature surface. • Uses piece-wise functions fitted to a small number of data points, but joins are continuous, hence can modify one part of curve without having to recompute whole • Overall function is continuous with continuous first and second derivatives.

14. Kriging Method • Kriging is a geostatistical method and a probabilistic method, unlike the others, which are deterministic. That is, there is a probability associated with each prediction. Kriging has both a deterministic and probabilistic component, respectively Z(s) = μ(s) + ε(s), where both are functions of distance • Like IDW in that taking weighted moving average, but the weights (λ) are based on statistically derived autocorrelation measures. • Interpolation parameters (e.g. weights) are chosen to optimize fn • Assumes that variable in space can be modeled as sum of three components: 1) structure/deterministic part, 2) random but spatially correlated part and 3) spatially uncorrelated random part

15. Semivariance • Semivariogram(distance h) = 0.5 * average [ (value at location i– value at location j)2] OR • Based on the scatter of points, the computer (Geostatistical analyst) fits a curve through those points • The inverse is the covariance matrix which shows correlation over space

16. Kriging Method • Hence, foundation of Kriging is notion of spatial autocorrelation, or tendency of values of entities closer in space to be related. • Autocorrelation can be assessed using a semivariogram. • Where autocorrelation exists, the semivariance should increase until certain distance where SV= variance around mean, so flattens out. That value is called a “sill.” The sloped area, or “range” is where values are related to each other. Intercept is nugget

17. Kriging Method • Semivariograms measure the strength of statistical correlation as a function of distance; they quantify spatial autocorrelation • Because Kriging is based on the semivariogram, it is probabilistic, while IDW and Spline are deterministic • Kriging associates some probability with each prediction, hence it provides not just a surface, but some measure of the accuracy of that surface • Kriging equations are determined by fitting line through points so as to minimize weighted sum of squares between points and line • These equations are weighted based on spatial autocorrelation, which is determined from the semivariograms

18. Steps • Variogram cloud; can use bins to make box plot • Empirical variogram: choose bins and lags • Model variogram: fit function through empirical variogram • Functional forms?

19. Variogram • Plots semi-variance against distance between points • Where autocorrelation exists, the semivariance should have slope • Is binned to simplify • Binned values generated by grouping empirical points using square cells one lag wide. • To show local variation around mean Binning with average only Binning with average and grouping

21. Functional Forms From Fortin and Dale Spatial Analysis

22. Kriging Method • We can then use a scatter plot of predicted versus actual values to see the extent to which our model actually predicts the values • If the blue line and the points lie along the 1:1 line this indicates that the kriging model predicts the data well

23. Kriging Method • Produces four types of prediction maps: • Prediction Map: Predicted values • Probability Map: Probability that value over x • Prediction Standard Error Map: fit of model • Quantile maps: Probability that value over certain quantile

24. Kriging: Ordinary vs. Universal Z(s) = µ(s) + ε(s), • Known as Kriging in the presence of universal trends. • Universal kriging is used where there is an underlying trend beyond the simple spatial autocorrelation • Generally this trend occurs at a different scale • Trend may be fn of some geographic feature that occurs on one part of the map